diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index bf473eb..138ab9a 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.7.2","generation_timestamp":"2024-02-08T18:41:37","documenter_version":"1.2.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.7.2","generation_timestamp":"2024-03-05T15:07:37","documenter_version":"1.3.0"}} \ No newline at end of file diff --git a/dev/SIOs/index.html b/dev/SIOs/index.html index d3c3d4b..6229e21 100644 --- a/dev/SIOs/index.html +++ b/dev/SIOs/index.html @@ -1,2 +1,2 @@ -Integral equations · IFSintegrals.jl
GitHub
IFSintegrals.DiscreteSIO — Type
DiscreteSIO(SIO::SIO; h_mesh::Real, h_quad::Real, h_quad_diag::Real)

is the constructor for a discretisation of a singular integral operator, 'SIO'. hmesh is the meshwidth parameter for the discretisation of the underlying fractal hquad denotes the discretisation parameter for the integrals in the stiffness matrix. hquaddiag is the parameter used to compute the diagonal elements of the matrix

source
+Integral equations · IFSintegrals.jl
GitHub
IFSintegrals.DiscreteSIO — Type
DiscreteSIO(SIO::SIO; h_mesh::Real, h_quad::Real, h_quad_diag::Real)

is the constructor for a discretisation of a singular integral operator, 'SIO'. hmesh is the meshwidth parameter for the discretisation of the underlying fractal hquad denotes the discretisation parameter for the integrals in the stiffness matrix. hquaddiag is the parameter used to compute the diagonal elements of the matrix

