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+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-06T12:52:16","documenter_version":"1.5.0"}} \ No newline at end of file diff --git a/v0.16.1/api/index.html b/v0.16.1/api/index.html new file mode 100644 index 000000000..ce7c97067 --- /dev/null +++ b/v0.16.1/api/index.html @@ -0,0 +1,37 @@ + +API · TaylorIntegration.jl
GitHub

Library

Exported functions

TaylorIntegration.taylorinteg — Function
taylorinteg(f, x0, t0, tmax, order, abstol, params[=nothing]; kwargs... )

General-purpose Taylor integrator for the explicit ODE $\dot{x}=f(x, p, t)$, where p are the parameters encoded in params. The initial conditions are specified by x0 at time t0; x0 may be of type T<:Number or Vector{T}, with T including TaylorN{T}; the latter case is of interest for jet transport applications.

The equations of motion are specified by the function f; we follow the same convention of DifferentialEquations.jl to define this function, i.e., f(x, p, t) or f!(dx, x, p, t); see the examples below.

The functions returns a TaylorSolution, whose fields are t and x; they represent, respectively, a vector with the values of time (independent variable), and a vector with the computed values of the dependent variable(s). When the keyword argument dense is set to true, it also outputs in the field p the Taylor polynomial expansion computed at each time step. The integration stops when time is larger than tmax, in which case the last returned value(s) correspond to tmax, or when the number of saved steps is larger than maxsteps.

The integration method uses polynomial expansions on the independent variable of order order; the parameter abstol serves to define the time step using the last two Taylor coefficients of the expansions. Make sure you use a large enough order to assure convergence.

Currently, the recognized keyword arguments are:

  • maxsteps[=500]: maximum number of integration steps.
  • parse_eqs[=true]: use the specialized method of jetcoeffs! created with @taylorize.
  • dense[=true]: output the Taylor polynomial expansion at each time step.

Examples

For one dependent variable the function f defines the RHS of the equation of motion, returning the value of $\dot{x}$. The arguments of this function are (x, p, t), where x are the dependent variables, p are the paremeters and t is the independent variable.

For several (two or more) dependent variables, the function f! defines the RHS of the equations of motion, mutating (in-place) the (preallocated) vector with components of $\dot{x}$. The arguments of this function are (dx, x, p, t), where dx is the preallocated vector of $\dot{x}$, x are the dependent variables, p are the paremeters entering the ODEs and t is the independent variable. The function may return this vector or simply nothing.

using TaylorIntegration
+
+f(x, p, t) = x^2
+
+sol = taylorinteg(f, 3, 0.0, 0.3, 25, 1.0e-20, maxsteps=100 )
+
+function f!(dx, x, p, t)
+    for i in eachindex(x)
+        dx[i] = x[i]^2
+    end
+    return nothing
+end
+
+sol = taylorinteg(f!, [3, 3], 0.0, 0.3, 25, 1.0e-20, maxsteps=100 )
+
+sol = taylorinteg(f!, [3, 3], 0.0, 0.3, 25, 1.0e-20, maxsteps=100, dense=true )
source
TaylorIntegration.lyap_taylorinteg — Function
lyap_taylorinteg(f!, q0, t0, tmax, order, abstol[, jacobianfunc!=nothing];
+    maxsteps::Int=500, parse_eqs::Bool=true)
+lyap_taylorinteg(f!, q0, trange, order, abstol[, jacobianfunc!=nothing];
+    maxsteps::Int=500, parse_eqs::Bool=true)

Similar to taylorinteg for the calculation of the Lyapunov spectrum. Note that the number of TaylorN variables should be set previously by the user (e.g., by means of TaylorSeries.set_variables) and should be equal to the length of the vector of initial conditions q0. Otherwise, whenever length(q0) != TaylorSeries.get_numvars(), then lyap_taylorinteg throws an AssertionError. Optionally, the user may provide a Jacobian function jacobianfunc! to evaluate the current value of the Jacobian. Otherwise, the current value of the Jacobian is computed via automatic differentiation using TaylorSeries.jl.

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Exported types

TaylorIntegration.TaylorSolution — Type

TaylorSolution{T, U, N, VT<:AbstractVector{T}, AX<:AbstractArray{U,N}, P<:Union{Nothing, AbstractArray{Taylor1{U}, N}}, VTE<:Union{Nothing, AbstractVector{U}}, AXE<:Union{Nothing, AbstractArray{U, N}}, VΛ<:Union{Nothing, AbstractArray{U,N}}} <: AbstractTaylorSolution{T, U}

This struct represents the return type for taylorinteg. Fields t and x represent, respectively, a vector with the values of time (independent variable), and a vector with the computed values of the dependent variable(s). When taylorinteg is called with dense=true, then field p stores the Taylor polynomial expansion computed at each time step. Fields tevents, xevents and gresids are related to root-finding methods of taylorinteg, while λ is related to the output of lyap_taylorinteg.

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Internal

TaylorIntegration.BookKeeping — Type
BookKeeping

Mutable struct that contains all the bookkeeping vectors/dictionaries used within _make_parsed_jetcoeffs: - d_indx : Dictionary mapping new variables (symbols) to old (perhaps indexed) symbols - d_assign : Dictionary with the numeric assignments (that are substituted) - d_decl : Dictionary declared arrays - v_newvars : Symbols of auxiliary indexed vars - v_arraydecl: Symbols which are explicitly declared as Array or Vector - v_array1 : Symbols which are explicitly declared as Array{Taylor1{T},1} - v_array2 : Symbols which are explicitly declared as Array{Taylor1{T},2} - v_array3 : Symbols which are explicitly declared as Array{Taylor1{T},3} - v_array4 : Symbols which are explicitly declared as Array{Taylor1{T},4} - v_preamb : Symbols or Expr used in the preamble (declarations, etc) - retvar : Guessed returned variable, which defines the LHS of the ODEs

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TaylorIntegration.RetAlloc — Type
RetAlloc{T <: Number}

Struct related to the returned variables that are pre-allocated when @taylorize is used. - v0 : Array{T,1} - v1 : Vector{Array{T,1}} - v2 : Vector{Array{T,2}} - v3 : Vector{Array{T,3}} - v4 : Vector{Array{T,4}}

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TaylorIntegration.__jetcoeffs! — Method
__jetcoeffs!(::Val{false}, f, t, x, params, rv)
+__jetcoeffs!(::Val{true}, f, t, x, params, rv)
+__jetcoeffs!(::Val{false}, f, t, x, dx, xaux, params, rv)
+__jetcoeffs!(::Val{true}, f, t, x, dx, xaux, params, rv)

Chooses a method of jetcoeffs! (hard-coded) or the generated by @taylorize) depending on Val{bool} (bool::Bool).

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TaylorIntegration._capture_fn_args_body! — Function

_capture_fn_args_body!(ex, dd::Dict{Symbol, Any})

Captures the name of a function, arguments, body and other properties, returning them as the values of the dictionary dd, which is updated in place.

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TaylorIntegration._defs_allocs! — Function

_defs_allocs!(preamble, fnargs, bkkeep, [inloop=false, ex_aux::Expr(:block,)])

Returns a vector with expressions defining the auxiliary variables in the preamble, and the declaration of the arrays. This function may modify bkkeep.d_indx if new variables are introduced. bkkeep.v_preamb is for bookkeeping the introduced variables.

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TaylorIntegration._determine_parsing! — Method
_determine_parsing!(parse_eqs::Bool, f, t, x, params)
+_determine_parsing!(parse_eqs::Bool, f, t, x, dx, params)

Check if the parsed method of jetcoeffs! exists and check it runs without error.

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TaylorIntegration._extract_parts — Method

_extract_parts(ex::Expr)

Returns the function name, the function arguments, and the body of a function passed as an Expr. The function may be provided as a one-line function, or in the long form (anonymous functions do not work).

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TaylorIntegration._make_parsed_jetcoeffs — Method

_make_parsed_jetcoeffs( ex )

This function constructs the expressions of two new methods, the first equivalent to the differential equations (jetcoeffs!), which exploits the mutating functions of TaylorSeries.jl, and the second one (allocatejetcoeffs) preallocates any auxiliary Taylor1 or Vector{Taylor1{T}} needed.

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TaylorIntegration._newfnbody! — Method

_newfnbody!(fnbody, fnargs, bkkeep)

Returns a new (modified) body of the function, a priori unfolding the expression graph (AST) as unary and binary calls, and updates the bookkeeping structure bkkeep.

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TaylorIntegration._newhead — Method

_newhead(fn, fnargs)

Creates the head of the new method of jetcoeffs! and _allocate_jetcoeffs. fn is the name of the passed function and fnargs is a vector with its arguments defning the function (which are either three or four).

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TaylorIntegration._parse_newfnbody! — Method

parsenewfnbody!(ex::Expr, preex::Expr, prealloc::Expr, bkkeep::BookKeeping, inloop::Bool)

Parses ex (the new body of the function) replacing the expressions to use the mutating functions of TaylorSeries, and building the preamble preex and prealloc expressions. This is done by traversing recursively (again) the args of ex, updating the bookkeeping struct bkkeep, in particular the fieldnames v_newvars and d_assign.

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TaylorIntegration._preamble_body — Method

_preamble_body(fnbody, fnargs)

Returns expressions for the preamble, the declaration of arrays, the body and the bookkeeping struct, which will be used to build the new functions. fnbody is the expression with the body of the original function (already adapted), fnargs is a vector of symbols of the original diferential equations function.

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TaylorIntegration._rename_indexedvars — Method

_rename_indexedvars(fnbody)

Renames the indexed variables (using Espresso.genname()) that exists in fnbody. Returns fnbody with the renamed variables and a dictionary that links the new variables to the old indexed ones.

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TaylorIntegration._replace_expr! — Method

_replace_expr!(ex::Expr, preex::Expr, , prealloc::Expr, i::Int, aalhs, aarhs, bkkeep::BookKeeping)

Replaces the calls in ex.args[i], and updates preex and prealloc with the appropriate expressions, based on the the LHS (aalhs) and RHS (aarhs) of the base assignment. The bookkeeping struct is updated (v_newvars) within _replacecalls!. d_indx is used to bring back the indexed variables.

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TaylorIntegration._replacecalls! — Method

_replacecalls!(bkkeep, fnold, newvar)

Replaces the symbols of unary and binary calls of the expression fnold, which defines newvar, by the mutating functions in TaylorSeries.jl. The vector bkkeep.v_vars is updated if new auxiliary variables are introduced (bookkeeping).

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TaylorIntegration._split_arraydecl! — Method

_split_arraydecl!(bkkeep)

Split bkkeep.varraydecl in the vector (bkkeep.varray1), matrix (bkkeep.v_array2), etc, to properly construct the RetAlloc variable.

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TaylorIntegration.findroot! — Method
findroot!(t, x, dx, g_tupl_old, g_tupl, eventorder, tvS, xvS, gvS,
+    t0, δt_old, x_dx, x_dx_val, g_dg, g_dg_val, nrabstol,
+    newtoniter, nevents) -> nevents

Internal root-finding subroutine, based on Newton-Raphson process. If there is a crossing, then the crossing data is stored in tvS, xvS and gvS and nevents, the number of events/crossings, is updated. Here t is a Taylor1 polynomial which represents the independent variable; x is an array of Taylor1 variables which represent the vector of dependent variables; dx is an array of Taylor1 variables which represent the LHS of the ODE; g_tupl_old is the last-before-current value returned by event function g and g_tupl is the current one; eventorder is the order of the derivative of g whose roots the user is interested in finding; tvS stores the surface-crossing instants; xvS stores the value of the solution at each of the crossings; gvS stores the values of the event function g (or its eventorder-th derivative) at each of the crossings; t0 is the current time; δt_old is the last time-step size; x_dx, x_dx_val, g_dg, g_dg_val are auxiliary variables; nrabstol is the Newton-Raphson process tolerance; newtoniter is the maximum allowed number of Newton-Raphson iteration; nevents is the current number of detected events/crossings.

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TaylorIntegration.init_psol — Method
init_psol(::Val{true}, xv::Array{U,1}, x::Taylor1{U}) where {U<:Number}
+init_psol(::Val{true}, xv::Array{U,2}, x::Array{Taylor1{U},1}) where {U<:Number}
+init_psol(::Val{false}, ::Array{U,1}, ::Taylor1{U}) where {U<:Number}
+init_psol(::Val{false}, ::Array{U,2}, ::Array{Taylor1{U},1}) where {U<:Number}

Auxiliary function to initialize psol during a call to taylorinteg. When the first argument in the call signature is Val(false) this function simply returns nothing. Otherwise, when the first argument in the call signature is Val(true), then the appropriate array is allocated and returned; this array is where the Taylor polynomials associated to the solution will be stored, corresponding to field :p in TaylorSolution.

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TaylorIntegration.jetcoeffs! — Method
jetcoeffs!(eqsdiff!::Function, t, x, dx, xaux, params)

Mutates x in-place using the recursion relation of the derivatives obtained from the differential equations $\dot{x}=dx/dt=f(x, p, t)$.

eqsdiff! is the function defining the RHS of the ODE, x contains the Taylor1 expansion of the dependent variables and t is the independent variable, and params are the parameters appearing on the function defining the differential equation. See taylorinteg for examples and convention for eqsdiff. Note that x is of type Vector{Taylor1{U}} where U<:Number; t is of type Taylor1{T} where T<:Real. In this case, two auxiliary containers dx and xaux (both of the same type as x) are needed to avoid allocations.

Initially, x contains only the 0-th order Taylor coefficient of the current system state (the initial conditions), and jetcoeffs! computes recursively the high-order derivates back into x.

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TaylorIntegration.jetcoeffs! — Method
jetcoeffs!(eqsdiff::Function, t, x, params)

Returns an updated x using the recursion relation of the derivatives obtained from the differential equations $\dot{x}=dx/dt=f(x, p, t)$.

eqsdiff is the function defining the RHS of the ODE, x contains the Taylor1 expansion of the dependent variable(s) and t is the independent variable, and params are the parameters appearing on the function defining the differential equation. See taylorinteg for examples and convention for eqsdiff. Note that x is of type Taylor1{U} where U<:Number; t is of type Taylor1{T} where T<:Real.

Initially, x contains only the 0-th order Taylor coefficient of the current system state (the initial conditions), and jetcoeffs! computes recursively the high-order derivates back into x.

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TaylorIntegration.lyap_jetcoeffs! — Method
lyap_jetcoeffs!(t, x, dx, jac, varsaux)

Similar to jetcoeffs! for the calculation of the Lyapunov spectrum. Updates only the elements of x which correspond to the solution of the 1st-order variational equations $\dot{\xi}=J \cdot \xi$, where $J$ is the Jacobian matrix, i.e., the linearization of the equations of motion. jac is the Taylor expansion of $J$ wrt the independent variable, around the current initial condition. varsaux is an auxiliary array of type Array{eltype(jac),3} to avoid allocations. Calling this method assumes that jac has been computed previously using stabilitymatrix!.

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TaylorIntegration.nrconvergencecriterion — Method
nrconvergencecriterion(g_val, nrabstol::T, nriter::Int, newtoniter::Int) where {T<:Real}

A rudimentary convergence criterion for the Newton-Raphson root-finding process. g_val may be either a Real, Taylor1{T} or a TaylorN{T}, where T<:Real. Returns true if: 1) the absolute value of g_val, the value of the event function g evaluated at the current estimated root by the Newton-Raphson process, is less than the nrabstol tolerance; and 2) the number of iterations nriter of the Newton-Raphson process is less than the maximum allowed number of iterations, newtoniter; otherwise, returns false.

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TaylorIntegration.set_psol! — Method
set_psol!(::Val{true}, psol::Array{Taylor1{U},1}, nsteps::Int, x::Taylor1{U}) where {U<:Number}
+set_psol!(::Val{true}, psol::Array{Taylor1{U},2}, nsteps::Int, x::Vector{Taylor1{U}}) where {U<:Number}
+set_psol!(::Val{false}, args...)

Auxiliary function to save Taylor polynomials in a call to taylorinteg. When the first argument in the call signature is Val(true), sets appropriate elements of argument psol. Otherwise, when the first argument in the call signature is Val(false), this function simply returns nothing. Argument psol is the array where the Taylor polynomials associated to the solution will be stored, corresponding to field :p in TaylorSolution. See also init_psol.

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TaylorIntegration.stabilitymatrix! — Method
stabilitymatrix!(eqsdiff!, t, x, δx, dδx, jac, _δv, params[, jacobianfunc!=nothing])

Updates the matrix jac::Matrix{Taylor1{U}} (linearized equations of motion) computed from the equations of motion (eqsdiff!), at time t at x; x is of type Vector{Taylor1{U}}, where U<:Number. δx, dδx and _δv are auxiliary arrays of type Vector{TaylorN{Taylor1{U}}} to avoid allocations. Optionally, the user may provide a Jacobian function jacobianfunc! to compute jac. Otherwise, jac is computed via automatic differentiation using TaylorSeries.jl.

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TaylorIntegration.stepsize — Method
stepsize(x, epsilon) -> h

Returns a maximum time-step for a the Taylor expansion x using a prescribed absolute tolerance epsilon and the last two Taylor coefficients of (each component of) x.

Note that x is of type Taylor1{U} or Vector{Taylor1{U}}, including also the cases Taylor1{TaylorN{U}} and Vector{Taylor1{TaylorN{U}}}.

Depending of eltype(x), i.e., U<:Number, it may be necessary to overload stepsize, specializing it on the type U, to avoid type instabilities.

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TaylorIntegration.surfacecrossing — Method
surfacecrossing(g_old, g_now, eventorder::Int)

Detect if the solution crossed a root of event function g. g_old represents the last-before-current value of event function g, and g_now represents the current one; these are Tuple{Bool,Taylor1{T}}s. eventorder is the order of the derivative of the event function g whose root we are trying to find. Returns true if the constant terms of g_old[2] and g_now[2] have different signs (i.e., if one is positive and the other one is negative). Otherwise, if g_old[2] and g_now[2] have the same sign or if the first component of either of them is false, then it returns false.

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TaylorIntegration.taylorstep! — Method
taylorstep!(::Val{false}, f, t, x, abstol, params, rv, parse_eqs=true) -> δt
+taylorstep!(::Val{true}, f, t, x, abstol, params, rv, parse_eqs=true) -> δt
+taylorstep!(::Val{false}, f!, t, x, dx, xaux, abstol, params, rv, parse_eqs=true) -> δt
+taylorstep!(::Val{true}, f!, t, x, dx, xaux, abstol, params, rv, parse_eqs=true) -> δt

One-step Taylor integration for the one-dependent variable ODE $\dot{x}=dx/dt=f(x, p, t)$ with initial conditions $x(t_0)=x_0$. Returns the time-step δt of the actual integration carried out (δt is positive).

Here, f represents the function defining the RHS of the ODE (see taylorinteg), t is the independent variable, x contains the Taylor expansion of the dependent variable, order is the degree used for the Taylor1 polynomials during the integration abstol is the absolute tolerance used to determine the time step of the integration, and params are the parameters entering the ODE functions. For several variables, dx and xaux, both of the same type as x, are needed to save allocations. Finally, parse_eqs is a switch to force not using (parse_eqs=false) the specialized method of jetcoeffs! created with @taylorize; the default is true (parse the equations). Finally, parse_eqs is a switch to force not using (parse_eqs=false) the specialized method of jetcoeffs! created with @taylorize; the default is true (parse the equations). The first argument in the function call Val{bool} (bool::Bool) controls whether a specialized `jetcoeffs! method is being used or not.

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TaylorIntegration.timeindex — Method
timeindex(sol::TaylorSolution, t::TT) where TT

Return the index of sol.t corresponding to t and the time elapsed from sol.t0 to t.

source

Index

diff --git a/v0.16.1/assets/documenter.js b/v0.16.1/assets/documenter.js new file mode 100644 index 000000000..b2bdd43ee --- /dev/null +++ b/v0.16.1/assets/documenter.js @@ -0,0 +1,1059 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.16.1/assets/figures/poincareanim1.gif b/v0.16.1/assets/figures/poincareanim1.gif new file mode 100644 index 0000000000000000000000000000000000000000..ae5c0abd0b0195687407a99d268d9a46f52db538 GIT binary 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe 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.button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe 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.button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe 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.button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe 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.button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-frappe .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-frappe div.icon-text{display:flex}html.theme--catppuccin-frappe .image,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-frappe .image img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-frappe .image img.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-frappe .image.is-fullwidth,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-frappe .image.is-square img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-frappe .image.is-square .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-frappe .image.is-1by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-frappe .image.is-1by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-frappe .image.is-5by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-frappe .image.is-5by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-frappe .image.is-4by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-frappe .image.is-4by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-frappe .image.is-3by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-frappe .image.is-5by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-frappe .image.is-5by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-frappe .image.is-16by9 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-frappe .image.is-16by9 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-frappe .image.is-2by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-frappe .image.is-2by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-frappe .image.is-1by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-frappe .image.is-1by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-frappe .image.is-square,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-frappe .image.is-1by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-frappe .image.is-5by4,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-frappe .image.is-4by3,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-frappe .image.is-3by2,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--catppuccin-frappe .image.is-5by3,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--catppuccin-frappe .image.is-16by9,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--catppuccin-frappe .image.is-2by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-frappe .image.is-3by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-frappe .image.is-4by5,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-frappe .image.is-3by4,html.theme--catppuccin-frappe #documenter .docs-sidebar 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.button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-outlined.is-focused{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-latte .icon.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-latte .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-latte .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-latte .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-latte .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-latte .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-latte div.icon-text{display:flex}html.theme--catppuccin-latte .image,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-latte .image img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-latte .image img.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-latte .image.is-fullwidth,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-latte .image.is-square img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-latte .image.is-square .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-latte .image.is-1by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-latte .image.is-1by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-latte .image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte .image.is-3by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-latte .image.is-4by5 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-latte .image.is-4by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-latte .image.is-3by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-latte .image.is-3by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-latte .image.is-2by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-latte .image.is-2by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by5 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-latte .image.is-square,html.theme--catppuccin-latte #documenter 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a.navbar-item{padding-right:3rem}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:focus,html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#8c8fa1}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}.navbar.is-spaced html.theme--catppuccin-latte .navbar-dropdown,html.theme--catppuccin-latte .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-latte .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-latte .navbar-divider{display:block}html.theme--catppuccin-latte .navbar>.container .navbar-brand,html.theme--catppuccin-latte .container>.navbar 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.button.is-white.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-white.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-dark,html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-dark.is-inverted[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-primary,html.theme--catppuccin-macchiato .docstring>section>a.button.docs-sourcelink{background-color:#8aadf4;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-primary:hover,html.theme--catppuccin-macchiato .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-macchiato .button.is-primary.is-hovered,html.theme--catppuccin-macchiato .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#7ea5f3;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato 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.button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-macchiato .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-macchiato .button.is-responsive,html.theme--catppuccin-macchiato .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-macchiato .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-macchiato 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.button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:hover,html.theme--catppuccin-mocha .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:focus,html.theme--catppuccin-mocha .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:focus:not(:active),html.theme--catppuccin-mocha .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-mocha .button.is-black:active,html.theme--catppuccin-mocha .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-mocha .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted:hover,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-outlined:hover,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-black.is-outlined:focus,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:hover,html.theme--catppuccin-mocha .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:focus,html.theme--catppuccin-mocha .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:focus:not(:active),html.theme--catppuccin-mocha .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-mocha .button.is-light:active,html.theme--catppuccin-mocha .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-mocha .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted:hover,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-outlined:hover,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-light.is-outlined:focus,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-dark,html.theme--catppuccin-mocha .content kbd.button{background-color:#313244;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:hover,html.theme--catppuccin-mocha .content kbd.button:hover,html.theme--catppuccin-mocha .button.is-dark.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-hovered{background-color:#2c2d3d;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus,html.theme--catppuccin-mocha .content kbd.button:focus,html.theme--catppuccin-mocha .button.is-dark.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus:not(:active),html.theme--catppuccin-mocha .content kbd.button:focus:not(:active),html.theme--catppuccin-mocha .button.is-dark.is-focused:not(:active),html.theme--catppuccin-mocha .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(49,50,68,0.25)}html.theme--catppuccin-mocha .button.is-dark:active,html.theme--catppuccin-mocha .content kbd.button:active,html.theme--catppuccin-mocha .button.is-dark.is-active,html.theme--catppuccin-mocha .content kbd.button.is-active{background-color:#262735;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark[disabled],html.theme--catppuccin-mocha .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button{background-color:#313244;border-color:#313244;box-shadow:none}html.theme--catppuccin-mocha .button.is-dark.is-inverted,html.theme--catppuccin-mocha .content kbd.button.is-inverted{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted:hover,html.theme--catppuccin-mocha .content kbd.button.is-inverted:hover,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-dark.is-inverted[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-loading::after,html.theme--catppuccin-mocha .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined,html.theme--catppuccin-mocha .content kbd.button.is-outlined{background-color:transparent;border-color:#313244;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-outlined:hover,html.theme--catppuccin-mocha .content kbd.button.is-outlined:hover,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-focused{background-color:#313244;border-color:#313244;color:#fff}html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-outlined{background-color:transparent;border-color:#313244;box-shadow:none;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-primary,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:hover,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-mocha .button.is-primary.is-hovered,html.theme--catppuccin-mocha .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#7dacf9;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:focus,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:focus,html.theme--catppuccin-mocha .button.is-primary.is-focused,html.theme--catppuccin-mocha .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:focus:not(:active),html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--catppuccin-mocha .button.is-primary.is-focused:not(:active),html.theme--catppuccin-mocha .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(137,180,250,0.25)}html.theme--catppuccin-mocha .button.is-primary:active,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:active,html.theme--catppuccin-mocha .button.is-primary.is-active,html.theme--catppuccin-mocha .docstring>section>a.button.is-active.docs-sourcelink{background-color:#71a4f9;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary[disabled],html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-primary,fieldset[disabled] html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:#89b4fa;box-shadow:none}html.theme--catppuccin-mocha .button.is-primary.is-inverted,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;color:#89b4fa}html.theme--catppuccin-mocha .button.is-primary.is-inverted:hover,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink:hover,html.theme--catppuccin-mocha .button.is-primary.is-inverted.is-hovered,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.is-hovered.docs-sourcelink{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-primary.is-inverted[disabled],html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-primary.is-inverted,fieldset[disabled] html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#89b4fa}html.theme--catppuccin-mocha .button.is-primary.is-loading::after,html.theme--catppuccin-mocha .docstring>section>a.button.is-loading.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-primary.is-outlined,html.theme--catppuccin-mocha 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha 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!important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content 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input[type="radio"]{vertical-align:baseline}html.theme--documenter-dark small{font-size:.875em}html.theme--documenter-dark span{font-style:inherit;font-weight:inherit}html.theme--documenter-dark strong{color:#f2f2f2;font-weight:700}html.theme--documenter-dark fieldset{border:none}html.theme--documenter-dark pre{-webkit-overflow-scrolling:touch;background-color:#282f2f;color:#fff;font-size:.875em;overflow-x:auto;padding:1.25rem 1.5rem;white-space:pre;word-wrap:normal}html.theme--documenter-dark pre code{background-color:transparent;color:currentColor;font-size:1em;padding:0}html.theme--documenter-dark table td,html.theme--documenter-dark table th{vertical-align:top}html.theme--documenter-dark table td:not([align]),html.theme--documenter-dark table th:not([align]){text-align:inherit}html.theme--documenter-dark table th{color:#f2f2f2}html.theme--documenter-dark .box{background-color:#343c3d;border-radius:8px;box-shadow:none;color:#fff;display:block;padding:1.25rem}html.theme--documenter-dark a.box:hover,html.theme--documenter-dark a.box:focus{box-shadow:0 0.5em 1em -0.125em rgba(10,10,10,0.1),0 0 0 1px #1abc9c}html.theme--documenter-dark a.box:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2),0 0 0 1px #1abc9c}html.theme--documenter-dark .button{background-color:#282f2f;border-color:#4c5759;border-width:1px;color:#375a7f;cursor:pointer;justify-content:center;padding-bottom:calc(0.5em - 1px);padding-left:1em;padding-right:1em;padding-top:calc(0.5em - 1px);text-align:center;white-space:nowrap}html.theme--documenter-dark .button strong{color:inherit}html.theme--documenter-dark .button .icon,html.theme--documenter-dark .button .icon.is-small,html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search>input.icon,html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search>input.icon,html.theme--documenter-dark .button .icon.is-medium,html.theme--documenter-dark .button .icon.is-large{height:1.5em;width:1.5em}html.theme--documenter-dark .button .icon:first-child:not(:last-child){margin-left:calc(-0.5em - 1px);margin-right:.25em}html.theme--documenter-dark .button .icon:last-child:not(:first-child){margin-left:.25em;margin-right:calc(-0.5em - 1px)}html.theme--documenter-dark .button .icon:first-child:last-child{margin-left:calc(-0.5em - 1px);margin-right:calc(-0.5em - 1px)}html.theme--documenter-dark .button:hover,html.theme--documenter-dark .button.is-hovered{border-color:#8c9b9d;color:#f2f2f2}html.theme--documenter-dark .button:focus,html.theme--documenter-dark .button.is-focused{border-color:#8c9b9d;color:#17a689}html.theme--documenter-dark .button:focus:not(:active),html.theme--documenter-dark .button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(26,188,156,0.25)}html.theme--documenter-dark .button:active,html.theme--documenter-dark .button.is-active{border-color:#343c3d;color:#f2f2f2}html.theme--documenter-dark .button.is-text{background-color:transparent;border-color:transparent;color:#fff;text-decoration:underline}html.theme--documenter-dark .button.is-text:hover,html.theme--documenter-dark .button.is-text.is-hovered,html.theme--documenter-dark .button.is-text:focus,html.theme--documenter-dark .button.is-text.is-focused{background-color:#282f2f;color:#f2f2f2}html.theme--documenter-dark .button.is-text:active,html.theme--documenter-dark .button.is-text.is-active{background-color:#1d2122;color:#f2f2f2}html.theme--documenter-dark .button.is-text[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-text{background-color:transparent;border-color:transparent;box-shadow:none}html.theme--documenter-dark .button.is-ghost{background:none;border-color:rgba(0,0,0,0);color:#1abc9c;text-decoration:none}html.theme--documenter-dark .button.is-ghost:hover,html.theme--documenter-dark .button.is-ghost.is-hovered{color:#1abc9c;text-decoration:underline}html.theme--documenter-dark .button.is-white{background-color:#fff;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:hover,html.theme--documenter-dark .button.is-white.is-hovered{background-color:#f9f9f9;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:focus,html.theme--documenter-dark .button.is-white.is-focused{border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:focus:not(:active),html.theme--documenter-dark .button.is-white.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--documenter-dark .button.is-white:active,html.theme--documenter-dark .button.is-white.is-active{background-color:#f2f2f2;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content 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.button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark 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Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. 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The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.16.1/assets/warner.js b/v0.16.1/assets/warner.js new file mode 100644 index 000000000..3f6f5d008 --- /dev/null +++ b/v0.16.1/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. 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    GitHub

