also update index.md

docs/src/index.md

@@ -2,19 +2,19 @@
 
 *A full-featured Julia interface to the Generalised Truncated Power Series Algebra library.*
 
-This package provides a full-featured Julia interface to the [Generalised Truncated Power Series Algebra (GTPSA) library](https://mad.web.cern.ch/mad/releases/madng/html/mad_mod_diffalg.html), which computes Taylor expansions, or Truncated Power Series (TPSs) of real and complex multivariable functions to arbitrary orders in the variables. 
+This package provides a full-featured Julia interface to the [Generalised Truncated Power Series Algebra (GTPSA) library](https://mad.web.cern.ch/mad/releases/madng/html/mad_mod_diffalg.html), which computes Taylor expansions, or Truncated Power Series (TPSs) of real and complex multivariable functions to arbitrary orders. 
 
-Truncated Power Series Algebra (TPSA) performs forward-mode automatic differentation (AD) similar to the dual-number implementation of `ForwardDiff.jl`. However, instead of nesting derivatives for higher orders, TPSA naturally extends to arbitary orders by directly using the power series expansions. This, paired with a highly optimized monomial indexing function/storage for propagating the partial derivatives, gives `GTPSA.jl` similar performance as `ForwardDiff.jl` to 1st-order, but significantly faster performance for 2nd-order calculations and above.
+Truncated Power Series Algebra (TPSA) performs forward-mode automatic differentation (AD) similar to the dual-number implementation of `ForwardDiff.jl`. However, instead of nesting derivatives for higher orders, TPSA naturally extends to arbitary orders by directly using the power series expansions. This, paired with a highly optimized monomial indexing function/storage for propagating the partial derivatives, makes `GTPSA.jl` significantly faster for 2nd-order calculations and above (for 1st-order
+calculations the performance is similar to `ForwardDiff.jl`).
+See the [`benchmark/track.jl`](https://github.com/bmad-sim/GTPSA.jl/blob/main/benchmark/track.jl) example for a speed comparison of `GTPSA.jl` with `ForwardDiff.jl` in calculating the partial derivatives for a system with 58 inputs and 6 outputs. **In this example, GTPSA was nearly x3 faster than ForwardDiff to 2nd order, and x15 faster to 3rd order.**
 
 GTPSA provides several advantages over current Julia AD packages:
 
 1. **Speed**: `GTPSA.jl` is significantly faster than `ForwardDiff.jl` for 2nd-order calculations and above, and has similar performance at 1st-order
-2. **Custom Orders in Individual Variables**: For example, computing the Taylor expansion of ``f(x_1,x_2)`` to 2nd order in ``x_1`` and 6th order in ``x_2`` is done trivially in GTPSA
-3. **Complex Numbers**: GTPSA natively supports complex numbers, and allows for mixing of complex and real truncated power series
-4. **Distinction Between State Variables and Parameters**: Distinguishing between dependent variables and parameters in the solution of a differential equation expressed as a power series in the dependent variables/parameters is very advantageous in analysis
-5. **Fast JIT Compilation**: Because most of the "heavy-lifting" is done in the precompiled C library, the JIT compilation for `GTPSA.jl` is fast, easing iterative REPL development
-
-See the [`benchmark/track.jl`](https://github.com/bmad-sim/GTPSA.jl/blob/main/benchmark/track.jl) example for a speed comparison of `GTPSA.jl` with `ForwardDiff.jl` in calculating the partial derivatives for a system with 58 inputs and 6 outputs. **We observed GTPSA to be nearly x3 faster than ForwardDiff to 2nd order, and x15 faster to 3rd order in our example.**
+2. **Custom Orders in Individual Variables**: Other packages use a single maximum order for all variables. With GTPSA, the
+maximum order can be set differently for different variables. For example, computing the Taylor expansion of $f(x_1,x_2)$ to 2nd order in $x_1$ and 6th order in $x_2$ is possible
+3. **Complex Numbers**: GTPSA natively supports complex numbers and allows for mixing of complex and real truncated power series
+4. **Distinction Between State Variables and Parameters**: Distinguishing between dependent variables and parameters in the solution of a differential equation expressed as a power series in the dependent variables/parameters can be advantageous in analysis
 
 ## Setup
 To use `GTPSA.jl`, in the Julia REPL run
@@ -23,30 +23,41 @@ To use `GTPSA.jl`, in the Julia REPL run
 import Pkg; Pkg.add("GTPSA")
 ```
 
-For developers,
-
-```julia
-import Pkg; Pkg.develop("GTPSA")
-```
-
 ## Basic Usage
 First, a `Descriptor` must be created specifying the number of variables, number of parameters, and truncation order for each variable/parameter in the TPSA. A `TPS` or `ComplexTPS` can then be created based on the `Descriptor`. TPSs can be manipulated using all of the arithmetic operators (`+`,`-`,`*`,`/`,`^`) and math functions (e.g. `abs`, `sqrt`, `sin`, `exp`, `log`, `tanh`, etc.).
 
-TPSs can be viewed as structures containing the coefficients for all of the monomials of a multivariable Taylor expansion up to the orders specified in the `Descriptor`. As an example, to compute the truncated power series of a function ``f(x_1, x_2) = \cos{(x_1)}+i\sin{(x_2)}`` to 6th order in ``x_1`` and ``x_2``:
-```@example
+TPSs can be viewed as structures containing the coefficients for all of the monomials of a multivariable Taylor expansion up to the orders specified in the `Descriptor`. As an example, to compute the truncated power series of a function $f(x_1, x_2) = \cos{(x_1)}+i\sin{(x_2)}$ to 6th order in $x_1$ and $x_2$:
+```julia
 using GTPSA
 
 # Descriptor for TPSA with 2 variables to 6th order
-d = Descriptor(2, 6);
+d = Descriptor(2, 6)
 
 # Get the TPSs corresponding to each variable based on the Descriptor
-x = vars();
+x = vars()
+# x[1] corresponds to the first variable and x[2] corresponds to the second variable
 
 # Manipulate the TPSs as you would any other mathematical variable in Julia
-f = cos(x[1]) + 1im*sin(x[2])
+f = cos(x[1]) + im*sin(x[2])
+# This creates a new TPS called f
 ```
 
-Advanced users are referred to [this paper](https://inspirehep.net/files/286f2ab60e1e7c372cec485337ab5eb6) written by the developers of the GTPSA library for more details.
+Note that scalars do not need to be defined as TPSs when writing expressions. Running `print(f)` gives the output
+
+```
+ComplexTPS:
+ Real Imag Order Exponent
+ 1.0000000000000000e+00 0.0000000000000000e+00 0 0 0
+ 0.0000000000000000e+00 1.0000000000000000e+00 1 0 1
+ -5.0000000000000000e-01 0.0000000000000000e+00 2 2 0
+ 0.0000000000000000e+00 -1.6666666666666666e-01 3 0 3
+ 4.1666666666666664e-02 0.0000000000000000e+00 4 4 0
+ 0.0000000000000000e+00 8.3333333333333332e-03 5 0 5
+ -1.3888888888888887e-03 0.0000000000000000e+00 6 6 0
+```
 
 ## Acknowledgements
-We thank Laurent Deniau, the creator of GTPSA, for very detailed and lengthy discussions on using his C library. 
+Much thanks must be given to Laurent Deniau, the creator of the C GTPSA library, for his time and great patience in explaining his code. 
+
+Advanced users are referred to [this paper](https://inspirehep.net/files/286f2ab60e1e7c372cec485337ab5eb6) discussing
+the inner workings of the C GTPSA library.