source
diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index f531160..c6562b5 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -4,7 +4,6 @@ requirejs.config({ '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', - 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.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', @@ -103,9 +102,10 @@ $(document).on("click", ".docstring header", function () { }); }); -$(document).on("click", ".docs-article-toggle-button", function () { +$(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) { @@ -116,7 +116,7 @@ $(document).on("click", ".docs-article-toggle-button", function () { isExpanded = false; - $(".docstring section").slideUp(); + $(".docstring section").slideUp(animationSpeed); } else { $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); $(".docstring-article-toggle-button") @@ -127,7 +127,7 @@ $(document).on("click", ".docs-article-toggle-button", function () { articleToggleTitle = "Collapse docstring"; navArticleToggleTitle = "Collapse all docstrings"; - $(".docstring section").slideDown(); + $(".docstring section").slideDown(animationSpeed); } $(this).prop("title", navArticleToggleTitle); @@ -224,224 +224,465 @@ $(document).ready(function () { }) //////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'minisearch'], function($, minisearch) { - -// In general, most search related things will have "search" as a prefix. -// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc +require(['jquery'], function($) { -let results = []; -let timer = undefined; +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); -let data = documenterSearchIndex["docs"].map((x, key) => { - x["id"] = key; // minisearch requires a unique for each object - return x; + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } }); -// list below is the lunr 2.1.3 list minus the intersect with names(Base) -// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) -// ideally we'd just filter the original list but it's not available as a variable -const stopWords = new Set([ - "a", - "able", - "about", - "across", - "after", - "almost", - "also", - "am", - "among", - "an", - "and", - "are", - "as", - "at", - "be", - "because", - "been", - "but", - "by", - "can", - "cannot", - "could", - "dear", - "did", - "does", - "either", - "ever", - "every", - "from", - "got", - "had", - "has", - "have", - "he", - "her", - "hers", - "him", - "his", - "how", - "however", - "i", - "if", - "into", - "it", - "its", - "just", - "least", - "like", - "likely", - "may", - "me", - "might", - "most", - "must", - "my", - "neither", - "no", - "nor", - "not", - "of", - "off", - "often", - "on", - "or", - "other", - "our", - "own", - "rather", - "said", - "say", - "says", - "she", - "should", - "since", - "so", - "some", - "than", - "that", - "the", - "their", - "them", - "then", - "there", - "these", - "they", - "this", - "tis", - "to", - "too", - "twas", - "us", - "wants", - "was", - "we", - "were", - "what", - "when", - "who", - "whom", - "why", - "will", - "would", - "yet", - "you", - "your", -]); - -let index = new minisearch({ - fields: ["title", "text"], // fields to index for full-text search - storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results - processTerm: (term) => { - let word = stopWords.has(term) ? null : term; - if (word) { - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - word = word - .replace(/^[^a-zA-Z0-9@!]+/, "") - .replace(/[^a-zA-Z0-9@!]+$/, ""); - } +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { - 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: { - boost: { title: 100 }, - fuzzy: 2, +/* +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); + 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 || ""; + } -let filters = [...new Set(data.map((x) => x.category))]; -var modal_filters = make_modal_body_filters(filters); -var filter_results = []; + /** + * 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 += ` (${result.page})`; + } -$(document).on("keyup", ".documenter-search-input", function (event) { - // Adding a debounce to prevent disruptions from super-speed typing! - debounce(() => update_search(filter_results), 300); + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "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; + }, + }); + + // 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")) { - $(this).removeClass("search-filter-selected"); + selected_filter = ""; } else { - $(this).addClass("search-filter-selected"); + selected_filter = $(this).text().toLowerCase(); } - // Adding a debounce to prevent disruptions from crazy clicking! - debounce(() => get_filters(), 300); + // This updates search results and toggles classes for UI: + update_search(); }); -/** - * A debounce function, takes a function and an optional timeout in milliseconds - * - * @function callback - * @param {number} timeout - */ -function debounce(callback, timeout = 300) { - clearTimeout(timer); - timer = setTimeout(callback, timeout); -} - /** * Make/Update the search component - * - * @param {string[]} selected_filters */ -function update_search(selected_filters = []) { - let initial_search_body = ` -
Type something to get started!
- `; - +function update_search() { let querystring = $(".documenter-search-input").val(); if (querystring.trim()) { - results = index.search(querystring, { - filter: (result) => { - // Filtering results - if (selected_filters.length === 0) { - return result.score >= 1; - } else { - return ( - result.score >= 1 && selected_filters.includes(result.category) - ); - } - }, - }); + 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) { @@ -449,19 +690,23 @@ function update_search(selected_filters = []) { let count = 0; let search_results = ""; - results.forEach(function (result) { - if (result.location) { - // Checking for duplication of results for the same page - if (!links.includes(result.location)) { - search_results += make_search_result(result, querystring); - count++; - } - + 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); } - }); + } - let result_count = `
${count} result(s)
`; + 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 = `
@@ -490,125 +735,37 @@ function update_search(selected_filters = []) { $(".search-modal-card-body").html(search_result_container); } else { - filter_results = []; - modal_filters = make_modal_body_filters(filters, filter_results); - if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { $(".search-modal-card-body").addClass("is-justify-content-center"); } - $(".search-modal-card-body").html(initial_search_body); + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); } } /** * Make the modal filter html * - * @param {string[]} filters - * @param {string[]} selected_filters * @returns string */ -function make_modal_body_filters(filters, selected_filters = []) { - let str = ``; - - filters.forEach((val) => { - if (selected_filters.includes(val)) { - str += `${val}`; - } else { - str += `${val}`; - } - }); +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); - let filter_html = ` + return `
Filters: ${str} -
- `; - - return filter_html; -} - -/** - * Make the result component given a minisearch result data object and the value of the search input as queryString. - * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult - * - * @param {object} result - * @param {string} querystring - * @returns string - */ -function make_search_result(result, querystring) { - let search_divider = `
`; - let display_link = - result.location.slice(Math.max(0), Math.min(50, result.location.length)) + - (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div - - if (result.page !== "") { - display_link += ` (${result.page})`; - } - - let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); - let text = - textindex !== null - ? result.text.slice( - Math.max(textindex.index - 100, 0), - Math.min( - textindex.index + querystring.length + 100, - result.text.length - ) - ) - : ""; // cut-off text before and after from the match - - let display_result = text.length - ? "..." + - text.replace( - new RegExp(`\\b${querystring}\\b`, "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 = ` - -
-
${result.title}
-
${result.category}
-
-

- ${display_result} -

-
- ${display_link} -
- - ${search_divider} - `; - - return result_div; -} - -/** - * Get selected filters, remake the filter html and lastly update the search modal - */ -function get_filters() { - let ele = $(".search-filters .search-filter-selected").get(); - filter_results = ele.map((x) => $(x).text().toLowerCase()); - modal_filters = make_modal_body_filters(filters, filter_results); - update_search(filter_results); +
`; } }) @@ -635,103 +792,107 @@ $(document).ready(function () { //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -let search_modal_header = ` -
-
-

- - - - -

-
-
- -
-
-`; - -let initial_search_body = ` -
Type something to get started!
-`; - -let search_modal_footer = ` - -`; - -$(document.body).append( - ` - source