    Interoperability with DifferentialEquations.jl

    Here, we show an example of interoperability between TaylorIntegration.jl and DifferentialEquations.jl, i.e., how to use TaylorIntegration.jl from the DifferentialEquations ecosystem. The basic requirement is to load OrdinaryDiffEq.jl together with TaylorIntegration.jl, which sets-up the common interface. Below, we shall also use OrdinaryDiffEq.jl to compare the accuracy of TaylorIntegration.jl with respect to high-accuracy methods for non-stiff problems (Vern9 method). While DifferentialEquations offers many macros to simplify certain aspects, we do not rely on them simply because using properly @taylorize improves the performance.

    Note

    Currently, the only keyword arguments supported by DiffEqBase.solve that are implemented in TaylorIntegration.jl are :saveat and :tstops. There is also experimental support for :callback, both discrete and continuous; some examples may be found in test/common.jl. The keyword argument :parse_eqs is available in order to control the use of methods defined via @taylorize.

    The problem we will integrate in this example is the planar circular restricted three-body problem (PCR3BP, also capitalized as PCRTBP). The PCR3BP describes the motion of a body with negligible mass $m_3$ under the gravitational influence of two bodies with masses $m_1$ and $m_2$, such that $m_1 \ge m_2$. It is assumed that $m_3$ is much smaller than the other two masses so it does not influence their motion, and therefore it is simply considered as a massless test particle. The body with the greater mass $m_1$ is referred as the primary, and $m_2$ as the secondary. These bodies are together called the primaries and are assumed to describe Keplerian circular orbits about their center of mass, which is placed at the origin of the reference frame. It is further assumed that the orbit of the third body takes place in the orbital plane of the primaries. A full treatment of the PCR3BP may be found in [1].

    The ratio $\mu = m_2/(m_1+m_2)$ is known as the mass parameter. Using mass units such that $m_1+m_2=1$, we have $m_1=1-\mu$ and $m_2=\mu$. In this example, we assume the mass parameter to have a value $\mu=0.01$.

    using Plots
+
+const μ = 0.01

    The Hamiltonian for the PCR3BP in the synodic frame (i.e., a frame which rotates such that the primaries are at rest on the $x$ axis) is

    \[H(x, y, p_x, p_y) = \frac{1}{2}(p_x^2+p_y^2) - (x p_y - y p_x) + V(x, y), \tag{1}\]

    where

    \[V(x, y) = - \frac{1-\mu}{\sqrt{(x-\mu)^2+y^2}} - \frac{\mu}{\sqrt{(x+1-\mu)^2+y^2}}.\tag{2}\]

    is the gravitational potential associated to the primaries. The RHS of Eq. (1) is also known as the Jacobi constant, since it is a preserved quantity of motion in the PCR3BP. We will use this property to check the accuracy of the solutions computed.

    V(x, y) = - (1-μ)/sqrt((x-μ)^2+y^2) - μ/sqrt((x+1-μ)^2+y^2)
+H(x, y, px, py) = (px^2+py^2)/2 - (x*py-y*px) + V(x, y)
+H(x) = H(x...)

    The equations of motion for the PCR3BP are

    \[\begin{aligned} + \dot{x} & = p_x + y,\\ + \dot{y} & = p_y - x,\\ + \dot{p_x} & = - \frac{(1-\mu)(x-\mu)}{((x-\mu)^2+y^2)^{3/2}} - \frac{\mu(x+1-\mu)}{((x+1-\mu)^2+y^2)^{3/2}} + p_y,\\ + \dot{p_y} & = - \frac{(1-\mu)y }{((x-\mu)^2+y^2)^{3/2}} - \frac{\mu y }{((x+1-\mu)^2+y^2)^{3/2}} - p_x. +\end{aligned}\]

    We define this system of ODEs in a way that allows the use of the @taylorize macro from TaylorIntegration.jl, which for the present example allows important speed-ups. For more details about the specifics of the use of @taylorize, see this section.

    using TaylorIntegration
+@taylorize function pcr3bp!(dq, q, param, t)
+    local μ = param[1]
+    local onemμ = 1 - μ
+    x1 = q[1]-μ
+    x1sq = x1^2
+    y = q[2]
+    ysq = y^2
+    r1_1p5 = (x1sq+ysq)^1.5
+    x2 = q[1]+onemμ
+    x2sq = x2^2
+    r2_1p5 = (x2sq+ysq)^1.5
+    dq[1] = q[3] + q[2]
+    dq[2] = q[4] - q[1]
+    dq[3] = (-((onemμ*x1)/r1_1p5) - ((μ*x2)/r2_1p5)) + q[4]
+    dq[4] = (-((onemμ*y )/r1_1p5) - ((μ*y )/r2_1p5)) - q[3]
+    return nothing
+end

    We shall define the initial conditions $q_0 = (x_0, y_0, p_{x,0}, p_{y,0})$ such that $H(q_0) = J_0$, where $J_0$ is a prescribed value. In order to do this, we select $y_0 = p_{x,0} = 0$ and compute the value of $p_{y,0}$ for which $H(q_0) = J_0$ holds.

    We consider a value for $J_0$ such that the test particle is able to display close encounters with both primaries, but cannot escape to infinity. We may obtain a first approximation to the desired value of $J_0$ if we plot the projection of the zero-velocity curves on the $x$-axis.

    ZVC(x) =  -x^2/2 + V(x, zero(x)) # projection of the zero-velocity curves on the x-axis
+
+plot(ZVC, -2:0.001:2, label="zero-vel. curve", legend=:topleft, fmt = :png)
+plot!([-2, 2], [-1.58, -1.58], label="J0 = -1.58")
+ylims!(-1.7, -1.45)
+xlabel!("x")
+ylabel!("J")
+title!("Zero-velocity curves (x-axis projection)")
    Example block output

    Notice that the maxima in the plot correspond to the Lagrangian points $L_1$, $L_2$ and $L_3$; below we shall concentrate in the value $J_0 = -1.58$.

    J0 = -1.58

    We define a function py!, which depends on the initial condition $q_0 = (x_0, 0, 0, p_{y,0})$ and the Jacobi constant value $J_0$, such that it computes an adequate value $p_{y,0}$ for which we have $H(q_0)=J_0$ and updates (in-place) the initial condition accordingly.

    function py!(q0, J0)
+    @assert iszero(q0[2]) && iszero(q0[3]) # q0[2] and q0[3] have to be equal to zero
+    q0[4] = q0[1] + sqrt( q0[1]^2-2( V(q0[1], q0[2])-J0 ) )
+    nothing
+end

    We are now ready to generate an appropriate initial condition.

    q0 = [-0.8, 0.0, 0.0, 0.0]
+py!(q0, J0)
+q0
    4-element Vector{Float64}:
+ -0.8
+  0.0
+  0.0
+ -0.6276410653920694

    We note that the value of q0 has been updated. We can check that the value of the Hamiltonian evaluated at the initial condition is indeed equal to J0.

    H(q0) == J0
    true

    Following the DifferentialEquations.jl tutorial, we define an ODEProblem for the integration; TaylorIntegration can be used via its common interface bindings with OrdinaryDiffEq.jl; both packages need to be loaded explicitly.

    tspan = (0.0, 2000.0)
+p = [μ]
+
+using OrdinaryDiffEq
+prob = ODEProblem(pcr3bp!, q0, tspan, p)
    ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
+timespan: (0.0, 2000.0)
+u0: 4-element Vector{Float64}:
+ -0.8
+  0.0
+  0.0
+ -0.6276410653920694

    We solve prob using a 25-th order Taylor method, with a local absolute tolerance $\epsilon_\mathrm{tol} = 10^{-15}$.

    solT = solve(prob, TaylorMethod(25), abstol=1e-15);
    retcode: Success
+Interpolation: Taylor series polynomial evaluation
+t: 19608-element Vector{Float64}:
+    0.0
+    0.13196071872716866
+    0.27014676027774454
+    0.42393639891281987
+    0.6040489984083115
+    0.7470888302887118
+    0.8537154042859872
+    0.9339017026636446
+    0.9948376044666346
+    1.041388851851968
+    ⋮
+ 1999.5042725273079
+ 1999.5431825892883
+ 1999.585321353172
+ 1999.6333069204968
+ 1999.6900342956144
+ 1999.7583339712685
+ 1999.841241219841
+ 1999.9430374708056
+ 2000.0
+u: 19608-element Vector{Vector{Float64}}:
+ [-0.8, 0.0, 0.0, -0.6276410653920694]
+ [-0.7932072459999212, 0.021999013383924005, 0.08140695643428739, -0.6377638461316928]
+ [-0.7711573074953806, 0.04021587810521527, 0.17675113719300067, -0.6688864600381992]
+ [-0.7274049294817609, 0.04873797433924279, 0.30569454759447606, -0.7263176192504541]
+ [-0.6471962297961017, 0.0339035917847401, 0.5097162066272879, -0.8242465081730088]
+ [-0.555741961444557, -0.004539329245841953, 0.7525448101567436, -0.9233378179280098]
+ [-0.4645151896861849, -0.05237933172838659, 1.0331768994048929, -0.99547157889477]
+ [-0.37557401376068666, -0.09959841168708136, 1.3584539352908696, -1.016191194976285]
+ [-0.2895689539623897, -0.13989162192605029, 1.7261635478522546, -0.9558013178409663]
+ [-0.20780076710609213, -0.1693782791151218, 2.1143731596742112, -0.7823572471371022]
+ ⋮
+ [-0.17496376257492635, -0.17162024831531136, 1.3853825963776936, -1.9910183910679593]
+ [-0.12304118303795919, -0.23591024736030314, 1.664519586011635, -1.6072083780763986]
+ [-0.06079568084101738, -0.2910297290272401, 1.7937412752512865, -1.201602071416007]
+ [0.010706160716735615, -0.33799420974483546, 1.8000388400926512, -0.8222349820550685]
+ [0.09036540182670252, -0.3776915547287994, 1.7168539805662775, -0.4948839982771776]
+ [0.1759176142431495, -0.4109734546288778, 1.5764514213719614, -0.2305013931376877]
+ [0.2641294989793357, -0.4393446214896214, 1.4058003373126944, -0.02725826011770835]
+ [0.3515484829667911, -0.46520091680777775, 1.2228269820636821, 0.12590172434120908]
+ [0.39178452468014624, -0.47746330663809045, 1.1343053892122903, 0.1855857869403744]

    As mentioned above, we will now solve the same problem prob with the Vern9 method from OrdinaryDiffEq, which the DifferentialEquations documentation recommends for high-accuracy (i.e., very low tolerance) integrations of non-stiff problems. Note that, besides setting an absolute tolerance abstol=1e-15, we're setting a relative tolerance reltol=1e-15 [2]. We have found that for the current problem this is a good balance between speed and accuracy for the Vern9 method, i.e., the Vern9 integration becomes noticeably slower (although more accurate) if either abstol or reltol are set to lower values.

    using OrdinaryDiffEq
+
+solV = solve(prob, Vern9(), abstol=1e-15, reltol=1e-15); #solve `prob` with the `Vern9` method
    retcode: Success
+Interpolation: specialized 9th order lazy interpolation
+t: 109969-element Vector{Float64}:
+    0.0
+    0.014218880460979355
+    0.029254985035048104
+    0.043100405390433874
+    0.05734634963317922
+    0.07358281996048438
+    0.08944879825305627
+    0.10587664107366627
+    0.1226246832421739
+    0.1435160589250072
+    ⋮
+ 1999.8835158404124
+ 1999.8979213519083
+ 1999.9116672471735
+ 1999.9261589673808
+ 1999.941671239634
+ 1999.9563042419318
+ 1999.9717788474554
+ 1999.987747535878
+ 2000.0
+u: 109969-element Vector{Vector{Float64}}:
+ [-0.8, 0.0, 0.0, -0.6276410653920694]
+ [-0.7999214879562981, 0.0024498152778736912, 0.00859413588182428, -0.6277599916692193]
+ [-0.7996675843508191, 0.005034208510608978, 0.017696409719266754, -0.6281442610049114]
+ [-0.7992782930574767, 0.007402685037193327, 0.026103506293378813, -0.6287324566601369]
+ [-0.7987218689531319, 0.00982280232842162, 0.034792021624444995, -0.6295711582485729]
+ [-0.797894470875534, 0.012553097170943344, 0.04475750864054082, -0.63081399178736]
+ [-0.7968864732598321, 0.015184680375190484, 0.0545768793929235, -0.6323213310978815]
+ [-0.7956341732152393, 0.017863254033671463, 0.06484608541850159, -0.634183992679585]
+ [-0.7941379505860766, 0.02053692742751024, 0.07544018310964262, -0.636395532591486]
+ [-0.7919590653442087, 0.023777693641981894, 0.08885831570084259, -0.6395904165292945]
+ ⋮
+ [-0.10036556390021033, -0.40881948168063087, 1.3263848652497512, -1.0339244230210929]
+ [-0.08720892823246354, -0.4219534753803598, 1.3306307677028557, -0.9775508905357453]
+ [-0.07478774834940205, -0.43392235178080524, 1.332218341540104, -0.9262738327498371]
+ [-0.0618595087150646, -0.4459796381322718, 1.33164440577699, -0.8748026893703007]
+ [-0.048236112287607984, -0.4582871679437302, 1.328841777142106, -0.8225447583250936]
+ [-0.03560926760707582, -0.46936547175800486, 1.3244363154112246, -0.7758223093021686]
+ [-0.022510911045455013, -0.480556024781558, 1.318210211953628, -0.7289957477163415]
+ [-0.009284187203683182, -0.4915754146264585, 1.3103724961844636, -0.6833061494007471]
+ [0.0006569134087038978, -0.49968918682462987, 1.303542664918598, -0.6499623338440916]

    We plot in the $x-y$ synodic plane the solution obtained with TaylorIntegration:

    plot(solT, vars=(1, 2), linewidth=1, fmt = :png)
+scatter!([μ, -1+μ], [0,0], leg=false) # positions of the primaries
+xlims!(-1+μ-0.2, 1+μ+0.2)
+ylims!(-0.8, 0.8)
+xlabel!("x")
+ylabel!("y")
    Example block output

    Note that the orbit obtained displays the expected dynamics: the test particle explores the regions surrounding both primaries, located at the red dots, without escaping to infinity. For comparison, we now plot the orbit corresponding to the solution obtained with the Vern9() integration; note that the scales are identical.

    plot(solV, vars=(1, 2), linewidth=1, fmt = :png)
+scatter!([μ, -1+μ], [0,0], leg=false) # positions of the primaries
+xlims!(-1+μ-0.2, 1+μ+0.2)
+ylims!(-0.8, 0.8)
+xlabel!("x")
+ylabel!("y")
    Example block output

    We note that both orbits display the same qualitative features, and also some differences. For instance, the TaylorMethod(25) solution gets closer to the primary than that the Vern9(). We can obtain a quantitative comparison of the validity of both integrations through the preservation of the Jacobi constant:

    ET = H.(solT.u)
+EV = H.(solV.u)
+δET = ET .- J0
+δEV = EV .- J0

    We plot first the value of the Jacobi constant as function of time.

    plot(solT.t, H.(solT.u), label="TaylorMethod(25)", fmt = :png, yformatter = :plain)
+plot!(solV.t, H.(solV.u), label="Vern9()")
+xlabel!("t")
+ylabel!("H")
    Example block output

    In the scale shown we observe that, while both solutions display a preservation of the Jacobi constant to a certain degree, the Vern9() solution suffers sudden jumps during the integration.

    We now plot, in log scale, the abs of the absolute error in the Jacobi constant as a function of time, for both solutions:

    plot(solT.t, abs.(δET), yscale=:log10, label="TaylorMethod(25)", legend=:topleft, fmt = :png, yformatter = :plain)
+plot!(solV.t, abs.(δEV), label="Vern9()")
+ylims!(10^-16, 10^-10)
+xlabel!("t")
+ylabel!("dE")
    Example block output

    We notice that the Jacobi constant absolute error for the TaylorMethod(25) solution remains bounded below $10^{-13}$ throughout the integration. While the Vern9() solution at the end of the integration time has reached a similar value, it displays a larger Jacobi constant error earlier in time.

    Finally, we comment on the time spent by each integration.

    using BenchmarkTools
+bT = @benchmark solve($prob, $(TaylorMethod(25)), abstol=1e-15)
+bV = @benchmark solve($prob, $(Vern9()), abstol=1e-15, reltol=1e-15)
    bT # TaylorMethod(25) benchmark
    BenchmarkTools.Trial: 51 samples with 1 evaluation.
+ Range (min … max):  96.099 ms … 117.861 ms  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     99.151 ms               ┊ GC (median):    0.00%
+ Time  (mean ± σ):   99.502 ms ±   3.505 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
+
+    ▁         █                                                 
+  ▆██▁▁▁▁▁▁▅▆██▆▆█▃▃▃▃▃▃▃▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁
+  96.1 ms         Histogram: frequency by time          111 ms <
+
+ Memory estimate: 8.66 MiB, allocs estimate: 117851.
    bV # Vern9 benchmark
    BenchmarkTools.Trial: 20 samples with 1 evaluation.
+ Range (min … max):  214.910 ms … 646.478 ms  ┊ GC (min … max):  0.00% … 65.40%
+ Time  (median):     232.664 ms               ┊ GC (median):     0.00%
+ Time  (mean ± σ):   269.170 ms ±  97.274 ms  ┊ GC (mean ± σ):  15.87% ± 17.62%
+
+    █                                                            
+  ▅▁█▁▃▃▃▁▁▁▃▁▁▁▁▃▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁
+  215 ms           Histogram: frequency by time          646 ms <
+
+ Memory estimate: 130.11 MiB, allocs estimate: 1319694.

    We notice in this setup, where the TaylorMethod(25) and the Vern9() integrations perform similarly in terms of accuracy, the former performs better in terms of runtime.

    We can tune the abstol and reltol for the Vern9() method we so that performance is similar. Such situation has an accuracy cost, which then makes TaylorIntegration a sensible alternative for high-accuracy integrations of non-stiff ODEs in some cases; see [2].

    Finally, as mentioned above, a crucial way in which TaylorIntegration provides high accuracy at competitive speeds is through the use of the @taylorize macro; see this section for details. Currently, TaylorIntegration supports the use of @taylorize via the common interface with DifferentialEquations only for in-place ODEProblem.

    References and notes

    [1] Murray, Carl D., Stanley F. Dermott. Solar System dynamics. Cambridge University Press, 1999.

    [2] SciMLBenchmarks.jl/DynamicalODE

    diff --git a/v0.16.1/index.html b/v0.16.1/index.html new file mode 100644 index 000000000..1db2c96d7 --- /dev/null +++ b/v0.16.1/index.html @@ -0,0 +1,5 @@ + +Home · TaylorIntegration.jl
    GitHub

    TaylorIntegration.jl

    ODE integration using Taylor's method, and more, in Julia.

    Authors

    • Jorge A. Pérez, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)
    • Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)

    License

    TaylorIntegration is licensed under the MIT "Expat" license.

    Installation

    TaylorIntegration.jl is a registered package, and is simply installed by running

    pkg> add TaylorIntegration

    Supporting and citing

    This package is developed as part of academic research. If you would like to help supporting it, please star the repository as such metrics may help us secure funding. If you use this software, we would be grateful if you could cite our work as follows (Bibtex entry can be found here):

    J.A. Pérez-Hernández and L. Benet
+TaylorIntegration.jl: Taylor Integration in Julia
+https://github.com/PerezHz/TaylorIntegration.jl
+DOI:[10.5281/zenodo.2562352](https://doi.org/10.5281/zenodo.2562352)

    Acknowledgments

    We acknowledge financial support from the DGAPA-PAPIIT (UNAM) grant IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).

    diff --git a/v0.16.1/jet_transport/index.html b/v0.16.1/jet_transport/index.html new file mode 100644 index 000000000..026c33c2b --- /dev/null +++ b/v0.16.1/jet_transport/index.html @@ -0,0 +1,44 @@ + +Jet transport · TaylorIntegration.jl
    GitHub

    Jet transport

    In this section we describe the jet transport capabilities included in TaylorIntegration.jl. Jet transport is a tool that allows the propagation under the flow of a small neighborhood in phase space around a given initial condition, instead of propagating a single initial condition only.

    To compute the propagation of $\mathbf{x}_0 + \delta \mathbf{x}$, where $\delta \mathbf{x}$ are independent small displacements in phase space around the initial condition $\mathbf{x}_0$, one has to solve high-order variational equations. The idea is to treat $\mathbf{x}_0 + \delta \mathbf{x}$ as a truncated polynomial in the $\delta \mathbf{x}$ variables. The maximum order of this polynomial has to be fixed in advance.

    Jet transport works in general with any ordinary ODE solver, provided the chosen solver supports computations using multi-variate polynomial algebra.

    A simple example

    Following D. Pérez-Palau et al [1], let us consider the differential equations for the harmonic oscillator:

    \[\begin{aligned} +\dot{x} & = y, \\ +\dot{y} & = -x, +\end{aligned}\]

    with the initial condition $\mathbf{x}_0=[x_0, y_0]^T$. We illustrate jet transport techniques using Euler's method

    \[\mathbf{x}_{n+1} = \mathbf{x}_n + h \mathbf{f}(\mathbf{x}_n).\]

    Instead of considering the initial conditions $\mathbf{x}_0$, we consider the time evolution of the polynomial

    \[P_{0,\mathbf{x}_0}(\delta\mathbf{x}) = [x_0+\delta x, y_0 + \delta y]^T,\]

    where $\delta x$ and $\delta y$ are small displacements. Below we concentrate in polynomials of order 1 in $\delta x$ and $\delta y$; since the equations of motion of the harmonic oscillator are linear, there are no higher order terms.

    Using Euler's method we obtain

    \[\begin{aligned} + \mathbf{x}_1 & = + \left( + \begin{array}{c} + x_0 + h y_0 \\ + y_0 - h x_0 + \end{array} + \right) + + \left( + \begin{array}{cc} + 1 & h \\ + -h & 1 + \end{array} + \right) + \left( + \begin{array}{c} + \delta x\\ + \delta y + \end{array} + \right). \\ + \mathbf{x}_2 & = + \left( + \begin{array}{c} + 1-h^2 x_0 + 2 h y_0 \\ + 1-h^2 y_0 - 2 h x_0 + \end{array} + \right) + + \left( + \begin{array}{cc} + 1-h^2 & 2 h \\ + -2 h & 1-h^2 + \end{array} + \right) + \left( + \begin{array}{c} + \delta x\\ + \delta y + \end{array} + \right). +\end{aligned}\]

    The first terms in the expressions for $\mathbf{x}_1$ and $\mathbf{x}_2$ above correspond to the result of an Euler integration step using the initial conditions only. The other terms are the (linear) corrections which involve the small displacements $\delta x$ and $\delta y$.

    In general, for differential equations involving non-linear terms, the resulting expansions in $\delta x$ and $\delta y$ will reflect aspects of the non-linearities of the ODEs. Clearly, jet transport techniques allow to address stability properties beyond the linear case, though memory constraints may play a role. See this example illustrating the implementation for the simple pendulum, and this one illustrating the construction of a Poincaré map with Jet transport techniques.

    References

    [1] D. Pérez-Palau, Josep J. Masdemont, Gerard Gómez, 2015, Celest. Mech. Dyn. Astron. 123, 239.

    diff --git a/v0.16.1/kepler/4ca9fc7b.svg b/v0.16.1/kepler/4ca9fc7b.svg new file mode 100644 index 000000000..d30374b9a --- /dev/null +++ b/v0.16.1/kepler/4ca9fc7b.svg @@ -0,0 +1,48 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/kepler/c569d86a.svg b/v0.16.1/kepler/c569d86a.svg new file mode 100644 index 000000000..cd633fad8 --- /dev/null +++ b/v0.16.1/kepler/c569d86a.svg @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/kepler/fb89632a.svg b/v0.16.1/kepler/fb89632a.svg new file mode 100644 index 000000000..ad6dacca6 --- /dev/null +++ b/v0.16.1/kepler/fb89632a.svg @@ -0,0 +1,50 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/kepler/index.html b/v0.16.1/kepler/index.html new file mode 100644 index 000000000..de293867b --- /dev/null +++ b/v0.16.1/kepler/index.html @@ -0,0 +1,59 @@ + +The Kepler problem · TaylorIntegration.jl
    GitHub

    The Kepler problem

    The Kepler problem corresponds to the study of the motion of two bodies which are influenced by their mutual gravitational attraction. In the center of mass and relative coordinates, the problem is reduced to the motion of one body of mass $m = m_1 m_2 / M$, which we shall refer as particle below, attracted gravitationally by another located at rest at the origin of mass $M=m_1+m_2$.

    In cartesian coordinates, the equations of motion can be written as

    \[\begin{aligned} +\dot{x} & = v_x,\\ +\dot{y} & = v_y,\\ +\dot{v}_x & = - \frac{G M x}{(x^2 + y^2)^{3/2}},\\ +\dot{v}_y & = - \frac{G M y}{(x^2 + y^2)^{3/2}}. +\end{aligned}\]

    For concreteness, we fix $\mu = G M = 1$. The coordinates $x$ and $y$ are the relative coordinates (to the center of mass) of the particle, and $v_x$ and $v_y$ its velocity. The function kepler_eqs! mutates the vectors corresponding to the LHS of the equations of motion.

    function kepler_eqs!(dq, q, params, t)
+    dq[1] = q[3]
+    dq[2] = q[4]
+    rr = ( q[1]^2 + q[2]^2 )^(3/2)
+    dq[3] = - q[1] / rr
+    dq[4] = - q[2] / rr
+end;
    kepler_eqs! (generic function with 1 method)

    For suitable initial conditions (such that the total energy is negative) the solutions are ellipses with one focus at the origin, which can be parameterized in terms of its semi-major axis $a$ and its eccentricity $e$. We set the initial conditions for the particle at periapse, which we locate on the positive x-axis. Using the semimajor axis and the eccentricity, we write them as

    \[\begin{aligned} +x_0 & = a (1-e),\\ +y_0 & = 0,\\ +v_{x_0} & = 0,\\ +v_{y_0} & = \frac{l_z}{x_0} = m \frac{\sqrt{\mu a (1-e^2)}}{x_0}, +\end{aligned}\]

    where $l_z$ is the angular momentum. We set the mass of the particle $m=1$, the semi-major axis $a=1$ and the eccentricity $e=0.8$. Kepler's third law defines the period of the motion as $T= 2\pi a^{3/2}$.

    const mu = 1.0
+const mass = 1.0
+const aKep = 1.0
+const eKep = 0.8;
    0.8

    The initial conditions are then set using ini_cond

    function ini_cond(a, e)
+    x0  = a*(one(e)-e)
+    vy0 = mass * sqrt( mu * a * (1-e^2) ) / x0
+    y0  = zero(vy0)
+    vx0 = zero(vy0)
+    return [x0, y0, vx0, vy0]
+end
+q0 = ini_cond(aKep, eKep)
    4-element Vector{Float64}:
+ 0.19999999999999996
+ 0.0
+ 0.0
+ 3.0

    We now perform the integration, using a 25 order expansion and absolute tolerance of $10^{-20}$.

    using TaylorIntegration, Plots
+sol = taylorinteg(kepler_eqs!, q0, 0.0, 10000*2pi, 25, 1.0e-20, maxsteps=700_000);
+t, q = sol.t, sol.x;
+t[end], q[end,:]
    (62831.853071795864, [0.20000000000001486, 5.940560592488514e-9, -4.950456454488883e-8, 2.9999999999999])

    We first plot the orbit. (For performance reasons only the first 10000 points are considered.)

    x = view(q, :, 1)
+y = view(q, :, 2)
+vx = view(q, :, 3)
+vy = view(q, :, 4)
+plot(x[1:10_000], y[1:10_000], legend=false)
+scatter!([0], [0], shape=:circle, ms=5)
+xaxis!("x", (-2.0, 0.5))
+yaxis!("y", (-1.0, 1.0))
+title!("Fig. 1")
    Example block output

    The following functions allow us to calculate the energy and angular momentum using cartesian coordinates.

    function energy( x, y, vx, vy )
+    kinetic = 0.5 * (vx*vx + vy*vy)
+    r = sqrt( x*x + y*y)
+    potential = - mu * mass / r
+    return kinetic + potential
+end
+lz( x, y, vx, vy ) = mass * ( x*vy - y*vx ) ;
    lz (generic function with 1 method)

    We use the change in energy and angular momentum of the orbit with respect to the initial value of the corresponding quantity as a function of time. These quantities are expressed in units of the local epsilon of the initial energy or angular momentum, respectively. This serves to illustrate the accuracy of the calculation, shown in Figure 2 and 3.

    e0 = energy(q0...)
+δE = (energy.(x,y,vx,vy) .- e0) ./ eps(e0)
+plot(t[1:3:end], δE[1:3:end])
+xlabel!("t")
+ylabel!("dE")
+title!("Fig. 2")
    Example block output
    lz0 = lz(q0...)
+δlz = (lz.(x,y,vx,vy) .- lz0) ./ eps(lz0)
+plot(t[1:3:end], δlz[1:3:end])
+xlabel!("t")
+ylabel!("dlz")
+title!("Fig. 3")
    Example block output

    These errors are reminiscent of random walks.

    The maximum absolute errors of the energy and angular momentum are

    maximum( abs.(energy.(x,y,vx,vy) .- e0) ), maximum( abs.(lz.(x,y,vx,vy) .- lz0) )
    (9.947598300641403e-14, 2.808864252301646e-14)
    diff --git a/v0.16.1/lorenz_lyapunov/12e1d07f.svg b/v0.16.1/lorenz_lyapunov/12e1d07f.svg new file mode 100644 index 000000000..e7f5bee3a --- /dev/null +++ b/v0.16.1/lorenz_lyapunov/12e1d07f.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/lorenz_lyapunov/6d97d4d8.svg b/v0.16.1/lorenz_lyapunov/6d97d4d8.svg new file mode 100644 index 000000000..d8b0a96a7 --- /dev/null +++ b/v0.16.1/lorenz_lyapunov/6d97d4d8.svg @@ -0,0 +1,50 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/lorenz_lyapunov/index.html b/v0.16.1/lorenz_lyapunov/index.html new file mode 100644 index 000000000..fc080b468 --- /dev/null +++ b/v0.16.1/lorenz_lyapunov/index.html @@ -0,0 +1,50 @@ + +Lyapunov spectrum of Lorenz system · TaylorIntegration.jl
    GitHub

    Lyapunov spectrum of Lorenz system

    Here, we present the calculation of the Lyapunov spectrum of the Lorenz system, using TaylorIntegration.jl. The computation involves evaluating the 1st order variational equations $\dot \xi = J \cdot \xi$ for this system, where $J = \operatorname{D}f$ is the Jacobian. By default, the numerical value of the Jacobian is computed using automatic differentiation techniques implemented in TaylorSeries.jl, which saves us from writing down explicitly the Jacobian. Conversely, this can be used to check a function implementing the Jacobian. As an alternative, specially important if performance is critical, the user may provide a Jacobian function.

    The Lorenz system is the ODE defined as:

    \[\begin{aligned} + \dot{x}_1 & = \sigma(x_2-x_1), \\ + \dot{x}_2 & = x_1(\rho-x_3)-x_2, \\ + \dot{x}_3 & = x_1x_2-\beta x_3, +\end{aligned}\]

    where $\sigma$, $\rho$ and $\beta$ are constant parameters.

    First, we write a Julia function which evaluates (in-place) the Lorenz system:

    #Lorenz system ODE:
+function lorenz!(dq, q, params, t)
+    σ, ρ, β = params
+    x, y, z = q
+    dq[1] = σ*(y-x)
+    dq[2] = x*(ρ-z)-y
+    dq[3] = x*y-β*z
+    nothing
+end

    Below, we use the the parameters $\sigma = 16.0$, $\beta = 4$ and $\rho = 45.92$.

    #Lorenz system parameters
+#we use the `const` prefix in order to help the compiler speed things up
+const params = [16.0, 45.92, 4.0]

    We define the initial conditions, the initial and final integration times for the integration:

    const x0 = [19.0, 20.0, 50.0] #the initial condition
+const t0 = 0.0     #the initial time
+const tmax = 100.0 #final time of integration

    Since the diagonal of the Jacobian is constant, the sum of the Lyapunov spectrum has to be equal to that value. We calculate this trace using TaylorSeries.jl, and after the numerical integration, we will come back to check if this value is conserved (or approximately conserved) as a function of time.

    # Note that TaylorSeries.jl is @reexport-ed by TaylorIntegration.jl
+# Calculate trace of Lorenz system Jacobian via TaylorSeries.jacobian:
+import LinearAlgebra: tr
+using TaylorIntegration
+xi = set_variables("δ", order=1, numvars=length(x0))
+x0TN = x0 .+ xi
+dx0TN = similar(x0TN)
+lorenz!(dx0TN, x0TN, params, t0)
+jjac = TaylorSeries.jacobian(dx0TN)
+lorenztr = tr(jjac) #trace of Lorenz system Jacobian matrix

    As explained above, the user may provide a function which computes the Jacobian of the ODE in-place:

    #Lorenz system Jacobian (in-place):
+function lorenz_jac!(jac, x, params, t)
+    σ, ρ, β = params
+    jac[1,1] = -σ + zero(x[1])
+    jac[1,2] = σ + zero(x[1])
+    jac[1,3] = zero(x[1])
+    jac[2,1] = ρ - x[3]
+    jac[2,2] = -1.0 + zero(x[1])
+    jac[2,3] = -x[1]
+    jac[3,1] = x[2]
+    jac[3,2] = x[1]
+    jac[3,3] = -β + zero(x[1])
+    nothing
+end
    Note

    We use of zero(x[1]) in the function lorenz_jac! when the RHS consists of a numeric value; this is needed to allow the proper promotion of the variables to carry out Taylor's method.

    We can actually check the consistency of lorenz_jac! with the computation of the jacobian using automatic differentiation techniques. Below we use the initial conditions x0, but it is easy to generalize this.

    lorenz_jac!(jjac, x0, params, t0)  # update the matrix `jjac` using Jacobian provided by the user
+TaylorSeries.jacobian(dx0TN) == jjac    # `dx0TN` is obtained via automatic differentiation
    true

    Now, we are ready to perform the integration using lyap_taylorinteg function, which integrates the 1st variational equations and uses Oseledets' theorem. The expansion order will be $28$ and the local absolute tolerance will be $10^{-20}$. lyap_taylorinteg will return three arrays: one with the evaluation times, one with the values of the dependent variables (at the time of evaluation), and another one with the values of the Lyapunov spectrum.

    We first carry out the integration computing internally the Jacobian

    sol = lyap_taylorinteg(lorenz!, x0, t0, tmax, 28, 1e-20, params; maxsteps=2000000);
+tv, xv, λv = sol.t, sol.x, sol.λ

    Now, the integration is obtained exploiting lorenz_jac!:

    sol_ = lyap_taylorinteg(lorenz!, x0, t0, tmax, 28, 1e-20, params, lorenz_jac!; maxsteps=2000000);
+tv_, xv_, λv_ = sol_.t, sol_.x, sol_.λ

    In terms of performance the second method is about ~50% faster than the first.

    We check the consistency of the orbits computed by the two methods:

    tv == tv_, xv == xv_, λv == λv_
    (true, true, true)

    As mentioned above, a more subtle check is related to the fact that the trace of the Jacobian is constant in time, which must coincide with the sum of all Lyapunov exponents. Using its initial value lorenztr, we compare it with the final Lyapunov exponents of the computation, and obtain

    sum(λv[end,:]) ≈ lorenztr, sum(λv_[end,:]) ≈ lorenztr, sum(λv[end,:]) ≈ sum(λv_[end,:])
    (true, true, true)

    Above we checked the approximate equality; we now show that the relative error is quite small and comparable with the local machine epsilon value around lorenztr:

    abs(sum(λv[end,:])/lorenztr - 1), abs(sum(λv_[end,:])/lorenztr - 1), eps(lorenztr)
    (1.1102230246251565e-15, 1.1102230246251565e-15, 3.552713678800501e-15)

    Therefore, the numerical error is dominated by roundoff errors in the floating point arithmetic of the integration. We will now proceed to plot our results. First, we plot Lorenz attractor in phase space

    using Plots
+plot(xv[:,1], xv[:,2], xv[:,3], leg=false)
    Example block output

    We display now the Lyapunov exponents as a function of time:

    using Plots
+plot(tv, λv[:,1], label="L_1", legend=:right)
+plot!(tv, λv[:,2], label="L_2")
+plot!(tv, λv[:,3], label="L_3")
+xlabel!("time")
+ylabel!("L_i, i=1,2,3")
+title!("Lyapunov exponents vs time")
    Example block output

    This plot shows that the calculation of the Lyapunov exponents has converged.

    diff --git a/v0.16.1/lyapunov_spectrum/index.html b/v0.16.1/lyapunov_spectrum/index.html new file mode 100644 index 000000000..109137ff1 --- /dev/null +++ b/v0.16.1/lyapunov_spectrum/index.html @@ -0,0 +1,2 @@ + +Lyapunov spectrum · TaylorIntegration.jl
    GitHub

    Lyapunov spectrum

    Here we describe the background of the Lyapunov spectra computations in TaylorIntegration.jl. Our implementation follows the numerical method of Benettin et al. [1], [2], which itself is based on Oseledet's multiplicative ergodic theorem [3]. Namely, simultaneously to the integration of the equations of motion, we integrate the 1st-order variational equations associated to them.

    In general, given a dynamical system defined by the equations of motion

    \[\dot{x} = f(t, x),\tag{1}\]

    along with the initial condition $x(t_0) = x_0$, then the first-order variational equations associated to this system are

    \[\dot{\xi} = (\operatorname{D}f)(x(t))\cdot \xi,\tag{2}\]

    where $(\operatorname{D}f)(x(t))$ is the Jacobian of the function $f$ with respect to the dependent variable $x$, evaluated at time $t$, for a given solution $x(t)$ to the equations of motion. The variable $\xi$ denotes a matrix, whose initial condition is $\xi(t_0) = \mathbb{1}_{n}$, the $n\times n$ identity matrix, where $n$ is the degrees of freedom or number of dependent variables $x$.

    For the actual computation of the Lyapunov spectrum, we proceed as follows. During the simultaneous numerical integration of the equations of motion and the variational equations, at fixed time intervals $t_k = k\cdot \Delta t$, $k = 1, 2, \ldots$ we perform a $QR$ decomposition over $\xi(t_k)$, the solution of the variational equations at time $t_k$. That is, we factorize $\xi(t_k)$ as $\xi(t_k)=Q_k\cdot R_k$, where $Q_k$ is an orthogonal $n\times n$ matrix and $R_k$ is an upper triangular $n\times n$ matrix with positive diagonal elements. The diagonal elements $R_{ii,k}$ are the growth factors from which the $l$-th Lyapunov exponent is computed at time $t_k$

    \[\lambda_l = \sum_{m=1}^k \frac{\log (R_{ll,m})}{k\cdot \Delta t}.\tag{3}\]

    In turn, the matrix $Q$ is substituted into $\xi(t_k)$ as the new (scaled) initial condition.

    The equations of motion together with the variational equations are integrated up to time $t_{k+1}$ using Taylor's method. We note that each time step of the integration is determined using the normalized derivatives of $x$ and the tolerance $\epsilon_\textrm{tol}$. This process is repeated until a prescribed $t_\textrm{max}$ is reached.

    This example illustrates the computation of the Lyapunov spectrum for the Lorenz system.

    References

    [1] Benettin G., Galgani L., Giorgilli A., Strelcyn J.M., 1980, Meccanica, 15, 9

    [2] Benettin G., Galgani L., Giorgilli A., Strelcyn J.M., 1980, Meccanica, 15, 21

    [3] Oseledets V. I., 1968, Trudy Moskovskogo Matematicheskogo Obshchestva, 19, 179

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    GitHub

    Jet transport: the simple pendulum

    In this example we illustrate the use of jet transport techniques in TaylorIntegration.jl for the simple pendulum. We propagate a neighborhood $U_0$ around an initial condition $q_0$ parametrized by the sum $q_0+\xi$, where $q_0=(x_0,p_0)$ represents the coordinates of the initial condition in phase space, and $\xi=(\xi_1,\xi_2)$ represents an small variation with respect to this initial condition. We re-interpret each component of the sum $q_0+\xi$ as a multivariate polynomial in the variables $\xi_1$ and $\xi_2$; below, the maximum order of the multivariate polynomial is fixed at 8. We propagate these multivariate polynomials in time using Taylor's method.

    The simple pendulum is defined by the Hamiltonian $H(x, p) = \frac{1}{2}p^2-\cos x$; the corresponding equations of motion are given by

    \[\begin{aligned} +\dot{x} & = p, \\ +\dot{p} & = -\sin x. +\end{aligned}\]

    We integrate this problem for a neighborhood $U_0$ around the initial condition $q_0 = (x(t_0), p(t_0)) = (x_0, p_0)$. For concreteness we take $p_0=0$ and choose $x_0$ such that the pendulum librates; that is, we will choose a numerical value for the energy $E=H(x_0,p_0)=-\cos x_0$ such that the pendulum's motion in phase space is "below" (inside) the region bounded by the separatrix. In this case, the libration period $T$ of the pendulum is

    \[T=\frac{4}{\sqrt{2}}\int_0^{x_0}\frac{dx}{\sqrt{\cos x_0-\cos x}},\]

    which can be expressed in terms of the complete elliptic integral of the first kind, $K$:

    \[T=4K(\sin(x_0/2)).\]

    The Hamiltonian for the simple pendulum is:

    H(x) = 0.5x[2]^2-cos(x[1])

    The equations of motion are:

    function pendulum!(dx, x, p, t)
+    dx[1] = x[2]
+    dx[2] = -sin(x[1])
+end

    We define the TaylorN variables necessary to perform the jet transport; varorder represents the maximum order of expansion in the variations $\xi$.

    const varorder = 8
+using TaylorIntegration
+ξ = set_variables("ξ", numvars=2, order=varorder)
    2-element Vector{TaylorN{Float64}}:
+  1.0 ξ₁ + 𝒪(‖x‖⁹)
+  1.0 ξ₂ + 𝒪(‖x‖⁹)

    Note that TaylorSeries.jl is @reexport-ed internally by TaylorIntegration.jl.

    The nominal initial condition is:

    q0 = [1.3, 0.0]
    2-element Vector{Float64}:
+ 1.3
+ 0.0

    The corresponding initial value of the energy is:

    H0 = H(q0)
    -0.26749882862458735

    The parametrization of the neighborhood $U_0$ is represented by

    q0TN = q0 .+ ξ
    2-element Vector{TaylorN{Float64}}:
+  1.3 + 1.0 ξ₁ + 𝒪(‖x‖⁹)
+        1.0 ξ₂ + 𝒪(‖x‖⁹)

    To understand how the jet transport technique works, we shall evaluate the Hamiltonian at $q_0+\xi$ in order to obtain the 8-th order Taylor expansion of the Hamiltonian with respect to the variations $\xi$, around the initial condition $q_0$:

    H(q0TN)
     - 0.26749882862458735 + 0.963558185417193 ξ₁ + 0.13374941431229367 ξ₁² + 0.5 ξ₂² - 0.1605930309028655 ξ₁³ - 0.011145784526024473 ξ₁⁴ + 0.008029651545143275 ξ₁⁵ + 0.0003715261508674824 ξ₁⁶ - 0.00019118217964626843 ξ₁⁷ - 6.634395551205043e-6 ξ₁⁸ + 𝒪(‖x‖⁹)

    Note that the 0-th order term of the expression above is equal to the value H(q0), as expected.

    Below, we set some parameters for the Taylor integration. We use a method of taylorinteg which returns the solution at t0, t0+integstep, t0+2integstep,...,tmax, where t0 and tmax are the initial and final times of integration, whereas integstep is a time interval chosen by the user; we use the variable tv = t0:integstep:tmax for this purpose and choose integstep as $\frac{T}{8}$.

    order = 28     #the order of the Taylor expansion wrt time
+abstol = 1e-20 #the absolute tolerance of the integration
+using Elliptic # we use Elliptic.jl to evaluate the elliptic integral K
+T = 4*Elliptic.K(sin(q0[1]/2)^2) #the libration period
+t0 = 0.0        #the initial time
+tmax = 6T       #the final time
+integstep = T/8 #the time interval between successive evaluations of the solution vector

    We perform the Taylor integration using the initial condition x0TN, during 6 periods of the pendulum (i.e., $6T$), exploiting multiple dispatch:

    tv = t0:integstep:tmax # the times at which the solution will be evaluated
+sol = taylorinteg(pendulum!, q0TN, tv, order, abstol)

    The integration above works for any initial neighborhood $U_0$ around the nominal initial condition $q_0$, provided it is sufficiently small.

    We will consider the particular case where $U_0$ is a disk of radius $r = 0.05$, centered at $q_0$; that is $U_0=\{ q_0+\xi:\xi=(r\cos\phi,r\sin\phi); \phi\in[0,2\pi) \}$ for a given radius $r>0$. We will denote by $U_t$ the propagation of the initial neighborhood $U_0$ evaluated at time $t$. Also, we denote by $q(t)$ the coordinates of the nominal solution at time $t$: $q(t)=(x(t),p(t))$. Likewise, we will denote the propagation at time $t$ of a given initial variation $\xi_0$ by $\xi(t)$. Then, we can compute the propagation of the boundary $\partial U_t$ of the neighborhood $U_t$.

    polar2cart(r, ϕ) = [r*cos(ϕ), r*sin(ϕ)] # convert radius r and angle ϕ to cartesian coordinates
+r = 0.05 #the radius of the neighborhood
+ϕ = 0.0:0.1:(2π+0.1) #the values of the angle
+ξv = polar2cart.(r, ϕ)

    We evaluate the jet at $\partial U_{x(t)}$ (the boundary of $U_{x(t)}$) at each value of the solution array sol.x; we organize these values such that we can plot them later:

    xjet_plot = map(λ->λ.(ξv), sol.x[:,1])
+pjet_plot = map(λ->λ.(ξv), sol.x[:,2])

    Above, we have exploited the fact that Array{TaylorN{Float64}} variables are callable objects. Now, we evaluate the jet at the nominal solution, which corresponds to $\xi=(0,0)$, at each value of the solution vector sol.x:

    x_nom = sol.x[:,1]()
+p_nom = sol.x[:,2]()

    Finally, we shall plot the nominal solution (black dots), as well as the evolution of the neighborhood $U_0$ (in colors), each $\frac{1}{8}$th of a period $T$. The initial condition corresponds to the black dot situated at $q_0=(1.3,0)$

    using Plots
+plot( xjet_plot, pjet_plot,
+    xaxis=("x",), yaxis=("p",),
+    title="Simple pendulum phase space",
+    leg=false, aspect_ratio=1.0
+)
+scatter!( x_nom, p_nom,
+    color=:black,
+    m=(1,2.8,stroke(0))
+)
    Example block output
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    Poincaré maps

    In this example, we shall illustrate how to construct a Poincaré map associated with the surface of section $x=0$, $\dot x>0$, for $E=0.1025$ for the Hénon-Heiles system. This is equivalent to find the roots of an appropriate function g(dx, x, params, t). We illustrate the implementation using many initial conditions (Monte Carlo like implementation), and then compare the results with the use of jet transport techniques.

    Monte Carlo simulation

    The Hénon-Heiles system is a 2-dof Hamiltonian system used to model the (planar) motion of a star around a galactic center. The Hamiltonian is given by $H = (p_x^2+p_y^2)/2 + (x^2+y^2)/2 + \lambda (x^2y-y^3/3)$, from which the equations of motion can be obtained; below we concentrate in the case $\lambda=1$.

    # Hamiltonian
+V(x,y) = 0.5*( x^2 + y^2 )+( x^2*y - y^3/3)
+H(x,y,p,q) = 0.5*(p^2+q^2) + V(x, y)
+H(x) = H(x...)
+
+# Equations of motion
+function henonheiles!(dq, q, p, t)
+    x, y, px, py = q
+    dq[1] = px
+    dq[2] = py
+    dq[3] = -x-2y*x
+    dq[4] = -y-(x^2-y^2)
+    nothing
+end

    We set the initial energy, which is a conserved quantity; x0 corresponds to the initial condition, which will be properly adjusted to be in the correct energy surface.

    # initial energy and initial condition
+const E0 = 0.1025
+x0 = [0.0, 0.45335, 0.0, 0.0]

    In order to be able to generate (random) initial conditions with the appropriate energy, we write a function px, which depends on x, y, py and the energy E, that returns the value of px>0 for which the initial condition [x, y, px, py] has energy E:

    # px: select px0>0 such that E=E0
+px(x, E) = sqrt(2(E-V(x[1], x[2]))-x[4]^2)
+
+# px!: in-place version of px; returns the modified initial condition `x0`
+function px!(x, E)
+    mypx = px(x, E)
+    x[3] = mypx
+    return x
+end
+
+# run px!
+px!(x0, E0)
    4-element Vector{Float64}:
+ 0.0
+ 0.45335
+ 0.24817464176177093
+ 0.0

    Let's check that the initial condition x0 has actually energy equal to E0, up to roundoff accuracy:

    H(x0)
    0.1025

    The scalar function g, which may depend on the time t, the vector of dependent variables x, the velocities dx, and perhaps some parameters params, following again the convention of DifferentialEquations.jl, defines the surface of section by means of the condition $g(dx, x, params, t) = 0$. Internally, the function g is assumed to return a Tuple{Bool, Taylor1{T}}, where T corresponds to eltype(x[1]) (x::Vector{Taylor1{T}}). In the particular case that the user wishes to discard a particular crossing (or crossings), the function g must return a false value, as will be illustrated below.

    For the present example, we are looking for crossings through the surface $x=0$, which corresponds to x[1]==0, restricting the crossings to satisfy $\dot x > 0$. i.e., x[3]>0. We thus define the function g as

    # x=0, px>0 section
+function g(dx, x, p, t)
+    px_ = constant_term(x[3])
+    # if px > 0...
+    if px_ > zero(px_)
+        return (true, x[1])
+    else
+        #otherwise, discard the crossing
+        return (false, x[1])
+    end
+end
    Note

    Note that in the definition of g we want to make sure that we only take the "positive" crossings through the surface of section $x=0$; hence the if...else... block.

    We initialize some auxiliary arrays, where we shall save the solutions:

    # number of initial conditions
+nconds = 100
+tvSv = Vector{Vector{Float64}}(undef, nconds)
+xvSv = Vector{Matrix{Float64}}(undef, nconds)
+gvSv = Vector{Vector{Float64}}(undef, nconds)
+x_ini = similar(x0)

    We generate nconds random initial conditions in a small neighborhood around x0 and integrate the equations of motion from t0=0 to tmax=135, using a polynomial of order 25 and absolute tolerance 1e-25:

    using TaylorIntegration
+
+for i in 1:nconds
+    rand1 = rand()
+    rand2 = rand()
+    x_ini .= x0 .+ 0.005 .* [0.0, sqrt(rand1)*cos(2pi*rand2), 0.0, sqrt(rand1)*sin(2pi*rand2)]
+    px!(x_ini, E0)   # ensure initial energy is E0
+
+    sol_i = taylorinteg(henonheiles!, g, x_ini, 0.0, 135.0,
+        25, 1e-25, maxsteps=30000);
+    tvSv[i] = vcat(0.0, sol_i.t)
+    xvSv[i] = vcat(transpose(x_ini), sol_i.x)
+    gvSv[i] = vcat(0.0, sol_i.gresids)
+end

    We generate an animation with the solutions

    using Plots
+poincare_anim1 = @animate for i=1:21
+    scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), label="$(i-1)-th iterate",
+        m=(1,stroke(0)), ratio=:equal)
+    xlims!(0.09,0.51)
+    ylims!(-0.11,0.11)
+    xlabel!("y")
+    ylabel!("py")
+    title!("Hénon-Heiles Poincaré map (21 iterates)")
+end
    Plots.Animation("/tmp/jl_jCESW4", ["000001.png", "000002.png", "000003.png", "000004.png", "000005.png", "000006.png", "000007.png", "000008.png", "000009.png", "000010.png"  …  "000012.png", "000013.png", "000014.png", "000015.png", "000016.png", "000017.png", "000018.png", "000019.png", "000020.png", "000021.png"])
    gif(poincare_anim1, "poincareanim1.gif", fps = 2)
    [ Info: Saved animation to /home/runner/work/TaylorIntegration.jl/TaylorIntegration.jl/docs/build/poincareanim1.gif

    Poincaré map for the Hénon Heiles system

    Jet transport

    Now, we illustrate the use of jet transport techniques in the same example, that is, we propagate a neighborhood around x0, which will be plotted in the Poincaré map. We first define the vector of small increments of the phase space variables, xTN; we fix the maximum order of the polynomial expansion in these variables to be 4. Then, x0TN is the neighborhood in the 4-dimensional phase space around $x0$.

    xTN = set_variables("δx δy δpx δpy", numvars=length(x0), order=4)
+x0TN = x0 .+ xTN

    As it was shown above, $x0$ belongs to the energy surface $H(x0) = E_0 = 0.1025$; yet, as it was defined above, the set of phase space points denoted by x0TN includes points that belong to other energy surfaces. This can be noticed by computing H(x0TN)

    H(x0TN)
     0.1025 + 0.2478237775 δy + 0.24817464176177093 δpx + 0.9533499999999999 δx² + 0.046650000000000025 δy² + 0.5 δpx² + 0.5 δpy² + 1.0 δx² δy - 0.3333333333333333 δy³ + 𝒪(‖x‖⁵)

    Clearly, the expression above may contain points whose energy is different from E0. As it was done above, we shall fix the px component of x0TN so all points of the neighborhood are in the same energy surface.

    px!(x0TN, E0) # Impose that all variations are on the proper energy shell!
+H(x0TN)
     0.1025 - 5.551115123125783e-17 δy² - 5.551115123125783e-17 δy³ - 1.7763568394002505e-15 δx² δy² - 4.440892098500626e-16 δy⁴ + 8.881784197001252e-16 δy² δpy² + 𝒪(‖x‖⁵)

    We notice that the coefficients of all monomials whose order is not zero are very small, and the constant_term is E0.

    In order to properly handle this case, we need to extend the definition of g to be useful for Taylor1{TaylorN{T}} vectors.

    #specialized method of g for Taylor1{TaylorN{T}}'s
+function g(dx::Array{Taylor1{TaylorN{T}},1}, x::Array{Taylor1{TaylorN{T}},1},
+        p, t) where {T<:Number}
+    px_ = constant_term(constant_term(x[3]))
+    if px_ > zero( T )
+        return (true, x[1])
+    else
+        return (false, x[1])
+    end
+end

    We are now set to carry out the integration.

    solTN = taylorinteg(henonheiles!, g, x0TN, 0.0, 135.0, 25, 1e-25, maxsteps=30000);

    We define some auxiliary arrays, and then make an animation with the results for plotting.

    #some auxiliaries:
+tvTN, xvTN, tvSTN, xvSTN, gvSTN = solTN.t, solTN.x, solTN.tevents, solTN.xevents, solTN.gresids
+xvSTNaa = Array{Array{TaylorN{Float64},1}}(undef, length(tvSTN)+1 );
+xvSTNaa[1] = x0TN
+for ind in 2:length(tvSTN)+1
+    whatever = xvSTN[ind-1,:]
+    xvSTNaa[ind] = whatever
+end
+tvSTNaa = union([zero(tvSTN[1])], tvSTN);
+
+myrnd  = 0:0.01:1
+npoints = length(myrnd)
+ncrosses = length(tvSTN)
+yS = Array{Float64}(undef, ncrosses+1, npoints)
+pS = Array{Float64}(undef, ncrosses+1, npoints)
+
+myrad=0.005
+ξy = @. myrad * cos(2pi*myrnd)
+ξp = @. myrad * sin(2pi*myrnd)
+
+for indpoint in 1:npoints
+    yS[1,indpoint] = x0[2] + ξy[indpoint]
+    pS[1,indpoint] = x0[4] + ξp[indpoint]
+    mycond = [0.0, ξy[indpoint], 0.0, ξp[indpoint]]
+    for indS in 2:ncrosses+1
+        temp = evaluate(xvSTNaa[indS], mycond)
+        yS[indS,indpoint] = temp[2]
+        pS[indS,indpoint] = temp[4]
+    end
+end
+
+poincare_anim2 = @animate for i=1:21
+    scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), marker=(:circle, stroke(0)),
+        markersize=0.01, label="Monte Carlo")
+    plot!(yS[i,:], pS[i,:], width=0.1, label="Jet transport")
+    xlims!(0.09,0.51)
+    ylims!(-0.11,0.11)
+    xlabel!("y")
+    ylabel!("py")
+    title!("Poincaré map: 4th-order jet transport vs Monte Carlo")
+end
    Plots.Animation("/tmp/jl_6eEcFL", ["000001.png", "000002.png", "000003.png", "000004.png", "000005.png", "000006.png", "000007.png", "000008.png", "000009.png", "000010.png"  …  "000012.png", "000013.png", "000014.png", "000015.png", "000016.png", "000017.png", "000018.png", "000019.png", "000020.png", "000021.png"])
    gif(poincare_anim2, "poincareanim2.gif", fps = 2)
    [ Info: Saved animation to /home/runner/work/TaylorIntegration.jl/TaylorIntegration.jl/docs/build/poincareanim2.gif

    Poincaré map: Jet transport vs Monte Carlo

    The next animation is the same as before, adapting the scale.

    poincare_anim3 = @animate for i=1:21
+    scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), marker=(:circle, stroke(0)),
+        markersize=0.1, label="Monte Carlo")
+    plot!(yS[i,:], pS[i,:], width=0.5, label="Jet transport")
+    xlabel!("y")
+    ylabel!("py")
+    title!("Poincaré map: 4th-order jet transport vs Monte Carlo")
+end
    Plots.Animation("/tmp/jl_Ok4fhx", ["000001.png", "000002.png", "000003.png", "000004.png", "000005.png", "000006.png", "000007.png", "000008.png", "000009.png", "000010.png"  …  "000012.png", "000013.png", "000014.png", "000015.png", "000016.png", "000017.png", "000018.png", "000019.png", "000020.png", "000021.png"])
    gif(poincare_anim3, "poincareanim3.gif", fps = 2)
    [ Info: Saved animation to /home/runner/work/TaylorIntegration.jl/TaylorIntegration.jl/docs/build/poincareanim3.gif

    Poincaré map: Jet transport vs Monte Carlo

    diff --git a/v0.16.1/search_index.js b/v0.16.1/search_index.js new file mode 100644 index 000000000..9b797ef04 --- /dev/null +++ b/v0.16.1/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"root_finding/#rootfinding","page":"Poincaré maps","title":"Poincaré maps","text":"","category":"section"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"In this example, we shall illustrate how to construct a Poincaré map associated with the surface of section x=0, dot x0, for E=01025 for the Hénon-Heiles system. This is equivalent to find the roots of an appropriate function g(dx, x, params, t). We illustrate the implementation using many initial conditions (Monte Carlo like implementation), and then compare the results with the use of jet transport techniques.","category":"page"},{"location":"root_finding/#Monte-Carlo-simulation","page":"Poincaré maps","title":"Monte Carlo simulation","text":"","category":"section"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"The Hénon-Heiles system is a 2-dof Hamiltonian system used to model the (planar) motion of a star around a galactic center. The Hamiltonian is given by H = (p_x^2+p_y^2)2 + (x^2+y^2)2 + lambda (x^2y-y^33), from which the equations of motion can be obtained; below we concentrate in the case lambda=1.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"# Hamiltonian\nV(x,y) = 0.5*( x^2 + y^2 )+( x^2*y - y^3/3)\nH(x,y,p,q) = 0.5*(p^2+q^2) + V(x, y)\nH(x) = H(x...)\n\n# Equations of motion\nfunction henonheiles!(dq, q, p, t)\n x, y, px, py = q\n dq[1] = px\n dq[2] = py\n dq[3] = -x-2y*x\n dq[4] = -y-(x^2-y^2)\n nothing\nend\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We set the initial energy, which is a conserved quantity; x0 corresponds to the initial condition, which will be properly adjusted to be in the correct energy surface.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"# initial energy and initial condition\nconst E0 = 0.1025\nx0 = [0.0, 0.45335, 0.0, 0.0]\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"In order to be able to generate (random) initial conditions with the appropriate energy, we write a function px, which depends on x, y, py and the energy E, that returns the value of px>0 for which the initial condition [x, y, px, py] has energy E:","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"# px: select px0>0 such that E=E0\npx(x, E) = sqrt(2(E-V(x[1], x[2]))-x[4]^2)\n\n# px!: in-place version of px; returns the modified initial condition `x0`\nfunction px!(x, E)\n mypx = px(x, E)\n x[3] = mypx\n return x\nend\n\n# run px!\npx!(x0, E0)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"Let's check that the initial condition x0 has actually energy equal to E0, up to roundoff accuracy:","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"H(x0)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"The scalar function g, which may depend on the time t, the vector of dependent variables x, the velocities dx, and perhaps some parameters params, following again the convention of DifferentialEquations.jl, defines the surface of section by means of the condition g(dx x params t) = 0. Internally, the function g is assumed to return a Tuple{Bool, Taylor1{T}}, where T corresponds to eltype(x[1]) (x::Vector{Taylor1{T}}). In the particular case that the user wishes to discard a particular crossing (or crossings), the function g must return a false value, as will be illustrated below.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"For the present example, we are looking for crossings through the surface x=0, which corresponds to x[1]==0, restricting the crossings to satisfy dot x 0. i.e., x[3]>0. We thus define the function g as","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"# x=0, px>0 section\nfunction g(dx, x, p, t)\n px_ = constant_term(x[3])\n # if px > 0...\n if px_ > zero(px_)\n return (true, x[1])\n else\n #otherwise, discard the crossing\n return (false, x[1])\n end\nend\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"note: Note\nNote that in the definition of g we want to make sure that we only take the \"positive\" crossings through the surface of section x=0; hence the if...else... block.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We initialize some auxiliary arrays, where we shall save the solutions:","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"# number of initial conditions\nnconds = 100\ntvSv = Vector{Vector{Float64}}(undef, nconds)\nxvSv = Vector{Matrix{Float64}}(undef, nconds)\ngvSv = Vector{Vector{Float64}}(undef, nconds)\nx_ini = similar(x0)\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We generate nconds random initial conditions in a small neighborhood around x0 and integrate the equations of motion from t0=0 to tmax=135, using a polynomial of order 25 and absolute tolerance 1e-25:","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"using TaylorIntegration\n\nfor i in 1:nconds\n rand1 = rand()\n rand2 = rand()\n x_ini .= x0 .+ 0.005 .* [0.0, sqrt(rand1)*cos(2pi*rand2), 0.0, sqrt(rand1)*sin(2pi*rand2)]\n px!(x_ini, E0) # ensure initial energy is E0\n\n sol_i = taylorinteg(henonheiles!, g, x_ini, 0.0, 135.0,\n 25, 1e-25, maxsteps=30000);\n tvSv[i] = vcat(0.0, sol_i.t)\n xvSv[i] = vcat(transpose(x_ini), sol_i.x)\n gvSv[i] = vcat(0.0, sol_i.gresids)\nend\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We generate an animation with the solutions","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"using Plots\npoincare_anim1 = @animate for i=1:21\n scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), label=\"$(i-1)-th iterate\",\n m=(1,stroke(0)), ratio=:equal)\n xlims!(0.09,0.51)\n ylims!(-0.11,0.11)\n xlabel!(\"y\")\n ylabel!(\"py\")\n title!(\"Hénon-Heiles Poincaré map (21 iterates)\")\nend","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"gif(poincare_anim1, \"poincareanim1.gif\", fps = 2)\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"(Image: Poincaré map for the Hénon Heiles system)","category":"page"},{"location":"root_finding/#jettransport2","page":"Poincaré maps","title":"Jet transport","text":"","category":"section"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"Now, we illustrate the use of jet transport techniques in the same example, that is, we propagate a neighborhood around x0, which will be plotted in the Poincaré map. We first define the vector of small increments of the phase space variables, xTN; we fix the maximum order of the polynomial expansion in these variables to be 4. Then, x0TN is the neighborhood in the 4-dimensional phase space around x0.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"xTN = set_variables(\"δx δy δpx δpy\", numvars=length(x0), order=4)\nx0TN = x0 .+ xTN\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"As it was shown above, x0 belongs to the energy surface H(x0) = E_0 = 01025; yet, as it was defined above, the set of phase space points denoted by x0TN includes points that belong to other energy surfaces. This can be noticed by computing H(x0TN)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"H(x0TN)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"Clearly, the expression above may contain points whose energy is different from E0. As it was done above, we shall fix the px component of x0TN so all points of the neighborhood are in the same energy surface.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"px!(x0TN, E0) # Impose that all variations are on the proper energy shell!\nH(x0TN)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We notice that the coefficients of all monomials whose order is not zero are very small, and the constant_term is E0.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"In order to properly handle this case, we need to extend the definition of g to be useful for Taylor1{TaylorN{T}} vectors.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"#specialized method of g for Taylor1{TaylorN{T}}'s\nfunction g(dx::Array{Taylor1{TaylorN{T}},1}, x::Array{Taylor1{TaylorN{T}},1},\n p, t) where {T<:Number}\n px_ = constant_term(constant_term(x[3]))\n if px_ > zero( T )\n return (true, x[1])\n else\n return (false, x[1])\n end\nend\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We are now set to carry out the integration.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"solTN = taylorinteg(henonheiles!, g, x0TN, 0.0, 135.0, 25, 1e-25, maxsteps=30000);\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"We define some auxiliary arrays, and then make an animation with the results for plotting.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"#some auxiliaries:\ntvTN, xvTN, tvSTN, xvSTN, gvSTN = solTN.t, solTN.x, solTN.tevents, solTN.xevents, solTN.gresids\nxvSTNaa = Array{Array{TaylorN{Float64},1}}(undef, length(tvSTN)+1 );\nxvSTNaa[1] = x0TN\nfor ind in 2:length(tvSTN)+1\n whatever = xvSTN[ind-1,:]\n xvSTNaa[ind] = whatever\nend\ntvSTNaa = union([zero(tvSTN[1])], tvSTN);\n\nmyrnd = 0:0.01:1\nnpoints = length(myrnd)\nncrosses = length(tvSTN)\nyS = Array{Float64}(undef, ncrosses+1, npoints)\npS = Array{Float64}(undef, ncrosses+1, npoints)\n\nmyrad=0.005\nξy = @. myrad * cos(2pi*myrnd)\nξp = @. myrad * sin(2pi*myrnd)\n\nfor indpoint in 1:npoints\n yS[1,indpoint] = x0[2] + ξy[indpoint]\n pS[1,indpoint] = x0[4] + ξp[indpoint]\n mycond = [0.0, ξy[indpoint], 0.0, ξp[indpoint]]\n for indS in 2:ncrosses+1\n temp = evaluate(xvSTNaa[indS], mycond)\n yS[indS,indpoint] = temp[2]\n pS[indS,indpoint] = temp[4]\n end\nend\n\npoincare_anim2 = @animate for i=1:21\n scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), marker=(:circle, stroke(0)),\n markersize=0.01, label=\"Monte Carlo\")\n plot!(yS[i,:], pS[i,:], width=0.1, label=\"Jet transport\")\n xlims!(0.09,0.51)\n ylims!(-0.11,0.11)\n xlabel!(\"y\")\n ylabel!(\"py\")\n title!(\"Poincaré map: 4th-order jet transport vs Monte Carlo\")\nend","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"gif(poincare_anim2, \"poincareanim2.gif\", fps = 2)\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"(Image: Poincaré map: Jet transport vs Monte Carlo)","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"The next animation is the same as before, adapting the scale.","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"poincare_anim3 = @animate for i=1:21\n scatter(map(x->x[i,2], xvSv), map(x->x[i,4], xvSv), marker=(:circle, stroke(0)),\n markersize=0.1, label=\"Monte Carlo\")\n plot!(yS[i,:], pS[i,:], width=0.5, label=\"Jet transport\")\n xlabel!(\"y\")\n ylabel!(\"py\")\n title!(\"Poincaré map: 4th-order jet transport vs Monte Carlo\")\nend","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"gif(poincare_anim3, \"poincareanim3.gif\", fps = 2)\nnothing # hide","category":"page"},{"location":"root_finding/","page":"Poincaré maps","title":"Poincaré maps","text":"(Image: Poincaré map: Jet transport vs Monte Carlo)","category":"page"},{"location":"lorenz_lyapunov/#lyap_lorenz","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"","category":"section"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Here, we present the calculation of the Lyapunov spectrum of the Lorenz system, using TaylorIntegration.jl. The computation involves evaluating the 1st order variational equations dot xi = J cdot xi for this system, where J = operatornameDf is the Jacobian. By default, the numerical value of the Jacobian is computed using automatic differentiation techniques implemented in TaylorSeries.jl, which saves us from writing down explicitly the Jacobian. Conversely, this can be used to check a function implementing the Jacobian. As an alternative, specially important if performance is critical, the user may provide a Jacobian function.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"The Lorenz system is the ODE defined as:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"beginaligned\n dotx_1 = sigma(x_2-x_1) \n dotx_2 = x_1(rho-x_3)-x_2 \n dotx_3 = x_1x_2-beta x_3\nendaligned","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"where sigma, rho and beta are constant parameters.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"First, we write a Julia function which evaluates (in-place) the Lorenz system:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"#Lorenz system ODE:\nfunction lorenz!(dq, q, params, t)\n σ, ρ, β = params\n x, y, z = q\n dq[1] = σ*(y-x)\n dq[2] = x*(ρ-z)-y\n dq[3] = x*y-β*z\n nothing\nend\nnothing #hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Below, we use the the parameters sigma = 160, beta = 4 and rho = 4592.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"#Lorenz system parameters\n#we use the `const` prefix in order to help the compiler speed things up\nconst params = [16.0, 45.92, 4.0]\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"We define the initial conditions, the initial and final integration times for the integration:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"const x0 = [19.0, 20.0, 50.0] #the initial condition\nconst t0 = 0.0 #the initial time\nconst tmax = 100.0 #final time of integration\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Since the diagonal of the Jacobian is constant, the sum of the Lyapunov spectrum has to be equal to that value. We calculate this trace using TaylorSeries.jl, and after the numerical integration, we will come back to check if this value is conserved (or approximately conserved) as a function of time.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"# Note that TaylorSeries.jl is @reexport-ed by TaylorIntegration.jl\n# Calculate trace of Lorenz system Jacobian via TaylorSeries.jacobian:\nimport LinearAlgebra: tr\nusing TaylorIntegration\nxi = set_variables(\"δ\", order=1, numvars=length(x0))\nx0TN = x0 .+ xi\ndx0TN = similar(x0TN)\nlorenz!(dx0TN, x0TN, params, t0)\njjac = TaylorSeries.jacobian(dx0TN)\nlorenztr = tr(jjac) #trace of Lorenz system Jacobian matrix\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"As explained above, the user may provide a function which computes the Jacobian of the ODE in-place:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"#Lorenz system Jacobian (in-place):\nfunction lorenz_jac!(jac, x, params, t)\n σ, ρ, β = params\n jac[1,1] = -σ + zero(x[1])\n jac[1,2] = σ + zero(x[1])\n jac[1,3] = zero(x[1])\n jac[2,1] = ρ - x[3]\n jac[2,2] = -1.0 + zero(x[1])\n jac[2,3] = -x[1]\n jac[3,1] = x[2]\n jac[3,2] = x[1]\n jac[3,3] = -β + zero(x[1])\n nothing\nend\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"note: Note\nWe use of zero(x[1]) in the function lorenz_jac! when the RHS consists of a numeric value; this is needed to allow the proper promotion of the variables to carry out Taylor's method.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"We can actually check the consistency of lorenz_jac! with the computation of the jacobian using automatic differentiation techniques. Below we use the initial conditions x0, but it is easy to generalize this.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"lorenz_jac!(jjac, x0, params, t0) # update the matrix `jjac` using Jacobian provided by the user\nTaylorSeries.jacobian(dx0TN) == jjac # `dx0TN` is obtained via automatic differentiation","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Now, we are ready to perform the integration using lyap_taylorinteg function, which integrates the 1st variational equations and uses Oseledets' theorem. The expansion order will be 28 and the local absolute tolerance will be 10^-20. lyap_taylorinteg will return three arrays: one with the evaluation times, one with the values of the dependent variables (at the time of evaluation), and another one with the values of the Lyapunov spectrum.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"We first carry out the integration computing internally the Jacobian","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"sol = lyap_taylorinteg(lorenz!, x0, t0, tmax, 28, 1e-20, params; maxsteps=2000000);\ntv, xv, λv = sol.t, sol.x, sol.λ\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Now, the integration is obtained exploiting lorenz_jac!:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"sol_ = lyap_taylorinteg(lorenz!, x0, t0, tmax, 28, 1e-20, params, lorenz_jac!; maxsteps=2000000);\ntv_, xv_, λv_ = sol_.t, sol_.x, sol_.λ\nnothing # hide","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"In terms of performance the second method is about ~50% faster than the first.","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"We check the consistency of the orbits computed by the two methods:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"tv == tv_, xv == xv_, λv == λv_","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"As mentioned above, a more subtle check is related to the fact that the trace of the Jacobian is constant in time, which must coincide with the sum of all Lyapunov exponents. Using its initial value lorenztr, we compare it with the final Lyapunov exponents of the computation, and obtain","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"sum(λv[end,:]) ≈ lorenztr, sum(λv_[end,:]) ≈ lorenztr, sum(λv[end,:]) ≈ sum(λv_[end,:])","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Above we checked the approximate equality; we now show that the relative error is quite small and comparable with the local machine epsilon value around lorenztr:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"abs(sum(λv[end,:])/lorenztr - 1), abs(sum(λv_[end,:])/lorenztr - 1), eps(lorenztr)","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"Therefore, the numerical error is dominated by roundoff errors in the floating point arithmetic of the integration. We will now proceed to plot our results. First, we plot Lorenz attractor in phase space","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"using Plots\nplot(xv[:,1], xv[:,2], xv[:,3], leg=false)","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"We display now the Lyapunov exponents as a function of time:","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"using Plots\nnothing # hide\nplot(tv, λv[:,1], label=\"L_1\", legend=:right)\nplot!(tv, λv[:,2], label=\"L_2\")\nplot!(tv, λv[:,3], label=\"L_3\")\nxlabel!(\"time\")\nylabel!(\"L_i, i=1,2,3\")\ntitle!(\"Lyapunov exponents vs time\")","category":"page"},{"location":"lorenz_lyapunov/","page":"Lyapunov spectrum of Lorenz system","title":"Lyapunov spectrum of Lorenz system","text":"This plot shows that the calculation of the Lyapunov exponents has converged.","category":"page"},{"location":"pendulum/#pendulum","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"","category":"section"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"In this example we illustrate the use of jet transport techniques in TaylorIntegration.jl for the simple pendulum. We propagate a neighborhood U_0 around an initial condition q_0 parametrized by the sum q_0+xi, where q_0=(x_0p_0) represents the coordinates of the initial condition in phase space, and xi=(xi_1xi_2) represents an small variation with respect to this initial condition. We re-interpret each component of the sum q_0+xi as a multivariate polynomial in the variables xi_1 and xi_2; below, the maximum order of the multivariate polynomial is fixed at 8. We propagate these multivariate polynomials in time using Taylor's method.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The simple pendulum is defined by the Hamiltonian H(x p) = frac12p^2-cos x; the corresponding equations of motion are given by","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"beginaligned\ndotx = p \ndotp = -sin x\nendaligned","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"We integrate this problem for a neighborhood U_0 around the initial condition q_0 = (x(t_0) p(t_0)) = (x_0 p_0). For concreteness we take p_0=0 and choose x_0 such that the pendulum librates; that is, we will choose a numerical value for the energy E=H(x_0p_0)=-cos x_0 such that the pendulum's motion in phase space is \"below\" (inside) the region bounded by the separatrix. In this case, the libration period T of the pendulum is","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"T=frac4sqrt2int_0^x_0fracdxsqrtcos x_0-cos x","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"which can be expressed in terms of the complete elliptic integral of the first kind, K:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"T=4K(sin(x_02))","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The Hamiltonian for the simple pendulum is:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"H(x) = 0.5x[2]^2-cos(x[1])\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The equations of motion are:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"function pendulum!(dx, x, p, t)\n dx[1] = x[2]\n dx[2] = -sin(x[1])\nend\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"We define the TaylorN variables necessary to perform the jet transport; varorder represents the maximum order of expansion in the variations xi.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"const varorder = 8\nusing TaylorIntegration\nξ = set_variables(\"ξ\", numvars=2, order=varorder)","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"Note that TaylorSeries.jl is @reexport-ed internally by TaylorIntegration.jl.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The nominal initial condition is:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"q0 = [1.3, 0.0]","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The corresponding initial value of the energy is:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"H0 = H(q0)","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The parametrization of the neighborhood U_0 is represented by","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"q0TN = q0 .+ ξ","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"To understand how the jet transport technique works, we shall evaluate the Hamiltonian at q_0+xi in order to obtain the 8-th order Taylor expansion of the Hamiltonian with respect to the variations xi, around the initial condition q_0:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"H(q0TN)","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"Note that the 0-th order term of the expression above is equal to the value H(q0), as expected.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"Below, we set some parameters for the Taylor integration. We use a method of taylorinteg which returns the solution at t0, t0+integstep, t0+2integstep,...,tmax, where t0 and tmax are the initial and final times of integration, whereas integstep is a time interval chosen by the user; we use the variable tv = t0:integstep:tmax for this purpose and choose integstep as fracT8.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"order = 28 #the order of the Taylor expansion wrt time\nabstol = 1e-20 #the absolute tolerance of the integration\nusing Elliptic # we use Elliptic.jl to evaluate the elliptic integral K\nT = 4*Elliptic.K(sin(q0[1]/2)^2) #the libration period\nt0 = 0.0 #the initial time\ntmax = 6T #the final time\nintegstep = T/8 #the time interval between successive evaluations of the solution vector\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"We perform the Taylor integration using the initial condition x0TN, during 6 periods of the pendulum (i.e., 6T), exploiting multiple dispatch:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"tv = t0:integstep:tmax # the times at which the solution will be evaluated\nsol = taylorinteg(pendulum!, q0TN, tv, order, abstol)\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"The integration above works for any initial neighborhood U_0 around the nominal initial condition q_0, provided it is sufficiently small.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"We will consider the particular case where U_0 is a disk of radius r = 005, centered at q_0; that is U_0= q_0+xixi=(rcosphirsinphi) phiin02pi) for a given radius r0. We will denote by U_t the propagation of the initial neighborhood U_0 evaluated at time t. Also, we denote by q(t) the coordinates of the nominal solution at time t: q(t)=(x(t)p(t)). Likewise, we will denote the propagation at time t of a given initial variation xi_0 by xi(t). Then, we can compute the propagation of the boundary partial U_t of the neighborhood U_t.","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"polar2cart(r, ϕ) = [r*cos(ϕ), r*sin(ϕ)] # convert radius r and angle ϕ to cartesian coordinates\nr = 0.05 #the radius of the neighborhood\nϕ = 0.0:0.1:(2π+0.1) #the values of the angle\nξv = polar2cart.(r, ϕ)\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"We evaluate the jet at partial U_x(t) (the boundary of U_x(t)) at each value of the solution array sol.x; we organize these values such that we can plot them later:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"xjet_plot = map(λ->λ.(ξv), sol.x[:,1])\npjet_plot = map(λ->λ.(ξv), sol.x[:,2])\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"Above, we have exploited the fact that Array{TaylorN{Float64}} variables are callable objects. Now, we evaluate the jet at the nominal solution, which corresponds to xi=(00), at each value of the solution vector sol.x:","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"x_nom = sol.x[:,1]()\np_nom = sol.x[:,2]()\nnothing # hide","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"Finally, we shall plot the nominal solution (black dots), as well as the evolution of the neighborhood U_0 (in colors), each frac18th of a period T. The initial condition corresponds to the black dot situated at q_0=(130)","category":"page"},{"location":"pendulum/","page":"Jet transport: the simple pendulum","title":"Jet transport: the simple pendulum","text":"using Plots\nplot( xjet_plot, pjet_plot,\n xaxis=(\"x\",), yaxis=(\"p\",),\n title=\"Simple pendulum phase space\",\n leg=false, aspect_ratio=1.0\n)\nscatter!( x_nom, p_nom,\n color=:black,\n m=(1,2.8,stroke(0))\n)","category":"page"},{"location":"jet_transport/#jettransport","page":"Jet transport","title":"Jet transport","text":"","category":"section"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"In this section we describe the jet transport capabilities included in TaylorIntegration.jl. Jet transport is a tool that allows the propagation under the flow of a small neighborhood in phase space around a given initial condition, instead of propagating a single initial condition only.","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"To compute the propagation of mathbfx_0 + delta mathbfx, where delta mathbfx are independent small displacements in phase space around the initial condition mathbfx_0, one has to solve high-order variational equations. The idea is to treat mathbfx_0 + delta mathbfx as a truncated polynomial in the delta mathbfx variables. The maximum order of this polynomial has to be fixed in advance.","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"Jet transport works in general with any ordinary ODE solver, provided the chosen solver supports computations using multi-variate polynomial algebra.","category":"page"},{"location":"jet_transport/#A-simple-example","page":"Jet transport","title":"A simple example","text":"","category":"section"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"Following D. Pérez-Palau et al [1], let us consider the differential equations for the harmonic oscillator:","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"beginaligned\ndotx = y \ndoty = -x\nendaligned","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"with the initial condition mathbfx_0=x_0 y_0^T. We illustrate jet transport techniques using Euler's method","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"mathbfx_n+1 = mathbfx_n + h mathbff(mathbfx_n)","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"Instead of considering the initial conditions mathbfx_0, we consider the time evolution of the polynomial","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"P_0mathbfx_0(deltamathbfx) = x_0+delta x y_0 + delta y^T","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"where delta x and delta y are small displacements. Below we concentrate in polynomials of order 1 in delta x and delta y; since the equations of motion of the harmonic oscillator are linear, there are no higher order terms.","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"Using Euler's method we obtain","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"beginaligned\n mathbfx_1 =\n left(\n beginarrayc\n x_0 + h y_0 \n y_0 - h x_0\n endarray\n right)\n + left(\n beginarraycc\n 1 h \n -h 1\n endarray\n right)\n left(\n beginarrayc\n delta x\n delta y\n endarray\n right) \n mathbfx_2 =\n left(\n beginarrayc\n 1-h^2 x_0 + 2 h y_0 \n 1-h^2 y_0 - 2 h x_0\n endarray\n right)\n + left(\n beginarraycc\n 1-h^2 2 h \n -2 h 1-h^2\n endarray\n right)\n left(\n beginarrayc\n delta x\n delta y\n endarray\n right)\nendaligned","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"The first terms in the expressions for mathbfx_1 and mathbfx_2 above correspond to the result of an Euler integration step using the initial conditions only. The other terms are the (linear) corrections which involve the small displacements delta x and delta y.","category":"page"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"In general, for differential equations involving non-linear terms, the resulting expansions in delta x and delta y will reflect aspects of the non-linearities of the ODEs. Clearly, jet transport techniques allow to address stability properties beyond the linear case, though memory constraints may play a role. See this example illustrating the implementation for the simple pendulum, and this one illustrating the construction of a Poincaré map with Jet transport techniques.","category":"page"},{"location":"jet_transport/#refsJT","page":"Jet transport","title":"References","text":"","category":"section"},{"location":"jet_transport/","page":"Jet transport","title":"Jet transport","text":"[1] D. Pérez-Palau, Josep J. Masdemont, Gerard Gómez, 2015, Celest. Mech. Dyn. Astron. 123, 239.","category":"page"},{"location":"api/#Library","page":"API","title":"Library","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"CurrentModule = TaylorIntegration","category":"page"},{"location":"api/#Exported-functions","page":"API","title":"Exported functions","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"taylorinteg\nlyap_taylorinteg\n@taylorize","category":"page"},{"location":"api/#TaylorIntegration.taylorinteg","page":"API","title":"TaylorIntegration.taylorinteg","text":"taylorinteg(f, x0, t0, tmax, order, abstol, params[=nothing]; kwargs... )\n\nGeneral-purpose Taylor integrator for the explicit ODE dotx=f(x p t), where p are the parameters encoded in params. The initial conditions are specified by x0 at time t0; x0 may be of type T<:Number or Vector{T}, with T including TaylorN{T}; the latter case is of interest for jet transport applications.\n\nThe equations of motion are specified by the function f; we follow the same convention of DifferentialEquations.jl to define this function, i.e., f(x, p, t) or f!(dx, x, p, t); see the examples below.\n\nThe functions returns a TaylorSolution, whose fields are t and x; they represent, respectively, a vector with the values of time (independent variable), and a vector with the computed values of the dependent variable(s). When the keyword argument dense is set to true, it also outputs in the field p the Taylor polynomial expansion computed at each time step. The integration stops when time is larger than tmax, in which case the last returned value(s) correspond to tmax, or when the number of saved steps is larger than maxsteps.\n\nThe integration method uses polynomial expansions on the independent variable of order order; the parameter abstol serves to define the time step using the last two Taylor coefficients of the expansions. Make sure you use a large enough order to assure convergence.\n\nCurrently, the recognized keyword arguments are:\n\nmaxsteps[=500]: maximum number of integration steps.\nparse_eqs[=true]: use the specialized method of jetcoeffs! created with @taylorize.\ndense[=true]: output the Taylor polynomial expansion at each time step.\n\nExamples\n\nFor one dependent variable the function f defines the RHS of the equation of motion, returning the value of dotx. The arguments of this function are (x, p, t), where x are the dependent variables, p are the paremeters and t is the independent variable.\n\nFor several (two or more) dependent variables, the function f! defines the RHS of the equations of motion, mutating (in-place) the (preallocated) vector with components of dotx. The arguments of this function are (dx, x, p, t), where dx is the preallocated vector of dotx, x are the dependent variables, p are the paremeters entering the ODEs and t is the independent variable. The function may return this vector or simply nothing.\n\nusing TaylorIntegration\n\nf(x, p, t) = x^2\n\nsol = taylorinteg(f, 3, 0.0, 0.3, 25, 1.0e-20, maxsteps=100 )\n\nfunction f!(dx, x, p, t)\n for i in eachindex(x)\n dx[i] = x[i]^2\n end\n return nothing\nend\n\nsol = taylorinteg(f!, [3, 3], 0.0, 0.3, 25, 1.0e-20, maxsteps=100 )\n\nsol = taylorinteg(f!, [3, 3], 0.0, 0.3, 25, 1.0e-20, maxsteps=100, dense=true )\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorIntegration.lyap_taylorinteg","page":"API","title":"TaylorIntegration.lyap_taylorinteg","text":"lyap_taylorinteg(f!, q0, t0, tmax, order, abstol[, jacobianfunc!=nothing];\n maxsteps::Int=500, parse_eqs::Bool=true)\nlyap_taylorinteg(f!, q0, trange, order, abstol[, jacobianfunc!=nothing];\n maxsteps::Int=500, parse_eqs::Bool=true)\n\nSimilar to taylorinteg for the calculation of the Lyapunov spectrum. Note that the number of TaylorN variables should be set previously by the user (e.g., by means of TaylorSeries.set_variables) and should be equal to the length of the vector of initial conditions q0. Otherwise, whenever length(q0) != TaylorSeries.get_numvars(), then lyap_taylorinteg throws an AssertionError. Optionally, the user may provide a Jacobian function jacobianfunc! to evaluate the current value of the Jacobian. Otherwise, the current value of the Jacobian is computed via automatic differentiation using TaylorSeries.jl.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorIntegration.@taylorize","page":"API","title":"TaylorIntegration.@taylorize","text":"@taylorize expr\n\nThis macro evals the function given by expr and defines a new method of jetcoeffs! which is specialized on that function. Integrating via taylorinteg of lyap_taylorinteg after using the macro yields better performance.\n\nSee the documentation for more details and limitations.\n\nwarning: Warning\nThis macro is on an experimental stage; check the integration results carefully.\n\n\n\n\n\n","category":"macro"},{"location":"api/#Exported-types","page":"API","title":"Exported types","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"TaylorSolution","category":"page"},{"location":"api/#TaylorIntegration.TaylorSolution","page":"API","title":"TaylorIntegration.TaylorSolution","text":"TaylorSolution{T, U, N, VT<:AbstractVector{T}, AX<:AbstractArray{U,N}, P<:Union{Nothing, AbstractArray{Taylor1{U}, N}}, VTE<:Union{Nothing, AbstractVector{U}}, AXE<:Union{Nothing, AbstractArray{U, N}}, VΛ<:Union{Nothing, AbstractArray{U,N}}} <: AbstractTaylorSolution{T, U}\n\nThis struct represents the return type for taylorinteg. Fields t and x represent, respectively, a vector with the values of time (independent variable), and a vector with the computed values of the dependent variable(s). When taylorinteg is called with dense=true, then field p stores the Taylor polynomial expansion computed at each time step. Fields tevents, xevents and gresids are related to root-finding methods of taylorinteg, while λ is related to the output of lyap_taylorinteg.\n\n\n\n\n\n","category":"type"},{"location":"api/#Internal","page":"API","title":"Internal","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Modules = [TaylorIntegration]\nPublic = false","category":"page"},{"location":"api/#TaylorIntegration.BookKeeping","page":"API","title":"TaylorIntegration.BookKeeping","text":"BookKeeping\n\nMutable struct that contains all the bookkeeping vectors/dictionaries used within _make_parsed_jetcoeffs: - d_indx : Dictionary mapping new variables (symbols) to old (perhaps indexed) symbols - d_assign : Dictionary with the numeric assignments (that are substituted) - d_decl : Dictionary declared arrays - v_newvars : Symbols of auxiliary indexed vars - v_arraydecl: Symbols which are explicitly declared as Array or Vector - v_array1 : Symbols which are explicitly declared as Array{Taylor1{T},1} - v_array2 : Symbols which are explicitly declared as Array{Taylor1{T},2} - v_array3 : Symbols which are explicitly declared as Array{Taylor1{T},3} - v_array4 : Symbols which are explicitly declared as Array{Taylor1{T},4} - v_preamb : Symbols or Expr used in the preamble (declarations, etc) - retvar : Guessed returned variable, which defines the LHS of the ODEs\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorIntegration.RetAlloc","page":"API","title":"TaylorIntegration.RetAlloc","text":"RetAlloc{T <: Number}\n\nStruct related to the returned variables that are pre-allocated when @taylorize is used. - v0 : Array{T,1} - v1 : Vector{Array{T,1}} - v2 : Vector{Array{T,2}} - v3 : Vector{Array{T,3}} - v4 : Vector{Array{T,4}}\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorIntegration.__jetcoeffs!-Union{Tuple{U}, Tuple{Val{false}, Any, Any, Taylor1{U}, Any, TaylorIntegration.RetAlloc{Taylor1{U}}}} where U","page":"API","title":"TaylorIntegration.__jetcoeffs!","text":"__jetcoeffs!(::Val{false}, f, t, x, params, rv)\n__jetcoeffs!(::Val{true}, f, t, x, params, rv)\n__jetcoeffs!(::Val{false}, f, t, x, dx, xaux, params, rv)\n__jetcoeffs!(::Val{true}, f, t, x, dx, xaux, params, rv)\n\nChooses a method of jetcoeffs! (hard-coded) or the generated by @taylorize) depending on Val{bool} (bool::Bool).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._allocated_defs!-Tuple{Expr, TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._allocated_defs!","text":"_allocated_defs!(new_jetcoeffs, bkkeep)\n\nAdd allocated variable definitions to new_jetcoeffs, to make it more human readable.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._capture_fn_args_body!","page":"API","title":"TaylorIntegration._capture_fn_args_body!","text":"_capture_fn_args_body!(ex, dd::Dict{Symbol, Any})\n\nCaptures the name of a function, arguments, body and other properties, returning them as the values of the dictionary dd, which is updated in place.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorIntegration._defs_allocs!","page":"API","title":"TaylorIntegration._defs_allocs!","text":"_defs_allocs!(preamble, fnargs, bkkeep, [inloop=false, ex_aux::Expr(:block,)])\n\nReturns a vector with expressions defining the auxiliary variables in the preamble, and the declaration of the arrays. This function may modify bkkeep.d_indx if new variables are introduced. bkkeep.v_preamb is for bookkeeping the introduced variables.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorIntegration._determine_parsing!-Tuple{Bool, Vararg{Any, 4}}","page":"API","title":"TaylorIntegration._determine_parsing!","text":"_determine_parsing!(parse_eqs::Bool, f, t, x, params)\n_determine_parsing!(parse_eqs::Bool, f, t, x, dx, params)\n\nCheck if the parsed method of jetcoeffs! exists and check it runs without error.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._extract_parts-Tuple{Expr}","page":"API","title":"TaylorIntegration._extract_parts","text":"_extract_parts(ex::Expr)\n\nReturns the function name, the function arguments, and the body of a function passed as an Expr. The function may be provided as a one-line function, or in the long form (anonymous functions do not work).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._make_parsed_jetcoeffs-Tuple{Expr}","page":"API","title":"TaylorIntegration._make_parsed_jetcoeffs","text":"_make_parsed_jetcoeffs( ex )\n\nThis function constructs the expressions of two new methods, the first equivalent to the differential equations (jetcoeffs!), which exploits the mutating functions of TaylorSeries.jl, and the second one (allocatejetcoeffs) preallocates any auxiliary Taylor1 or Vector{Taylor1{T}} needed.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._newfnbody!-Tuple{Any, Any, TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._newfnbody!","text":"_newfnbody!(fnbody, fnargs, bkkeep)\n\nReturns a new (modified) body of the function, a priori unfolding the expression graph (AST) as unary and binary calls, and updates the bookkeeping structure bkkeep.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._newhead-Tuple{Any, Any}","page":"API","title":"TaylorIntegration._newhead","text":"_newhead(fn, fnargs)\n\nCreates the head of the new method of jetcoeffs! and _allocate_jetcoeffs. fn is the name of the passed function and fnargs is a vector with its arguments defning the function (which are either three or four).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._parse_newfnbody!-Tuple{Expr, Expr, Expr, TaylorIntegration.BookKeeping, Bool}","page":"API","title":"TaylorIntegration._parse_newfnbody!","text":"parsenewfnbody!(ex::Expr, preex::Expr, prealloc::Expr, bkkeep::BookKeeping, inloop::Bool)\n\nParses ex (the new body of the function) replacing the expressions to use the mutating functions of TaylorSeries, and building the preamble preex and prealloc expressions. This is done by traversing recursively (again) the args of ex, updating the bookkeeping struct bkkeep, in particular the fieldnames v_newvars and d_assign.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._preamble_body-Tuple{Any, Any}","page":"API","title":"TaylorIntegration._preamble_body","text":"_preamble_body(fnbody, fnargs)\n\nReturns expressions for the preamble, the declaration of arrays, the body and the bookkeeping struct, which will be used to build the new functions. fnbody is the expression with the body of the original function (already adapted), fnargs is a vector of symbols of the original diferential equations function.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._recursionloop-Tuple{Any, TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._recursionloop","text":"_recursionloop(fnargs, bkkeep)\n\nBuild the expression for the recursion-loop.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._rename_indexedvars-Tuple{Any}","page":"API","title":"TaylorIntegration._rename_indexedvars","text":"_rename_indexedvars(fnbody)\n\nRenames the indexed variables (using Espresso.genname()) that exists in fnbody. Returns fnbody with the renamed variables and a dictionary that links the new variables to the old indexed ones.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._replace_expr!-Tuple{Expr, Expr, Expr, Int64, Any, Any, TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._replace_expr!","text":"_replace_expr!(ex::Expr, preex::Expr, , prealloc::Expr, i::Int, aalhs, aarhs, bkkeep::BookKeeping)\n\nReplaces the calls in ex.args[i], and updates preex and prealloc with the appropriate expressions, based on the the LHS (aalhs) and RHS (aarhs) of the base assignment. The bookkeeping struct is updated (v_newvars) within _replacecalls!. d_indx is used to bring back the indexed variables.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._replacecalls!-Tuple{TaylorIntegration.BookKeeping, Expr, Symbol}","page":"API","title":"TaylorIntegration._replacecalls!","text":"_replacecalls!(bkkeep, fnold, newvar)\n\nReplaces the symbols of unary and binary calls of the expression fnold, which defines newvar, by the mutating functions in TaylorSeries.jl. The vector bkkeep.v_vars is updated if new auxiliary variables are introduced (bookkeeping).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._returned_expr-Tuple{TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._returned_expr","text":"_returned_expr(bkkeep)\n\nConstructs the expression to be returned by TaylorIntegration._allocate_jetcoeffs!\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._second_stepsize-Union{Tuple{U}, Tuple{T}, Tuple{Taylor1{U}, T}} where {T<:Real, U<:Number}","page":"API","title":"TaylorIntegration._second_stepsize","text":"_second_stepsize(x, epsilon)\n\nCorresponds to the \"second stepsize control\" in Jorba and Zou (2005) paper. We use it if stepsize returns Inf.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._split_arraydecl!-Tuple{TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration._split_arraydecl!","text":"_split_arraydecl!(bkkeep)\n\nSplit bkkeep.varraydecl in the vector (bkkeep.varray1), matrix (bkkeep.v_array2), etc, to properly construct the RetAlloc variable.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration._stepsize-Union{Tuple{U}, Tuple{T}, Tuple{U, T, Int64}} where {T<:Real, U<:Number}","page":"API","title":"TaylorIntegration._stepsize","text":"_stepsize(aux1, epsilon, k)\n\nHelper function to avoid code repetition. Returns (epsilon/aux1)^(1/k).\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.build_lyap_solution-Union{Tuple{U}, Tuple{T}, Tuple{AbstractVector{T}, Matrix{U}, Matrix{U}, Int64}} where {T, U}","page":"API","title":"TaylorIntegration.build_lyap_solution","text":"build_lyap_solution(t, x, λ, nsteps)\nbuild_lyap_solution(t, x, λ)\n\nHelper function to build a TaylorSolution from a call to lyap_taylorinteg.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.build_solution-Union{Tuple{P}, Tuple{U}, Tuple{T}, Tuple{AbstractVector{T}, Matrix{U}, P, AbstractVector{U}, Matrix{U}, AbstractVector{U}, Int64, Int64}} where {T, U, P<:Union{Nothing, Array{Taylor1{U}, 2}}}","page":"API","title":"TaylorIntegration.build_solution","text":"build_solution(t, x, p, tevents, xevents, gresids, nsteps, nevents)\nbuild_solution(t, x, tevents, xevents, gresids, nsteps, nevents)\n\nHelper function to build a TaylorSolution from a call to a root-finding method of taylorinteg.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.build_solution-Union{Tuple{U}, Tuple{T}, Tuple{AbstractVector{T}, Vector{U}, Nothing, Int64}} where {T, U}","page":"API","title":"TaylorIntegration.build_solution","text":"build_solution(t, x, p, nsteps)\nbuild_solution(t, x)\n\nHelper function to build a TaylorSolution from a call to taylorinteg.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.findroot!-NTuple{18, Any}","page":"API","title":"TaylorIntegration.findroot!","text":"findroot!(t, x, dx, g_tupl_old, g_tupl, eventorder, tvS, xvS, gvS,\n t0, δt_old, x_dx, x_dx_val, g_dg, g_dg_val, nrabstol,\n newtoniter, nevents) -> nevents\n\nInternal root-finding subroutine, based on Newton-Raphson process. If there is a crossing, then the crossing data is stored in tvS, xvS and gvS and nevents, the number of events/crossings, is updated. Here t is a Taylor1 polynomial which represents the independent variable; x is an array of Taylor1 variables which represent the vector of dependent variables; dx is an array of Taylor1 variables which represent the LHS of the ODE; g_tupl_old is the last-before-current value returned by event function g and g_tupl is the current one; eventorder is the order of the derivative of g whose roots the user is interested in finding; tvS stores the surface-crossing instants; xvS stores the value of the solution at each of the crossings; gvS stores the values of the event function g (or its eventorder-th derivative) at each of the crossings; t0 is the current time; δt_old is the last time-step size; x_dx, x_dx_val, g_dg, g_dg_val are auxiliary variables; nrabstol is the Newton-Raphson process tolerance; newtoniter is the maximum allowed number of Newton-Raphson iteration; nevents is the current number of detected events/crossings.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.inbookkeeping-Tuple{Any, TaylorIntegration.BookKeeping}","page":"API","title":"TaylorIntegration.inbookkeeping","text":"inbookkeeping(v, bkkeep::BookKeeping)\n\nChecks if v is declared in bkkeep, considering the d_indx, v_newvars and v_arraydecl fields.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.init_psol-Union{Tuple{U}, Tuple{Val{true}, Vector{U}, Taylor1{U}}} where U<:Number","page":"API","title":"TaylorIntegration.init_psol","text":"init_psol(::Val{true}, xv::Array{U,1}, x::Taylor1{U}) where {U<:Number}\ninit_psol(::Val{true}, xv::Array{U,2}, x::Array{Taylor1{U},1}) where {U<:Number}\ninit_psol(::Val{false}, ::Array{U,1}, ::Taylor1{U}) where {U<:Number}\ninit_psol(::Val{false}, ::Array{U,2}, ::Array{Taylor1{U},1}) where {U<:Number}\n\nAuxiliary function to initialize psol during a call to taylorinteg. When the first argument in the call signature is Val(false) this function simply returns nothing. Otherwise, when the first argument in the call signature is Val(true), then the appropriate array is allocated and returned; this array is where the Taylor polynomials associated to the solution will be stored, corresponding to field :p in TaylorSolution.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.jetcoeffs!-Union{Tuple{N}, Tuple{U}, Tuple{T}, Tuple{Function, Taylor1{T}, AbstractArray{Taylor1{U}, N}, AbstractArray{Taylor1{U}, N}, AbstractArray{Taylor1{U}, N}, Any}} where {T<:Real, U<:Number, N}","page":"API","title":"TaylorIntegration.jetcoeffs!","text":"jetcoeffs!(eqsdiff!::Function, t, x, dx, xaux, params)\n\nMutates x in-place using the recursion relation of the derivatives obtained from the differential equations dotx=dxdt=f(x p t).\n\neqsdiff! is the function defining the RHS of the ODE, x contains the Taylor1 expansion of the dependent variables and t is the independent variable, and params are the parameters appearing on the function defining the differential equation. See taylorinteg for examples and convention for eqsdiff. Note that x is of type Vector{Taylor1{U}} where U<:Number; t is of type Taylor1{T} where T<:Real. In this case, two auxiliary containers dx and xaux (both of the same type as x) are needed to avoid allocations.\n\nInitially, x contains only the 0-th order Taylor coefficient of the current system state (the initial conditions), and jetcoeffs! computes recursively the high-order derivates back into x.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.jetcoeffs!-Union{Tuple{U}, Tuple{T}, Tuple{Function, Taylor1{T}, Taylor1{U}, Any}} where {T<:Real, U<:Number}","page":"API","title":"TaylorIntegration.jetcoeffs!","text":"jetcoeffs!(eqsdiff::Function, t, x, params)\n\nReturns an updated x using the recursion relation of the derivatives obtained from the differential equations dotx=dxdt=f(x p t).\n\neqsdiff is the function defining the RHS of the ODE, x contains the Taylor1 expansion of the dependent variable(s) and t is the independent variable, and params are the parameters appearing on the function defining the differential equation. See taylorinteg for examples and convention for eqsdiff. Note that x is of type Taylor1{U} where U<:Number; t is of type Taylor1{T} where T<:Real.\n\nInitially, x contains only the 0-th order Taylor coefficient of the current system state (the initial conditions), and jetcoeffs! computes recursively the high-order derivates back into x.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.lyap_jetcoeffs!-Union{Tuple{S}, Tuple{T}, Tuple{Taylor1{T}, AbstractArray{Taylor1{S}, 1}, AbstractArray{Taylor1{S}, 1}, Array{Taylor1{S}, 2}, Array{Taylor1{S}, 3}}} where {T<:Real, S<:Number}","page":"API","title":"TaylorIntegration.lyap_jetcoeffs!","text":"lyap_jetcoeffs!(t, x, dx, jac, varsaux)\n\nSimilar to jetcoeffs! for the calculation of the Lyapunov spectrum. Updates only the elements of x which correspond to the solution of the 1st-order variational equations dotxi=J cdot xi, where J is the Jacobian matrix, i.e., the linearization of the equations of motion. jac is the Taylor expansion of J wrt the independent variable, around the current initial condition. varsaux is an auxiliary array of type Array{eltype(jac),3} to avoid allocations. Calling this method assumes that jac has been computed previously using stabilitymatrix!.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.nrconvergencecriterion-Union{Tuple{T}, Tuple{U}, Tuple{U, T, Int64, Int64}} where {U<:Number, T<:Real}","page":"API","title":"TaylorIntegration.nrconvergencecriterion","text":"nrconvergencecriterion(g_val, nrabstol::T, nriter::Int, newtoniter::Int) where {T<:Real}\n\nA rudimentary convergence criterion for the Newton-Raphson root-finding process. g_val may be either a Real, Taylor1{T} or a TaylorN{T}, where T<:Real. Returns true if: 1) the absolute value of g_val, the value of the event function g evaluated at the current estimated root by the Newton-Raphson process, is less than the nrabstol tolerance; and 2) the number of iterations nriter of the Newton-Raphson process is less than the maximum allowed number of iterations, newtoniter; otherwise, returns false.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.set_psol!-Union{Tuple{U}, Tuple{Val{true}, Array{Taylor1{U}, 1}, Int64, Taylor1{U}}} where U<:Number","page":"API","title":"TaylorIntegration.set_psol!","text":"set_psol!(::Val{true}, psol::Array{Taylor1{U},1}, nsteps::Int, x::Taylor1{U}) where {U<:Number}\nset_psol!(::Val{true}, psol::Array{Taylor1{U},2}, nsteps::Int, x::Vector{Taylor1{U}}) where {U<:Number}\nset_psol!(::Val{false}, args...)\n\nAuxiliary function to save Taylor polynomials in a call to taylorinteg. When the first argument in the call signature is Val(true), sets appropriate elements of argument psol. Otherwise, when the first argument in the call signature is Val(false), this function simply returns nothing. Argument psol is the array where the Taylor polynomials associated to the solution will be stored, corresponding to field :p in TaylorSolution. See also init_psol.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.stabilitymatrix!-Union{Tuple{U}, Tuple{T}, Tuple{Any, Taylor1{T}, Array{Taylor1{U}, 1}, Array{TaylorN{Taylor1{U}}, 1}, Array{TaylorN{Taylor1{U}}, 1}, Array{Taylor1{U}, 2}, Array{TaylorN{Taylor1{U}}, 1}, Any, Nothing}} where {T<:Real, U<:Number}","page":"API","title":"TaylorIntegration.stabilitymatrix!","text":"stabilitymatrix!(eqsdiff!, t, x, δx, dδx, jac, _δv, params[, jacobianfunc!=nothing])\n\nUpdates the matrix jac::Matrix{Taylor1{U}} (linearized equations of motion) computed from the equations of motion (eqsdiff!), at time t at x; x is of type Vector{Taylor1{U}}, where U<:Number. δx, dδx and _δv are auxiliary arrays of type Vector{TaylorN{Taylor1{U}}} to avoid allocations. Optionally, the user may provide a Jacobian function jacobianfunc! to compute jac. Otherwise, jac is computed via automatic differentiation using TaylorSeries.jl.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.stepsize-Union{Tuple{U}, Tuple{T}, Tuple{Taylor1{U}, T}} where {T<:Real, U<:Number}","page":"API","title":"TaylorIntegration.stepsize","text":"stepsize(x, epsilon) -> h\n\nReturns a maximum time-step for a the Taylor expansion x using a prescribed absolute tolerance epsilon and the last two Taylor coefficients of (each component of) x.\n\nNote that x is of type Taylor1{U} or Vector{Taylor1{U}}, including also the cases Taylor1{TaylorN{U}} and Vector{Taylor1{TaylorN{U}}}.\n\nDepending of eltype(x), i.e., U<:Number, it may be necessary to overload stepsize, specializing it on the type U, to avoid type instabilities.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.surfacecrossing-Union{Tuple{T}, Tuple{Tuple{Bool, Taylor1{T}}, Tuple{Bool, Taylor1{T}}, Int64}} where T<:Number","page":"API","title":"TaylorIntegration.surfacecrossing","text":"surfacecrossing(g_old, g_now, eventorder::Int)\n\nDetect if the solution crossed a root of event function g. g_old represents the last-before-current value of event function g, and g_now represents the current one; these are Tuple{Bool,Taylor1{T}}s. eventorder is the order of the derivative of the event function g whose root we are trying to find. Returns true if the constant terms of g_old[2] and g_now[2] have different signs (i.e., if one is positive and the other one is negative). Otherwise, if g_old[2] and g_now[2] have the same sign or if the first component of either of them is false, then it returns false.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.taylorstep!-Union{Tuple{V}, Tuple{U}, Tuple{T}, Tuple{Val{V}, Any, Taylor1{T}, Taylor1{U}, T, Any, TaylorIntegration.RetAlloc{Taylor1{U}}}} where {T<:Real, U<:Number, V}","page":"API","title":"TaylorIntegration.taylorstep!","text":"taylorstep!(::Val{false}, f, t, x, abstol, params, rv, parse_eqs=true) -> δt\ntaylorstep!(::Val{true}, f, t, x, abstol, params, rv, parse_eqs=true) -> δt\ntaylorstep!(::Val{false}, f!, t, x, dx, xaux, abstol, params, rv, parse_eqs=true) -> δt\ntaylorstep!(::Val{true}, f!, t, x, dx, xaux, abstol, params, rv, parse_eqs=true) -> δt\n\nOne-step Taylor integration for the one-dependent variable ODE dotx=dxdt=f(x p t) with initial conditions x(t_0)=x_0. Returns the time-step δt of the actual integration carried out (δt is positive).\n\nHere, f represents the function defining the RHS of the ODE (see taylorinteg), t is the independent variable, x contains the Taylor expansion of the dependent variable, order is the degree used for the Taylor1 polynomials during the integration abstol is the absolute tolerance used to determine the time step of the integration, and params are the parameters entering the ODE functions. For several variables, dx and xaux, both of the same type as x, are needed to save allocations. Finally, parse_eqs is a switch to force not using (parse_eqs=false) the specialized method of jetcoeffs! created with @taylorize; the default is true (parse the equations). Finally, parse_eqs is a switch to force not using (parse_eqs=false) the specialized method of jetcoeffs! created with @taylorize; the default is true (parse the equations). The first argument in the function call Val{bool} (bool::Bool) controls whether a specialized `jetcoeffs! method is being used or not.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorIntegration.timeindex-Union{Tuple{TT}, Tuple{TaylorSolution, TT}} where TT","page":"API","title":"TaylorIntegration.timeindex","text":"timeindex(sol::TaylorSolution, t::TT) where TT\n\nReturn the index of sol.t corresponding to t and the time elapsed from sol.t0 to t.\n\n\n\n\n\n","category":"method"},{"location":"api/#Index","page":"API","title":"Index","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Pages = [\"api.md\"]\nModule = [\"TaylorIntegration\"]\nOrder = [:function]\nPublic = true","category":"page"},{"location":"common/#diffeqinterface","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"","category":"section"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Here, we show an example of interoperability between TaylorIntegration.jl and DifferentialEquations.jl, i.e., how to use TaylorIntegration.jl from the DifferentialEquations ecosystem. The basic requirement is to load OrdinaryDiffEq.jl together with TaylorIntegration.jl, which sets-up the common interface. Below, we shall also use OrdinaryDiffEq.jl to compare the accuracy of TaylorIntegration.jl with respect to high-accuracy methods for non-stiff problems (Vern9 method). While DifferentialEquations offers many macros to simplify certain aspects, we do not rely on them simply because using properly @taylorize improves the performance.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"note: Note\nCurrently, the only keyword arguments supported by DiffEqBase.solve that are implemented in TaylorIntegration.jl are :saveat and :tstops. There is also experimental support for :callback, both discrete and continuous; some examples may be found in test/common.jl. The keyword argument :parse_eqs is available in order to control the use of methods defined via @taylorize.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"The problem we will integrate in this example is the planar circular restricted three-body problem (PCR3BP, also capitalized as PCRTBP). The PCR3BP describes the motion of a body with negligible mass m_3 under the gravitational influence of two bodies with masses m_1 and m_2, such that m_1 ge m_2. It is assumed that m_3 is much smaller than the other two masses so it does not influence their motion, and therefore it is simply considered as a massless test particle. The body with the greater mass m_1 is referred as the primary, and m_2 as the secondary. These bodies are together called the primaries and are assumed to describe Keplerian circular orbits about their center of mass, which is placed at the origin of the reference frame. It is further assumed that the orbit of the third body takes place in the orbital plane of the primaries. A full treatment of the PCR3BP may be found in [1].","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"The ratio mu = m_2(m_1+m_2) is known as the mass parameter. Using mass units such that m_1+m_2=1, we have m_1=1-mu and m_2=mu. In this example, we assume the mass parameter to have a value mu=001.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"using Plots\n\nconst μ = 0.01\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"The Hamiltonian for the PCR3BP in the synodic frame (i.e., a frame which rotates such that the primaries are at rest on the x axis) is","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"H(x y p_x p_y) = frac12(p_x^2+p_y^2) - (x p_y - y p_x) + V(x y) tag1","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"where","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"V(x y) = - frac1-musqrt(x-mu)^2+y^2 - fracmusqrt(x+1-mu)^2+y^2tag2","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"is the gravitational potential associated to the primaries. The RHS of Eq. (1) is also known as the Jacobi constant, since it is a preserved quantity of motion in the PCR3BP. We will use this property to check the accuracy of the solutions computed.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"V(x, y) = - (1-μ)/sqrt((x-μ)^2+y^2) - μ/sqrt((x+1-μ)^2+y^2)\nH(x, y, px, py) = (px^2+py^2)/2 - (x*py-y*px) + V(x, y)\nH(x) = H(x...)\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"The equations of motion for the PCR3BP are","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"beginaligned\n dotx = p_x + y\n doty = p_y - x\n dotp_x = - frac(1-mu)(x-mu)((x-mu)^2+y^2)^32 - fracmu(x+1-mu)((x+1-mu)^2+y^2)^32 + p_y\n dotp_y = - frac(1-mu)y ((x-mu)^2+y^2)^32 - fracmu y ((x+1-mu)^2+y^2)^32 - p_x\nendaligned","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We define this system of ODEs in a way that allows the use of the @taylorize macro from TaylorIntegration.jl, which for the present example allows important speed-ups. For more details about the specifics of the use of @taylorize, see this section.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"using TaylorIntegration\n@taylorize function pcr3bp!(dq, q, param, t)\n local μ = param[1]\n local onemμ = 1 - μ\n x1 = q[1]-μ\n x1sq = x1^2\n y = q[2]\n ysq = y^2\n r1_1p5 = (x1sq+ysq)^1.5\n x2 = q[1]+onemμ\n x2sq = x2^2\n r2_1p5 = (x2sq+ysq)^1.5\n dq[1] = q[3] + q[2]\n dq[2] = q[4] - q[1]\n dq[3] = (-((onemμ*x1)/r1_1p5) - ((μ*x2)/r2_1p5)) + q[4]\n dq[4] = (-((onemμ*y )/r1_1p5) - ((μ*y )/r2_1p5)) - q[3]\n return nothing\nend\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We shall define the initial conditions q_0 = (x_0 y_0 p_x0 p_y0) such that H(q_0) = J_0, where J_0 is a prescribed value. In order to do this, we select y_0 = p_x0 = 0 and compute the value of p_y0 for which H(q_0) = J_0 holds.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We consider a value for J_0 such that the test particle is able to display close encounters with both primaries, but cannot escape to infinity. We may obtain a first approximation to the desired value of J_0 if we plot the projection of the zero-velocity curves on the x-axis.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"ZVC(x) = -x^2/2 + V(x, zero(x)) # projection of the zero-velocity curves on the x-axis\n\nplot(ZVC, -2:0.001:2, label=\"zero-vel. curve\", legend=:topleft, fmt = :png)\nplot!([-2, 2], [-1.58, -1.58], label=\"J0 = -1.58\")\nylims!(-1.7, -1.45)\nxlabel!(\"x\")\nylabel!(\"J\")\ntitle!(\"Zero-velocity curves (x-axis projection)\")","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Notice that the maxima in the plot correspond to the Lagrangian points L_1, L_2 and L_3; below we shall concentrate in the value J_0 = -158.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"J0 = -1.58\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We define a function py!, which depends on the initial condition q_0 = (x_0 0 0 p_y0) and the Jacobi constant value J_0, such that it computes an adequate value p_y0 for which we have H(q_0)=J_0 and updates (in-place) the initial condition accordingly.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"function py!(q0, J0)\n @assert iszero(q0[2]) && iszero(q0[3]) # q0[2] and q0[3] have to be equal to zero\n q0[4] = q0[1] + sqrt( q0[1]^2-2( V(q0[1], q0[2])-J0 ) )\n nothing\nend\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We are now ready to generate an appropriate initial condition.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"q0 = [-0.8, 0.0, 0.0, 0.0]\npy!(q0, J0)\nq0","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We note that the value of q0 has been updated. We can check that the value of the Hamiltonian evaluated at the initial condition is indeed equal to J0.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"H(q0) == J0","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Following the DifferentialEquations.jl tutorial, we define an ODEProblem for the integration; TaylorIntegration can be used via its common interface bindings with OrdinaryDiffEq.jl; both packages need to be loaded explicitly.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"tspan = (0.0, 2000.0)\np = [μ]\n\nusing OrdinaryDiffEq\nprob = ODEProblem(pcr3bp!, q0, tspan, p)","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We solve prob using a 25-th order Taylor method, with a local absolute tolerance epsilon_mathrmtol = 10^-15.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"solT = solve(prob, TaylorMethod(25), abstol=1e-15);","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"As mentioned above, we will now solve the same problem prob with the Vern9 method from OrdinaryDiffEq, which the DifferentialEquations documentation recommends for high-accuracy (i.e., very low tolerance) integrations of non-stiff problems. Note that, besides setting an absolute tolerance abstol=1e-15, we're setting a relative tolerance reltol=1e-15 [2]. We have found that for the current problem this is a good balance between speed and accuracy for the Vern9 method, i.e., the Vern9 integration becomes noticeably slower (although more accurate) if either abstol or reltol are set to lower values.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"using OrdinaryDiffEq\n\nsolV = solve(prob, Vern9(), abstol=1e-15, reltol=1e-15); #solve `prob` with the `Vern9` method","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We plot in the x-y synodic plane the solution obtained with TaylorIntegration:","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"plot(solT, vars=(1, 2), linewidth=1, fmt = :png)\nscatter!([μ, -1+μ], [0,0], leg=false) # positions of the primaries\nxlims!(-1+μ-0.2, 1+μ+0.2)\nylims!(-0.8, 0.8)\nxlabel!(\"x\")\nylabel!(\"y\")","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Note that the orbit obtained displays the expected dynamics: the test particle explores the regions surrounding both primaries, located at the red dots, without escaping to infinity. For comparison, we now plot the orbit corresponding to the solution obtained with the Vern9() integration; note that the scales are identical.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"plot(solV, vars=(1, 2), linewidth=1, fmt = :png)\nscatter!([μ, -1+μ], [0,0], leg=false) # positions of the primaries\nxlims!(-1+μ-0.2, 1+μ+0.2)\nylims!(-0.8, 0.8)\nxlabel!(\"x\")\nylabel!(\"y\")","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We note that both orbits display the same qualitative features, and also some differences. For instance, the TaylorMethod(25) solution gets closer to the primary than that the Vern9(). We can obtain a quantitative comparison of the validity of both integrations through the preservation of the Jacobi constant:","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"ET = H.(solT.u)\nEV = H.(solV.u)\nδET = ET .- J0\nδEV = EV .- J0\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We plot first the value of the Jacobi constant as function of time.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"plot(solT.t, H.(solT.u), label=\"TaylorMethod(25)\", fmt = :png, yformatter = :plain)\nplot!(solV.t, H.(solV.u), label=\"Vern9()\")\nxlabel!(\"t\")\nylabel!(\"H\")","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"In the scale shown we observe that, while both solutions display a preservation of the Jacobi constant to a certain degree, the Vern9() solution suffers sudden jumps during the integration.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We now plot, in log scale, the abs of the absolute error in the Jacobi constant as a function of time, for both solutions:","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"plot(solT.t, abs.(δET), yscale=:log10, label=\"TaylorMethod(25)\", legend=:topleft, fmt = :png, yformatter = :plain)\nplot!(solV.t, abs.(δEV), label=\"Vern9()\")\nylims!(10^-16, 10^-10)\nxlabel!(\"t\")\nylabel!(\"dE\")","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We notice that the Jacobi constant absolute error for the TaylorMethod(25) solution remains bounded below 10^-13 throughout the integration. While the Vern9() solution at the end of the integration time has reached a similar value, it displays a larger Jacobi constant error earlier in time.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Finally, we comment on the time spent by each integration.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"using BenchmarkTools\nbT = @benchmark solve($prob, $(TaylorMethod(25)), abstol=1e-15)\nbV = @benchmark solve($prob, $(Vern9()), abstol=1e-15, reltol=1e-15)\nnothing # hide","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"bT # TaylorMethod(25) benchmark","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"bV # Vern9 benchmark","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We notice in this setup, where the TaylorMethod(25) and the Vern9() integrations perform similarly in terms of accuracy, the former performs better in terms of runtime.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"We can tune the abstol and reltol for the Vern9() method we so that performance is similar. Such situation has an accuracy cost, which then makes TaylorIntegration a sensible alternative for high-accuracy integrations of non-stiff ODEs in some cases; see [2].","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"Finally, as mentioned above, a crucial way in which TaylorIntegration provides high accuracy at competitive speeds is through the use of the @taylorize macro; see this section for details. Currently, TaylorIntegration supports the use of @taylorize via the common interface with DifferentialEquations only for in-place ODEProblem.","category":"page"},{"location":"common/#refsPCR3BP","page":"Interoperability with DifferentialEquations.jl","title":"References and notes","text":"","category":"section"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"[1] Murray, Carl D., Stanley F. Dermott. Solar System dynamics. Cambridge University Press, 1999.","category":"page"},{"location":"common/","page":"Interoperability with DifferentialEquations.jl","title":"Interoperability with DifferentialEquations.jl","text":"[2] SciMLBenchmarks.jl/DynamicalODE","category":"page"},{"location":"simple_example/#example1","page":"Infinity in finite time","title":"Infinity in finite time","text":"","category":"section"},{"location":"simple_example/#Illustration-of-the-method","page":"Infinity in finite time","title":"Illustration of the method","text":"","category":"section"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We shall illustrate first with a simple example how the method explicitly constructs the solution, and how to use the package to obtain it.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We consider the differential equation given by","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"dotx = x^2tag1","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"with the initial condition x(0)=x_0, whose exact solution reads","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"x(t) = fracx_01-x_0ttag2","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We shall implement the construction of this example explicitly, which illustrates the way TaylorIntegration.jl is conceived.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"The initial condition defines the 0-th order approximation, i.e., x(t) = x_0 + mathcalO(t^1). We now write the solution as x(t) = x_0 + x_1t + mathcalO(t^2), and we want to determine x_1. Substituting this solution into the RHS of (1), yields","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"x^2 = x_0^2 + 2 x_0 x_1 t + x_1^2 t^2 =\n x_0^2 + mathcalO(t^1)","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"where in the last equality we have kept all terms up to order 0, since we want to determine x_1, and the recursion formula requires for that the 0-th order term of the Taylor expansion of the RHS of the equation of motion. Hence, we have f_0=x_0^2, which using the recursion relation x_k+1 = f_k(k+1) yields x_1 = x_0^2.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Following the same procedure, we write x(t) = x_0 + x_1 t + x_2 t^2 + mathcalO(t^3), and","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"x^2 = x_0^2 + 2 x_0 x_1 t + mathcalO(t^2)","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"where we kept all terms up to order 1. We thus have f_1=2 x_0 x_1 = 2 x_0^3, which then yields x_2 = x_0^3. Repeating this calculation, we obtain","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"x(t) = x_0 + x_0^2 t + x_0^3 t^2 + cdots + x_0^k+1 t^k + cdotstag3","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"The solution given by Eq. (3) is a geometrical series, which is identical to the exact solution, Eq. (2). Yet, it is not obvious from the solution that it is only defined for t1x_0. To see this, we obtain the step size, as described previously, for the series truncated to order k. The Taylor coefficient of order k is x_k=x_0^k+1, so the time step is","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"h Big(fracepsilon_textrmtolx_0^k+1Big)^1k =\nfracepsilon_textrmtol^1kx_0^1+1k","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"In the limit ktoinfty we obtain h h_textrmmax=1x_0, which is the domain of existence of the exact solution.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Below, we shall fix a maximum order for the expansion. This entails a truncation error which is somewhat controlled through the absolute tolerance epsilon_textrmtol. The key to a correct use of Taylor's method is to impose a quite small value of epsilon_textrmtol together with a large enough order of the expansion.","category":"page"},{"location":"simple_example/#implementation_ex1","page":"Infinity in finite time","title":"Implementation","text":"","category":"section"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We shall illustrate how to use TaylorIntegration.jl to integrate Eq. (1) for the initial condition x(0)=3. Notice that according to the exact solution Eq. (2), the solution only exists for tt_mathrmmax =13; in addition, we note that this number can not be represented exactly as a floating-point number.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We first load the required packages and define a function which represents the equation of motion.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"using TaylorIntegration, Plots\ndiffeq(x, p, t) = x^2;","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"note: Note\nIn TaylorIntegration.jl we use the same convention of DifferentialEquations.jl when writing the function representing the equations of motion.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We now integrate the equations of motion using taylorinteg; note that, despite of the fact that the solution only exists for tt_textrmmax, below we shall try to compute it up to t_textrmend=034; as we shall see, Taylor's method takes care of this. For the integration presented below, we use a 25-th series expansion, with epsilon_textrmtol = 10^-20, and compute up to 150 integration steps. The type of the solution returned by taylorinteg is TaylorSolution.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"sol = taylorinteg(diffeq, 3.0, 0.0, 0.34, 25, 1e-20, maxsteps=150);","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"We first note that the last point of the calculation does not exceed t_textrmmax.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"sol.t[end]","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Increasing the maxsteps parameter pushes sol.t[end] closer to t_textrmmax but it actually does not reach this value.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Figure 1 displays the computed solution as a function of time, in log scale.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"plot(sol.t, log10.(sol.x), shape=:circle)\nxlabel!(\"t\")\nylabel!(\"log10(x(t))\")\nxlims!(0,0.34)\ntitle!(\"Fig. 1\")","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Clearly, the solution diverges without bound when tto t_textrmmax = 13, i.e., x(t) approaches infinity in finite time.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Figure 2 shows the relative difference between the numerical and the analytical solution in terms of time.","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"exactsol(t, x0) = x0 / (1 - x0 * t)\nδxT = abs.(sol.x .- exactsol.(sol.t, 3.0)) ./ exactsol.(sol.t, 3.0);\nplot(sol.t[6:end], log10.(sol.x[6:end]), shape=:circle)\nxlabel!(\"t\")\nylabel!(\"log10(dx(t))\")\nxlims!(0, 0.4)\ntitle!(\"Fig. 2\")","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"To put in perspective how good is the constructed solution, we impose (arbitrarily) a relative accuracy of 10^-13; the time until such accuracy is satisfied is given by:","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"indx = findfirst(δxT .> 1.0e-13);\nesol = exactsol(sol.t[indx-1],3.0);\nsol.t[indx-1], esol, eps(esol)","category":"page"},{"location":"simple_example/","page":"Infinity in finite time","title":"Infinity in finite time","text":"Note that, the accuracy imposed in terms of the actual value of the exact solution means that the difference of the computed and the exact solution is essentially due to the eps of the computed value.","category":"page"},{"location":"taylor_method/#taylormethod","page":"Taylor's method","title":"Taylor's method","text":"","category":"section"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Taylor's integration method is a quite powerful method to integrate ODEs which are smooth enough, allowing to reach a precision comparable to round-off errors per time-step. A high-order Taylor approximation of the solution (dependent variable) is constructed such that the error is quite small. A time-step is constructed which guarantees the validity of the series; this is used to sum up the Taylor expansion to obtain an approximation of the solution at a later time.","category":"page"},{"location":"taylor_method/#rec_rel","page":"Taylor's method","title":"The recurrence relation","text":"","category":"section"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Let us consider the following","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"dotx = f(t x)tag1","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"and define the initial value problem with the initial condition x(t_0) = x(0).","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"We write the solution of this equation as","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"x = x_0 + x_1 (t-t_0) + x_2 (t-t_0)^2 + cdots +\nx_k (t-t_0)^k + cdotstag2","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"where the initial condition imposes that x_0 = x(0). Below, we show how to obtain the coefficients x_k of the Taylor expansion of the solution.","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"We assume that the Taylor expansion around t_0 of f(t x(t)) is known, which we write as","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"f(t x(t)) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + cdots\n+ f_k (t-t_0)^k + cdotstag3","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Here, f_0=f(t_0x_0), and the Taylor coefficients f_k = f_k(t_0) are the k-th normalized derivatives at t_0 given by","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"f_k = frac1k fracrm d^k f rm d t^k(t_0)tag4","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Then, we are assuming that we know how to obtain f_k; these coefficients are obtained using TaylorSeries.jl.","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Substituting Eq. (2) in (1), and equating powers of t-t_0, we obtain","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"x_k+1 = fracf_k(t_0)k+1 quad k=01dotstag5","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Therefore, the coefficients of the Taylor expansion (2) are obtained recursively using Eq. (5).","category":"page"},{"location":"taylor_method/#time-step","page":"Taylor's method","title":"Time step","text":"","category":"section"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"In the computer, the expansion (2) has to be computed to a finite order. We shall denote by K the order of the series. In principle, the larger the order K, the more accurate the obtained solution is.","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"The theorem of existence and uniqueness of the solution of Eq.~(1) ensures that the Taylor expansion converges. Then, assuming that K is large enough to be within the convergent tail, we introduce the parameter epsilon_textrmtol 0 to control how large is the last term. The idea is to set this parameter to a small value, usually smaller than the machine-epsilon. Denoting by h = t_1-t_0 the time step, then imposing x_K h^K le epsilon_textrmtol we obtain","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"h le Big(fracepsilon_textrmtol x_K Big)^1Ktag6","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Equation (6) represents the maximum time-step which is consistent with epsilon_textrmtol, K and the assumption of being within the convergence tail. Notice that the arguments exposed above simply ensure that h is a maximum time-step, but any other smaller than h can be used since the series is convergent in the open interval tin(t_0-ht_0+h).","category":"page"},{"location":"taylor_method/","page":"Taylor's method","title":"Taylor's method","text":"Finally, from Eq. (2) with (6) we obtain x(t_1) = x(t_0+h), which is again an initial value problem.","category":"page"},{"location":"taylorize/#taylorize","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"","category":"section"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Here, we describe the use of the macro @taylorize, which parses the functions containing the ODEs to be integrated, allowing taylorinteg and lyap_taylorinteg to be sped up.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"warning: Warning\nThe macro @taylorize is still in an experimental state; be cautious of the resulting integration, which has to be tested carefully.","category":"page"},{"location":"taylorize/#idea","page":"Optimizing: @taylorize","title":"Some context and the idea","text":"","category":"section"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The way in which taylorinteg works by default is repeatedly calling the function where the ODEs of the problem are defined, in order to compute the recurrence relations that are used to construct the Taylor expansion of the solution. This is done for each order of the series in TaylorIntegration.jetcoeffs!. These computations are not optimized: they waste memory due to repeated allocations of some temporary arrays, and perform some operations whose result has already been previously computed.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Here we describe one way to optimize this: The idea is to replace the default method of TaylorIntegration.jetcoeffs! by another method (same function name) which is called by dispatch, and that in principle performs better. The new method is constructed specifically for the function defining the equations of motion by parsing its expression. This new method performs in principle exactly the same operations, but avoids repeating some operations and the extra allocations. To achieve the latter, the macro also creates an internal function TaylorIntegration._allocate_jetcoeffs!, which allocates all temporary Taylor1 objects as well as the declared Array{Taylor1,N}s, which are stored in a TaylorIntegration.RetAlloc struct for efficiency, and include arrays (of Taylor1{T} objects) with up-to-three indices.","category":"page"},{"location":"taylorize/#An-example","page":"Optimizing: @taylorize","title":"An example","text":"","category":"section"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"In order to explain how the macro works, we shall use as an example the mathematical pendulum. First, we carry out the integration using the default method, as described before.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"using TaylorIntegration\n\nfunction pendulumNP!(dx, x, p, t) # `pendulum!` ODEs, not parsed\n dx[1] = x[2]\n dx[2] = -sin(x[1])\n return dx\nend\n\n# Initial time (t0), final time (tf) and initial condition (q0)\nt0 = 0.0\ntf = 10000.0\nq0 = [1.3, 0.0]\n\n# The actual integration\nsol1 = taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000); # warm-up run\ne1 = @elapsed taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000);\nall1 = @allocated taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000);\ne1, all1","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The initial number of methods defined for TaylorIntegration.jetcoeffs! is 2; yet, since @taylorize was used in an example previously, the current number of methods is 3, as explained below.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"println(methods(TaylorIntegration.jetcoeffs!)) # default methods","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Similarly, the number of methods for TaylorIntegration._allocate_jetcoeffs! originally is 2, and for the same reasons it is currently 3.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"println(methods(TaylorIntegration._allocate_jetcoeffs!)) # default methods","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Using @taylorize will increase this number by creating a new method for these functions.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The macro @taylorize is intended to be used in front of the function that implements the equations of motion. The macro does the following: it first parses the function as it is, so the integration can still be computed using taylorinteg as above, by explicitly using the keyword argument parse_eqs=false; this also declares the function of the ODEs, whose name is used for parsing. It then creates and evaluates a new method of TaylorIntegration.jetcoeffs!, which is the specialized method (through Val) on the specific function passed to the macro as well as a specialized TaylorIntegration._allocate_jetcoeffs!.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"@taylorize function pendulum!(dx, x, p, t)\n dx[1] = x[2]\n dx[2] = -sin(x[1])\n return dx\nend\n\nprintln(methods(pendulum!))","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"println(methods(TaylorIntegration.jetcoeffs!)) # result should be 4","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"We see that there is only one method of pendulum!, and there is a new method (four in total) of TaylorIntegration.jetcoeffs!, whose signature appears in this documentation as Val{Main.pendulum!}. It is a specialized version for the function pendulum! (with some extra information about the module where the function was created). This method is selected internally if it exists (default), exploiting dispatch, when calling taylorinteg or lyap_taylorinteg. In order to integrate using the hard-coded standard (default) method of TaylorIntegration.jetcoeffs! of the integration above, the keyword argument parse_eqs has to be set to false. Similarly, one can check that there exists a new method of TaylorIntegration._allocate_jetcoeffs!.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Now we carry out the integration using the specialized method; note that we use the same instruction as above; the default value for the keyword argument parse_eqs is true, so we may omit it.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"sol2 = taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000); # warm-up run\ne2 = @elapsed taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000);\nall2 = @allocated taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000);\ne2, all2","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"We note the difference in the performance and allocations:","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"e1/e2, all1/all2","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"We can check that both integrations yield the same results.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"sol1.t == sol2.t && sol1.x == sol2.x","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"As stated above, in order to allow opting out of using the specialized method created by @taylorize, taylorinteg and lyap_taylorinteg recognize the keyword argument parse_eqs; setting it to false causes the standard method to be used.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false); # warm-up run\n\ne3 = @elapsed taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false);\nall3 = @allocated taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false);\ne1/e3, all1/all3","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"We now illustrate the possibility of exploiting the macro when using TaylorIntegration.jl from DifferentialEquations.jl.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"using OrdinaryDiffEq\n\nprob = ODEProblem(pendulum!, q0, (t0, tf), nothing) # no parameters\nsolT = solve(prob, TaylorMethod(25), abstol=1e-20, parse_eqs=true); # warm-up run\ne4 = @elapsed solve(prob, TaylorMethod(25), abstol=1e-20, parse_eqs=true);\n\ne1/e4","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Note that there is an additional cost to using solve in comparison with taylorinteg, but still @taylorize yields improved running times.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The speed-up obtained comes from the design of the new (specialized) method of TaylorIntegration.jetcoeffs! as described above: it avoids some allocations and some repeated computations. This is achieved by knowing the specific AST of the function of the ODEs integrated, which is walked through and translated into the actual implementation, where some required auxiliary arrays are created and reused, and the low-level functions defined in TaylorSeries.jl are used. For this, we heavily rely on Espresso.jl and some metaprogramming; we thank Andrei Zhabinski for his help and comments.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The new TaylorIntegration.jetcoeffs! and TaylorIntegration._allocate_jetcoeffs! methods can be inspected by constructing the expression corresponding to the function, and using TaylorIntegration._make_parsed_jetcoeffs:","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"ex = :(function pendulum!(dx, x, p, t)\n dx[1] = x[2]\n dx[2] = -sin(x[1])\n return dx\nend)\n\nnew_ex1, new_ex2 = TaylorIntegration._make_parsed_jetcoeffs(ex)","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The first function has a similar structure as the hard-coded method of TaylorIntegration.jetcoeffs!, but uses low-level functions in TaylorSeries (e.g., sincos! above). Temporary Taylor1 objects as well as declared arrays are allocated once by TaylorIntegration._allocate_jetcoeffs!. More complex functions quickly become very difficult to read. Note that, if necessary, one can further optimize new_ex manually.","category":"page"},{"location":"taylorize/#Limitations-and-some-advice","page":"Optimizing: @taylorize","title":"Limitations and some advice","text":"","category":"section"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"The construction of the internal function obtained by using @taylorize is somewhat complicated and limited. Here we list some limitations and provide some advice.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Expressions must involve two arguments at most, which requires using parentheses appropriately: For example, res = a+b+c should be written as res = (a+b)+c. This may lead to more parentheses used in compound expressions than would be typical outside of the @taylorize context. It also means the sequence of operations will be explicit for a compound expression rather than implicit.\nUpdating operators such as +=, *=, etc., are not supported. For example, the expression x += y is not recognized by @taylorize. Likewise, expressions such as x = x+y are not supported by @taylorize and should be replaced by equivalent expressions in which variables appear only on one side of the assignment; e.g. z = x+y; x = z. The introduction of such temporary variables z is left to the user.\nThe macro allows the use of array declarations through Array or Vector, but other ways (e.g. similar) are not yet implemented. Note that certain temporary arrays may be introduced to avoid re-computating certain expressions; only up-to-three indices expressions are currently handled.\nAvoid using variables prefixed by an underscore, in particular _T, _S, _N and __idx, as well as ord; using them may lead to name collisions with some internal variables used in the constructed expressions.\nBroadcasting is not recognized by @taylorize.\nThe macro may be used in combination with the common interface with DifferentialEquations.jl, for functions using the (du, u, p, t) in-place form, as we showed above. Other extensions allowed by DifferentialEquations may not be able to exploit it.\nif-else blocks are recognized in their long form, but short-circuit conditional operators (&& and ||) are not. When comparing to a Taylor expansion, use operators such as iszero for if-else tests rather than comparing against numeric literals.\nInput and output lengths should be determined at the time of @taylorize application (parse time), not at runtime. Avoid using the length of the input as an implicit indicator of whether to write all elements of the output. If conditional output of auxiliary equations is desired use explicit methods, such as through parameters or by setting auxiliary vector elements to zero, and assigning unwanted auxiliary outputs zero.\nExpressions which correspond to function calls (so the head field is :call) which are not recognized by the parser are simply copied. The heuristics used, especially for vectors, may not work for all cases.\nUse local for internal parameters, e.g., simple constant values; this improves performance. Do not use it if the variable is needed to be Taylor expanded during the integration step.\nTo examine the code generated for jetcoeffs! and _allocate_jetcoeffs! for a specific ODE function, follow the pendulum example above; create an expression by wrapping the ODE function (without @taylorize prefix) in a :()-block, and supply the expression to TaylorIntegration._make_parsed_jetcoeffs. This can help in debugging issues with either function generated by @taylorize.\n@taylorize supports multi-threading via Threads.@threads. WARNING: this feature is experimental. Since thread-safety depends on the definition of each ODE, we cannot guarantee the resulting code to be thread-safe in advance. The user should check the resulting code to ensure that it is indeed thread-safe. For more information about multi-threading, the reader is referred to the Julia documentation.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"We recommend to skim test/taylorize.jl, which implements different cases and highlights examples where the macro does not work, and how to solve the problem; read the information that is in the comments.","category":"page"},{"location":"taylorize/","page":"Optimizing: @taylorize","title":"Optimizing: @taylorize","text":"Please report any problems you may encounter.","category":"page"},{"location":"lyapunov_spectrum/#lyap","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"","category":"section"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"Here we describe the background of the Lyapunov spectra computations in TaylorIntegration.jl. Our implementation follows the numerical method of Benettin et al. [1], [2], which itself is based on Oseledet's multiplicative ergodic theorem [3]. Namely, simultaneously to the integration of the equations of motion, we integrate the 1st-order variational equations associated to them.","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"In general, given a dynamical system defined by the equations of motion","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"dotx = f(t x)tag1","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"along with the initial condition x(t_0) = x_0, then the first-order variational equations associated to this system are","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"dotxi = (operatornameDf)(x(t))cdot xitag2","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"where (operatornameDf)(x(t)) is the Jacobian of the function f with respect to the dependent variable x, evaluated at time t, for a given solution x(t) to the equations of motion. The variable xi denotes a matrix, whose initial condition is xi(t_0) = mathbb1_n, the ntimes n identity matrix, where n is the degrees of freedom or number of dependent variables x.","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"For the actual computation of the Lyapunov spectrum, we proceed as follows. During the simultaneous numerical integration of the equations of motion and the variational equations, at fixed time intervals t_k = kcdot Delta t, k = 1 2 ldots we perform a QR decomposition over xi(t_k), the solution of the variational equations at time t_k. That is, we factorize xi(t_k) as xi(t_k)=Q_kcdot R_k, where Q_k is an orthogonal ntimes n matrix and R_k is an upper triangular ntimes n matrix with positive diagonal elements. The diagonal elements R_iik are the growth factors from which the l-th Lyapunov exponent is computed at time t_k","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"lambda_l = sum_m=1^k fraclog (R_llm)kcdot Delta ttag3","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"In turn, the matrix Q is substituted into xi(t_k) as the new (scaled) initial condition.","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"The equations of motion together with the variational equations are integrated up to time t_k+1 using Taylor's method. We note that each time step of the integration is determined using the normalized derivatives of x and the tolerance epsilon_textrmtol. This process is repeated until a prescribed t_textrmmax is reached.","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"This example illustrates the computation of the Lyapunov spectrum for the Lorenz system.","category":"page"},{"location":"lyapunov_spectrum/#refsL","page":"Lyapunov spectrum","title":"References","text":"","category":"section"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"[1] Benettin G., Galgani L., Giorgilli A., Strelcyn J.M., 1980, Meccanica, 15, 9","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"[2] Benettin G., Galgani L., Giorgilli A., Strelcyn J.M., 1980, Meccanica, 15, 21","category":"page"},{"location":"lyapunov_spectrum/","page":"Lyapunov spectrum","title":"Lyapunov spectrum","text":"[3] Oseledets V. I., 1968, Trudy Moskovskogo Matematicheskogo Obshchestva, 19, 179","category":"page"},{"location":"#TaylorIntegration.jl","page":"Home","title":"TaylorIntegration.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"ODE integration using Taylor's method, and more, in Julia.","category":"page"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"#Authors","page":"Home","title":"Authors","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Jorge A. Pérez, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)\nLuis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)","category":"page"},{"location":"#License","page":"Home","title":"License","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorIntegration is licensed under the MIT \"Expat\" license.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorIntegration.jl is a registered package, and is simply installed by running","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add TaylorIntegration","category":"page"},{"location":"#Supporting-and-citing","page":"Home","title":"Supporting and citing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package is developed as part of academic research. If you would like to help supporting it, please star the repository as such metrics may help us secure funding. If you use this software, we would be grateful if you could cite our work as follows (Bibtex entry can be found here):","category":"page"},{"location":"","page":"Home","title":"Home","text":"J.A. Pérez-Hernández and L. Benet\nTaylorIntegration.jl: Taylor Integration in Julia\nhttps://github.com/PerezHz/TaylorIntegration.jl\nDOI:[10.5281/zenodo.2562352](https://doi.org/10.5281/zenodo.2562352)","category":"page"},{"location":"#Acknowledgments","page":"Home","title":"Acknowledgments","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"We acknowledge financial support from the DGAPA-PAPIIT (UNAM) grant IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).","category":"page"},{"location":"kepler/#kepler_problem","page":"The Kepler problem","title":"The Kepler problem","text":"","category":"section"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"The Kepler problem corresponds to the study of the motion of two bodies which are influenced by their mutual gravitational attraction. In the center of mass and relative coordinates, the problem is reduced to the motion of one body of mass m = m_1 m_2 M, which we shall refer as particle below, attracted gravitationally by another located at rest at the origin of mass M=m_1+m_2.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"In cartesian coordinates, the equations of motion can be written as","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"beginaligned\ndotx = v_x\ndoty = v_y\ndotv_x = - fracG M x(x^2 + y^2)^32\ndotv_y = - fracG M y(x^2 + y^2)^32\nendaligned","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"For concreteness, we fix mu = G M = 1. The coordinates x and y are the relative coordinates (to the center of mass) of the particle, and v_x and v_y its velocity. The function kepler_eqs! mutates the vectors corresponding to the LHS of the equations of motion.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"function kepler_eqs!(dq, q, params, t)\n dq[1] = q[3]\n dq[2] = q[4]\n rr = ( q[1]^2 + q[2]^2 )^(3/2)\n dq[3] = - q[1] / rr\n dq[4] = - q[2] / rr\nend;","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"For suitable initial conditions (such that the total energy is negative) the solutions are ellipses with one focus at the origin, which can be parameterized in terms of its semi-major axis a and its eccentricity e. We set the initial conditions for the particle at periapse, which we locate on the positive x-axis. Using the semimajor axis and the eccentricity, we write them as","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"beginaligned\nx_0 = a (1-e)\ny_0 = 0\nv_x_0 = 0\nv_y_0 = fracl_zx_0 = m fracsqrtmu a (1-e^2)x_0\nendaligned","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"where l_z is the angular momentum. We set the mass of the particle m=1, the semi-major axis a=1 and the eccentricity e=08. Kepler's third law defines the period of the motion as T= 2pi a^32.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"const mu = 1.0\nconst mass = 1.0\nconst aKep = 1.0\nconst eKep = 0.8;","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"The initial conditions are then set using ini_cond","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"function ini_cond(a, e)\n x0 = a*(one(e)-e)\n vy0 = mass * sqrt( mu * a * (1-e^2) ) / x0\n y0 = zero(vy0)\n vx0 = zero(vy0)\n return [x0, y0, vx0, vy0]\nend\nq0 = ini_cond(aKep, eKep)","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"We now perform the integration, using a 25 order expansion and absolute tolerance of 10^-20.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"using TaylorIntegration, Plots\nsol = taylorinteg(kepler_eqs!, q0, 0.0, 10000*2pi, 25, 1.0e-20, maxsteps=700_000);\nt, q = sol.t, sol.x;\nt[end], q[end,:]","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"We first plot the orbit. (For performance reasons only the first 10000 points are considered.)","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"x = view(q, :, 1)\ny = view(q, :, 2)\nvx = view(q, :, 3)\nvy = view(q, :, 4)\nplot(x[1:10_000], y[1:10_000], legend=false)\nscatter!([0], [0], shape=:circle, ms=5)\nxaxis!(\"x\", (-2.0, 0.5))\nyaxis!(\"y\", (-1.0, 1.0))\ntitle!(\"Fig. 1\")","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"The following functions allow us to calculate the energy and angular momentum using cartesian coordinates.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"function energy( x, y, vx, vy )\n kinetic = 0.5 * (vx*vx + vy*vy)\n r = sqrt( x*x + y*y)\n potential = - mu * mass / r\n return kinetic + potential\nend\nlz( x, y, vx, vy ) = mass * ( x*vy - y*vx ) ;","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"We use the change in energy and angular momentum of the orbit with respect to the initial value of the corresponding quantity as a function of time. These quantities are expressed in units of the local epsilon of the initial energy or angular momentum, respectively. This serves to illustrate the accuracy of the calculation, shown in Figure 2 and 3.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"e0 = energy(q0...)\nδE = (energy.(x,y,vx,vy) .- e0) ./ eps(e0)\nplot(t[1:3:end], δE[1:3:end])\nxlabel!(\"t\")\nylabel!(\"dE\")\ntitle!(\"Fig. 2\")","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"lz0 = lz(q0...)\nδlz = (lz.(x,y,vx,vy) .- lz0) ./ eps(lz0)\nplot(t[1:3:end], δlz[1:3:end])\nxlabel!(\"t\")\nylabel!(\"dlz\")\ntitle!(\"Fig. 3\")","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"These errors are reminiscent of random walks.","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"The maximum absolute errors of the energy and angular momentum are","category":"page"},{"location":"kepler/","page":"The Kepler problem","title":"The Kepler problem","text":"maximum( abs.(energy.(x,y,vx,vy) .- e0) ), maximum( abs.(lz.(x,y,vx,vy) .- lz0) )","category":"page"}] +} diff --git a/v0.16.1/simple_example/47ad181b.svg b/v0.16.1/simple_example/47ad181b.svg new file mode 100644 index 000000000..dd6644793 --- /dev/null +++ b/v0.16.1/simple_example/47ad181b.svg @@ -0,0 +1,197 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/simple_example/529b4583.svg b/v0.16.1/simple_example/529b4583.svg new file mode 100644 index 000000000..5bb8de2f8 --- /dev/null +++ b/v0.16.1/simple_example/529b4583.svg @@ -0,0 +1,200 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.16.1/simple_example/index.html b/v0.16.1/simple_example/index.html new file mode 100644 index 000000000..b593600b5 --- /dev/null +++ b/v0.16.1/simple_example/index.html @@ -0,0 +1,17 @@ + +Infinity in finite time · TaylorIntegration.jl
    GitHub

    Infinity in finite time

    Illustration of the method

    We shall illustrate first with a simple example how the method explicitly constructs the solution, and how to use the package to obtain it.

    We consider the differential equation given by

    \[\dot{x} = x^2,\tag{1}\]

    with the initial condition $x(0)=x_0$, whose exact solution reads

    \[x(t) = \frac{x_0}{1-x_0t}.\tag{2}\]

    We shall implement the construction of this example explicitly, which illustrates the way TaylorIntegration.jl is conceived.

    The initial condition defines the 0-th order approximation, i.e., $x(t) = x_0 + \mathcal{O}(t^1)$. We now write the solution as $x(t) = x_0 + x_{[1]}t + \mathcal{O}(t^2)$, and we want to determine $x_{[1]}$. Substituting this solution into the RHS of (1), yields

    \[x^2 = x_0^2 + 2 x_0 x_{[1]} t + x_{[1]}^2 t^2 = + x_0^2 + \mathcal{O}(t^1),\]

    where in the last equality we have kept all terms up to order 0, since we want to determine $x_{[1]}$, and the recursion formula requires for that the 0-th order term of the Taylor expansion of the RHS of the equation of motion. Hence, we have $f_{[0]}=x_0^2$, which using the recursion relation $x_{[k+1]} = f_{[k]}/(k+1)$ yields $x_{[1]} = x_0^2$.

    Following the same procedure, we write $x(t) = x_0 + x_{[1]} t + x_{[2]} t^2 + \mathcal{O}(t^3)$, and

    \[x^2 = x_0^2 + 2 x_0 x_{[1]} t + \mathcal{O}(t^2),\]

    where we kept all terms up to order 1. We thus have $f_{[1]}=2 x_0 x_{[1]} = 2 x_0^3$, which then yields $x_{[2]} = x_0^3$. Repeating this calculation, we obtain

    \[x(t) = x_0 + x_0^2 t + x_0^3 t^2 + \cdots + x_0^{k+1} t^k + \cdots.\tag{3}\]

    The solution given by Eq. (3) is a geometrical series, which is identical to the exact solution, Eq. (2). Yet, it is not obvious from the solution that it is only defined for $t<1/x_0$. To see this, we obtain the step size, as described previously, for the series truncated to order $k$. The Taylor coefficient of order $k$ is $x_{[k]}=x_0^{k+1}$, so the time step is

    \[h < \Big(\frac{\epsilon_\textrm{tol}}{|x_0^{k+1}|}\Big)^{1/k} = +\frac{\epsilon_\textrm{tol}^{1/k}}{|x_0|^{1+1/k}}.\]

    In the limit $k\to\infty$ we obtain $h < h_\textrm{max}=1/x_0$, which is the domain of existence of the exact solution.

    Below, we shall fix a maximum order for the expansion. This entails a truncation error which is somewhat controlled through the absolute tolerance $\epsilon_\textrm{tol}$. The key to a correct use of Taylor's method is to impose a quite small value of $\epsilon_\textrm{tol}$ together with a large enough order of the expansion.

    Implementation

    We shall illustrate how to use TaylorIntegration.jl to integrate Eq. (1) for the initial condition $x(0)=3$. Notice that according to the exact solution Eq. (2), the solution only exists for $t<t_\mathrm{max} =1/3$; in addition, we note that this number can not be represented exactly as a floating-point number.

    We first load the required packages and define a function which represents the equation of motion.

    using TaylorIntegration, Plots
+diffeq(x, p, t) = x^2;
    diffeq (generic function with 1 method)
    Note

    In TaylorIntegration.jl we use the same convention of DifferentialEquations.jl when writing the function representing the equations of motion.

    We now integrate the equations of motion using taylorinteg; note that, despite of the fact that the solution only exists for $t<t_\textrm{max}$, below we shall try to compute it up to $t_\textrm{end}=0.34$; as we shall see, Taylor's method takes care of this. For the integration presented below, we use a 25-th series expansion, with $\epsilon_\textrm{tol} = 10^{-20}$, and compute up to 150 integration steps. The type of the solution returned by taylorinteg is TaylorSolution.

    sol = taylorinteg(diffeq, 3.0, 0.0, 0.34, 25, 1e-20, maxsteps=150);
    tspan: (0.0, 0.33333328367608234), x: 1 Float64 variable

    We first note that the last point of the calculation does not exceed $t_\textrm{max}$.

    sol.t[end]
    0.33333328367608234

    Increasing the maxsteps parameter pushes sol.t[end] closer to $t_\textrm{max}$ but it actually does not reach this value.

    Figure 1 displays the computed solution as a function of time, in log scale.

    plot(sol.t, log10.(sol.x), shape=:circle)
+xlabel!("t")
+ylabel!("log10(x(t))")
+xlims!(0,0.34)
+title!("Fig. 1")
    Example block output

    Clearly, the solution diverges without bound when $t\to t_\textrm{max} = 1/3$, i.e., $x(t)$ approaches infinity in finite time.

    Figure 2 shows the relative difference between the numerical and the analytical solution in terms of time.

    exactsol(t, x0) = x0 / (1 - x0 * t)
+δxT = abs.(sol.x .- exactsol.(sol.t, 3.0)) ./ exactsol.(sol.t, 3.0);
+plot(sol.t[6:end], log10.(sol.x[6:end]), shape=:circle)
+xlabel!("t")
+ylabel!("log10(dx(t))")
+xlims!(0, 0.4)
+title!("Fig. 2")
    Example block output

    To put in perspective how good is the constructed solution, we impose (arbitrarily) a relative accuracy of $10^{-13}$; the time until such accuracy is satisfied is given by:

    indx = findfirst(δxT .> 1.0e-13);
+esol = exactsol(sol.t[indx-1],3.0);
+sol.t[indx-1], esol, eps(esol)
    (0.3323590833381362, 1026.4305926915836, 2.2737367544323206e-13)

    Note that, the accuracy imposed in terms of the actual value of the exact solution means that the difference of the computed and the exact solution is essentially due to the eps of the computed value.

    diff --git a/v0.16.1/siteinfo.js b/v0.16.1/siteinfo.js new file mode 100644 index 000000000..afd23cc7e --- /dev/null +++ b/v0.16.1/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.16.1"; diff --git a/v0.16.1/taylor_method/index.html b/v0.16.1/taylor_method/index.html new file mode 100644 index 000000000..eb842294c --- /dev/null +++ b/v0.16.1/taylor_method/index.html @@ -0,0 +1,4 @@ + +Taylor's method · TaylorIntegration.jl
    GitHub

    Taylor's method

    Taylor's integration method is a quite powerful method to integrate ODEs which are smooth enough, allowing to reach a precision comparable to round-off errors per time-step. A high-order Taylor approximation of the solution (dependent variable) is constructed such that the error is quite small. A time-step is constructed which guarantees the validity of the series; this is used to sum up the Taylor expansion to obtain an approximation of the solution at a later time.

    The recurrence relation

    Let us consider the following

    \[\dot{x} = f(t, x),\tag{1}\]

    and define the initial value problem with the initial condition $x(t_0) = x(0)$.

    We write the solution of this equation as

    \[x = x_{[0]} + x_{[1]} (t-t_0) + x_{[2]} (t-t_0)^2 + \cdots + +x_{[k]} (t-t_0)^k + \cdots,\tag{2}\]

    where the initial condition imposes that $x_{[0]} = x(0)$. Below, we show how to obtain the coefficients $x_{[k]}$ of the Taylor expansion of the solution.

    We assume that the Taylor expansion around $t_0$ of $f(t, x(t))$ is known, which we write as

    \[f(t, x(t)) = f_{[0]} + f_{[1]} (t-t_0) + f_{[2]} (t-t_0)^2 + \cdots ++ f_{[k]} (t-t_0)^k + \cdots.\tag{3}\]

    Here, $f_{[0]}=f(t_0,x_0)$, and the Taylor coefficients $f_{[k]} = f_{[k]}(t_0)$ are the $k$-th normalized derivatives at $t_0$ given by

    \[f_{[k]} = \frac{1}{k!} \frac{{\rm d}^k f} {{\rm d} t^k}(t_0).\tag{4}\]

    Then, we are assuming that we know how to obtain $f_{[k]}$; these coefficients are obtained using TaylorSeries.jl.

    Substituting Eq. (2) in (1), and equating powers of $t-t_0$, we obtain

    \[x_{[k+1]} = \frac{f_{[k]}(t_0)}{k+1}, \quad k=0,1,\dots.\tag{5}\]

    Therefore, the coefficients of the Taylor expansion (2) are obtained recursively using Eq. (5).

    Time step

    In the computer, the expansion (2) has to be computed to a finite order. We shall denote by $K$ the order of the series. In principle, the larger the order $K$, the more accurate the obtained solution is.

    The theorem of existence and uniqueness of the solution of Eq.~(1) ensures that the Taylor expansion converges. Then, assuming that $K$ is large enough to be within the convergent tail, we introduce the parameter $\epsilon_\textrm{tol} > 0$ to control how large is the last term. The idea is to set this parameter to a small value, usually smaller than the machine-epsilon. Denoting by $h = t_1-t_0$ the time step, then imposing $| x_{[K]} | h^K \le \epsilon_\textrm{tol}$ we obtain

    \[h \le \Big(\frac{\epsilon_\textrm{tol}}{| x_{[K]} |}\Big)^{1/K}.\tag{6}\]

    Equation (6) represents the maximum time-step which is consistent with $\epsilon_\textrm{tol}$, $K$ and the assumption of being within the convergence tail. Notice that the arguments exposed above simply ensure that $h$ is a maximum time-step, but any other smaller than $h$ can be used since the series is convergent in the open interval $t\in(t_0-h,t_0+h)$.

    Finally, from Eq. (2) with (6) we obtain $x(t_1) = x(t_0+h)$, which is again an initial value problem.

    diff --git a/v0.16.1/taylorize/index.html b/v0.16.1/taylorize/index.html new file mode 100644 index 000000000..360d57692 --- /dev/null +++ b/v0.16.1/taylorize/index.html @@ -0,0 +1,87 @@ + +Optimizing: @taylorize · TaylorIntegration.jl
    GitHub

    Optimizing: @taylorize

    Here, we describe the use of the macro @taylorize, which parses the functions containing the ODEs to be integrated, allowing taylorinteg and lyap_taylorinteg to be sped up.

    Warning

    The macro @taylorize is still in an experimental state; be cautious of the resulting integration, which has to be tested carefully.

    Some context and the idea

    The way in which taylorinteg works by default is repeatedly calling the function where the ODEs of the problem are defined, in order to compute the recurrence relations that are used to construct the Taylor expansion of the solution. This is done for each order of the series in TaylorIntegration.jetcoeffs!. These computations are not optimized: they waste memory due to repeated allocations of some temporary arrays, and perform some operations whose result has already been previously computed.

    Here we describe one way to optimize this: The idea is to replace the default method of TaylorIntegration.jetcoeffs! by another method (same function name) which is called by dispatch, and that in principle performs better. The new method is constructed specifically for the function defining the equations of motion by parsing its expression. This new method performs in principle exactly the same operations, but avoids repeating some operations and the extra allocations. To achieve the latter, the macro also creates an internal function TaylorIntegration._allocate_jetcoeffs!, which allocates all temporary Taylor1 objects as well as the declared Array{Taylor1,N}s, which are stored in a TaylorIntegration.RetAlloc struct for efficiency, and include arrays (of Taylor1{T} objects) with up-to-three indices.

    An example

    In order to explain how the macro works, we shall use as an example the mathematical pendulum. First, we carry out the integration using the default method, as described before.

    using TaylorIntegration
+
+function pendulumNP!(dx, x, p, t) # `pendulum!` ODEs, not parsed
+    dx[1] = x[2]
+    dx[2] = -sin(x[1])
+    return dx
+end
+
+# Initial time (t0), final time (tf) and initial condition (q0)
+t0 = 0.0
+tf = 10000.0
+q0 = [1.3, 0.0]
+
+# The actual integration
+sol1 = taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000); # warm-up run
+e1 = @elapsed taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000);
+all1 = @allocated taylorinteg(pendulumNP!, q0, t0, tf, 25, 1e-20, maxsteps=50000);
+e1, all1
    (1.058779165, 812244848)

    The initial number of methods defined for TaylorIntegration.jetcoeffs! is 2; yet, since @taylorize was used in an example previously, the current number of methods is 3, as explained below.

    println(methods(TaylorIntegration.jetcoeffs!)) # default methods
    # 3 methods for generic function "jetcoeffs!" from TaylorIntegration:
+ [1] jetcoeffs!(::Val{Main.__atexample__named__common.pcr3bp!}, t::Taylor1{_T}, q::AbstractArray{Taylor1{_S}, _N}, dq::AbstractArray{Taylor1{_S}, _N}, param, __ralloc::TaylorIntegration.RetAlloc{Taylor1{_S}}) where {_T<:Real, _S<:Number, _N}
+     @ Main.__atexample__named__common none:0
+ [2] jetcoeffs!(eqsdiff!::Function, t::Taylor1{T}, x::AbstractArray{Taylor1{U}, N}, dx::AbstractArray{Taylor1{U}, N}, xaux::AbstractArray{Taylor1{U}, N}, params) where {T<:Real, U<:Number, N}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:62
+ [3] jetcoeffs!(eqsdiff::Function, t::Taylor1{T}, x::Taylor1{U}, params) where {T<:Real, U<:Number}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:23

    Similarly, the number of methods for TaylorIntegration._allocate_jetcoeffs! originally is 2, and for the same reasons it is currently 3.

    println(methods(TaylorIntegration._allocate_jetcoeffs!)) # default methods
    # 3 methods for generic function "_allocate_jetcoeffs!" from TaylorIntegration:
+ [1] _allocate_jetcoeffs!(::Val{Main.__atexample__named__common.pcr3bp!}, t::Taylor1{_T}, q::AbstractArray{Taylor1{_S}, _N}, dq::AbstractArray{Taylor1{_S}, _N}, param) where {_T<:Real, _S<:Number, _N}
+     @ Main.__atexample__named__common none:0
+ [2] _allocate_jetcoeffs!(::Taylor1{T}, x::AbstractArray{Taylor1{S}, N}, ::AbstractArray{Taylor1{S}, N}, params) where {T, S, N}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:109
+ [3] _allocate_jetcoeffs!(::Taylor1{T}, x::Taylor1{S}, params) where {T, S}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:108

    Using @taylorize will increase this number by creating a new method for these functions.

    The macro @taylorize is intended to be used in front of the function that implements the equations of motion. The macro does the following: it first parses the function as it is, so the integration can still be computed using taylorinteg as above, by explicitly using the keyword argument parse_eqs=false; this also declares the function of the ODEs, whose name is used for parsing. It then creates and evaluates a new method of TaylorIntegration.jetcoeffs!, which is the specialized method (through Val) on the specific function passed to the macro as well as a specialized TaylorIntegration._allocate_jetcoeffs!.

    @taylorize function pendulum!(dx, x, p, t)
+    dx[1] = x[2]
+    dx[2] = -sin(x[1])
+    return dx
+end
+
+println(methods(pendulum!))
    # 1 method for generic function "pendulum!" from Main:
+ [1] pendulum!(dx, x, p, t)
+     @ none:0
    println(methods(TaylorIntegration.jetcoeffs!)) # result should be 4
    # 4 methods for generic function "jetcoeffs!" from TaylorIntegration:
+ [1] jetcoeffs!(::Val{Main.pendulum!}, t::Taylor1{_T}, x::AbstractArray{Taylor1{_S}, _N}, dx::AbstractArray{Taylor1{_S}, _N}, p, __ralloc::TaylorIntegration.RetAlloc{Taylor1{_S}}) where {_T<:Real, _S<:Number, _N}
+     @ Main none:0
+ [2] jetcoeffs!(::Val{Main.__atexample__named__common.pcr3bp!}, t::Taylor1{_T}, q::AbstractArray{Taylor1{_S}, _N}, dq::AbstractArray{Taylor1{_S}, _N}, param, __ralloc::TaylorIntegration.RetAlloc{Taylor1{_S}}) where {_T<:Real, _S<:Number, _N}
+     @ Main.__atexample__named__common none:0
+ [3] jetcoeffs!(eqsdiff!::Function, t::Taylor1{T}, x::AbstractArray{Taylor1{U}, N}, dx::AbstractArray{Taylor1{U}, N}, xaux::AbstractArray{Taylor1{U}, N}, params) where {T<:Real, U<:Number, N}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:62
+ [4] jetcoeffs!(eqsdiff::Function, t::Taylor1{T}, x::Taylor1{U}, params) where {T<:Real, U<:Number}
+     @ ~/work/TaylorIntegration.jl/TaylorIntegration.jl/src/integrator/jetcoeffs.jl:23

    We see that there is only one method of pendulum!, and there is a new method (four in total) of TaylorIntegration.jetcoeffs!, whose signature appears in this documentation as Val{Main.pendulum!}. It is a specialized version for the function pendulum! (with some extra information about the module where the function was created). This method is selected internally if it exists (default), exploiting dispatch, when calling taylorinteg or lyap_taylorinteg. In order to integrate using the hard-coded standard (default) method of TaylorIntegration.jetcoeffs! of the integration above, the keyword argument parse_eqs has to be set to false. Similarly, one can check that there exists a new method of TaylorIntegration._allocate_jetcoeffs!.

    Now we carry out the integration using the specialized method; note that we use the same instruction as above; the default value for the keyword argument parse_eqs is true, so we may omit it.

    sol2 = taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000); # warm-up run
+e2 = @elapsed taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000);
+all2 = @allocated taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000);
+e2, all2
    (0.117495775, 51763216)

    We note the difference in the performance and allocations:

    e1/e2, all1/all2
    (9.01121053076164, 15.69154528574886)

    We can check that both integrations yield the same results.

    sol1.t == sol2.t && sol1.x == sol2.x
    true

    As stated above, in order to allow opting out of using the specialized method created by @taylorize, taylorinteg and lyap_taylorinteg recognize the keyword argument parse_eqs; setting it to false causes the standard method to be used.

    taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false); # warm-up run
+
+e3 = @elapsed taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false);
+all3 = @allocated taylorinteg(pendulum!, q0, t0, tf, 25, 1e-20, maxsteps=50000, parse_eqs=false);
+e1/e3, all1/all3
    (1.0013556719726204, 1.0000024229205924)

    We now illustrate the possibility of exploiting the macro when using TaylorIntegration.jl from DifferentialEquations.jl.

    using OrdinaryDiffEq
+
+prob = ODEProblem(pendulum!, q0, (t0, tf), nothing) # no parameters
+solT = solve(prob, TaylorMethod(25), abstol=1e-20, parse_eqs=true); # warm-up run
+e4 = @elapsed solve(prob, TaylorMethod(25), abstol=1e-20, parse_eqs=true);
+
+e1/e4
    16.428707409855935

    Note that there is an additional cost to using solve in comparison with taylorinteg, but still @taylorize yields improved running times.

    The speed-up obtained comes from the design of the new (specialized) method of TaylorIntegration.jetcoeffs! as described above: it avoids some allocations and some repeated computations. This is achieved by knowing the specific AST of the function of the ODEs integrated, which is walked through and translated into the actual implementation, where some required auxiliary arrays are created and reused, and the low-level functions defined in TaylorSeries.jl are used. For this, we heavily rely on Espresso.jl and some metaprogramming; we thank Andrei Zhabinski for his help and comments.

    The new TaylorIntegration.jetcoeffs! and TaylorIntegration._allocate_jetcoeffs! methods can be inspected by constructing the expression corresponding to the function, and using TaylorIntegration._make_parsed_jetcoeffs:

    ex = :(function pendulum!(dx, x, p, t)
+    dx[1] = x[2]
+    dx[2] = -sin(x[1])
+    return dx
+end)
+
+new_ex1, new_ex2 = TaylorIntegration._make_parsed_jetcoeffs(ex)
    (:(function TaylorIntegration.jetcoeffs!(::Val{pendulum!}, t::Taylor1{_T}, x::AbstractArray{Taylor1{_S}, _N}, dx::AbstractArray{Taylor1{_S}, _N}, p, __ralloc::TaylorIntegration.RetAlloc{Taylor1{_S}}) where {_T <: Real, _S <: Number, _N}
+      order = t.order
+      tmp381 = __ralloc.v0[1]
+      tmp383 = __ralloc.v0[2]
+      for ord = 0:order - 1
+          ordnext = ord + 1
+          TaylorSeries.identity!(dx[1], x[2], ord)
+          TaylorSeries.sincos!(tmp381, tmp383, x[1], ord)
+          TaylorSeries.subst!(dx[2], tmp381, ord)
+          for __idx = eachindex(x)
+              TaylorIntegration.solcoeff!(x[__idx], dx[__idx], ordnext)
+          end
+      end
+      return nothing
+  end), :(function TaylorIntegration._allocate_jetcoeffs!(::Val{pendulum!}, t::Taylor1{_T}, x::AbstractArray{Taylor1{_S}, _N}, dx::AbstractArray{Taylor1{_S}, _N}, p) where {_T <: Real, _S <: Number, _N}
+      order = t.order
+      dx[1] = Taylor1(identity(constant_term(x[2])), order)
+      tmp381 = Taylor1(sin(constant_term(x[1])), order)
+      tmp383 = Taylor1(cos(constant_term(x[1])), order)
+      dx[2] = Taylor1(-(constant_term(tmp381)), order)
+      return TaylorIntegration.RetAlloc{Taylor1{_S}}([tmp381, tmp383], [Array{Taylor1{_S}, 1}(undef, 0)], [Array{Taylor1{_S}, 2}(undef, 0, 0)], [Array{Taylor1{_S}, 3}(undef, 0, 0, 0)], [Array{Taylor1{_S}, 4}(undef, 0, 0, 0, 0)])
+  end))

    The first function has a similar structure as the hard-coded method of TaylorIntegration.jetcoeffs!, but uses low-level functions in TaylorSeries (e.g., sincos! above). Temporary Taylor1 objects as well as declared arrays are allocated once by TaylorIntegration._allocate_jetcoeffs!. More complex functions quickly become very difficult to read. Note that, if necessary, one can further optimize new_ex manually.

    Limitations and some advice

    The construction of the internal function obtained by using @taylorize is somewhat complicated and limited. Here we list some limitations and provide some advice.

    • Expressions must involve two arguments at most, which requires using parentheses appropriately: For example, res = a+b+c should be written as res = (a+b)+c. This may lead to more parentheses used in compound expressions than would be typical outside of the @taylorize context. It also means the sequence of operations will be explicit for a compound expression rather than implicit.

    • Updating operators such as +=, *=, etc., are not supported. For example, the expression x += y is not recognized by @taylorize. Likewise, expressions such as x = x+y are not supported by @taylorize and should be replaced by equivalent expressions in which variables appear only on one side of the assignment; e.g. z = x+y; x = z. The introduction of such temporary variables z is left to the user.

    • The macro allows the use of array declarations through Array or Vector, but other ways (e.g. similar) are not yet implemented. Note that certain temporary arrays may be introduced to avoid re-computating certain expressions; only up-to-three indices expressions are currently handled.

    • Avoid using variables prefixed by an underscore, in particular _T, _S, _N and __idx, as well as ord; using them may lead to name collisions with some internal variables used in the constructed expressions.

    • Broadcasting is not recognized by @taylorize.

    • The macro may be used in combination with the common interface with DifferentialEquations.jl, for functions using the (du, u, p, t) in-place form, as we showed above. Other extensions allowed by DifferentialEquations may not be able to exploit it.

    • if-else blocks are recognized in their long form, but short-circuit conditional operators (&& and ||) are not. When comparing to a Taylor expansion, use operators such as iszero for if-else tests rather than comparing against numeric literals.

    • Input and output lengths should be determined at the time of @taylorize application (parse time), not at runtime. Avoid using the length of the input as an implicit indicator of whether to write all elements of the output. If conditional output of auxiliary equations is desired use explicit methods, such as through parameters or by setting auxiliary vector elements to zero, and assigning unwanted auxiliary outputs zero.

    • Expressions which correspond to function calls (so the head field is :call) which are not recognized by the parser are simply copied. The heuristics used, especially for vectors, may not work for all cases.

    • Use local for internal parameters, e.g., simple constant values; this improves performance. Do not use it if the variable is needed to be Taylor expanded during the integration step.

    • To examine the code generated for jetcoeffs! and _allocate_jetcoeffs! for a specific ODE function, follow the pendulum example above; create an expression by wrapping the ODE function (without @taylorize prefix) in a :()-block, and supply the expression to TaylorIntegration._make_parsed_jetcoeffs. This can help in debugging issues with either function generated by @taylorize.

    • @taylorize supports multi-threading via Threads.@threads. WARNING: this feature is experimental. Since thread-safety depends on the definition of each ODE, we cannot guarantee the resulting code to be thread-safe in advance. The user should check the resulting code to ensure that it is indeed thread-safe. For more information about multi-threading, the reader is referred to the Julia documentation.

    We recommend to skim test/taylorize.jl, which implements different cases and highlights examples where the macro does not work, and how to solve the problem; read the information that is in the comments.

    Please report any problems you may encounter.

    diff --git a/versions.js b/versions.js index 69f51fc08..dc79edf5a 100644 --- a/versions.js +++ b/versions.js @@ -15,5 +15,5 @@ var DOC_VERSIONS = [ "v0.4", "dev", ]; -var DOCUMENTER_NEWEST = "v0.16.0"; +var DOCUMENTER_NEWEST = "v0.16.1"; var DOCUMENTER_STABLE = "stable";