InvariantMeasure

The iterated function system may be interpreted as a set of these maps, $\{s_m\}_{m=1}^M$. The attractor $\Gamma$ is the unique non-empty compact set which satisfies

\[\Gamma = S(\Gamma):=\bigcup_{m=1}^M s_m(\Gamma).\]

We can construct the map $S$ defined above as follows.

using IFSintegrals
+    1.5707963267948966
source

InvariantMeasure

The iterated function system may be interpreted as a set of these maps, $\{s_m\}_{m=1}^M$. The attractor $\Gamma$ is the unique non-empty compact set which satisfies

\[\Gamma = S(\Gamma):=\bigcup_{m=1}^M s_m(\Gamma).\]

We can construct the map $S$ defined above as follows.

using IFSintegrals
 s₁ = Similarity(1/3,2/3)
 s₂ = Similarity(1/3,2/3)
 S = [s₁,s₂]
 x = rand()
 S(x) # applies the map S to the point x
2-element Vector{Float64}:
- 0.9039827419889248
- 0.9039827419889248

This is the IFS for the Cantor Set, and this can be converted into an InvariantMeasure with the command

 Γ = InvariantMeasure(S)
InvariantMeasure{Float64, Float64}(Similarity{Float64, Float64}[Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333), Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333)], 1, 0.6309297535714574, true, true, 0.9999999999999999, 0.0, 1.0, [0.5, 0.5], true, Bool[1 0; 0 1], IFSintegrals.AutomorphicMap{Float64, Float64}[IFSintegrals.AutomorphicMap{Float64, Float64}(1.0, 0.0)])

The outer constructor for InvariantMeasure constructs other properties, such as diameter and Hausdorff dimension, which describe this fractal measure.

The full type is described below:

IFSintegrals.InvariantMeasure — Type
struct InvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
+ 0.753691603836707
+ 0.753691603836707

This is the IFS for the Cantor Set, and this can be converted into an InvariantMeasure with the command

 Γ = InvariantMeasure(S)
InvariantMeasure{Float64, Float64}(Similarity{Float64, Float64}[Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333), Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333)], 1, 0.6309297535714574, true, true, 0.9999999999999999, 0.0, 1.0, [0.5, 0.5], true, Bool[1 0; 0 1], IFSintegrals.AutomorphicMap{Float64, Float64}[IFSintegrals.AutomorphicMap{Float64, Float64}(1.0, 0.0)])

The outer constructor for InvariantMeasure constructs other properties, such as diameter and Hausdorff dimension, which describe this fractal measure.

The full type is described below:

IFSintegrals.InvariantMeasure — Type
struct InvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
     IFS::Vector{Similarity{V,M}}
     spatial_dimension::Int64
     Hausdorff_dimension::Float64
@@ -27,7 +27,7 @@
     connectedness::Matrix{Bool}
     symmetry_group::Vector{AutomorphicMap{V,M}}
 end

Representation of a invariant measure, whose support is an iterated function system (IFS). Constructor requires only an IFS, which is of type Array{Similarity}. All other essential properties can be deduced from this, including barycentre, diameter and dimension, which are approximated numerically.

Has the outer constructor, which only requires IFS (a vector of Similarity) as an input.

InvariantMeasure(sims::Vector{Similarity}; measure::Real=1.0) = 
-    InvariantMeasure(sims, get_diameter(sims); measure=measure)

Fields

  • IFS: The iterated function system, a vector of Similarity, describing the fractal
  • spatial_dimension: The integer dimension of the smallest open set containing the fractal
  • Hausdorff_dimension: The Hausdorff dimenson of the fractal
  • homogeneous: A flag, true if all the contractions are of the same size
  • Hausdorff_weights: A flag, true if this is a Hausdorff measure
  • barycentre: The barycentre of the measure
  • diameter: The diemater of the fractal support
  • measure: The measure of the whole fractal, usually set to one
  • weights: The probability weights describing the invariant measure
  • disjoint: Flag for if the fractal support is disjoint
  • connectedness: Matrix describing which subcomponents are connected
  • symmetry_group: Vector of AutomorphicMap, describing symmetries of the measure
source

Now let's try a more complicated example. The second (translation) argument of Similarity can also be a vector. For example:

ρ = 0.41
+    InvariantMeasure(sims, get_diameter(sims); measure=measure)

Fields

  • IFS: The iterated function system, a vector of Similarity, describing the fractal
  • spatial_dimension: The integer dimension of the smallest open set containing the fractal
  • Hausdorff_dimension: The Hausdorff dimenson of the fractal
  • homogeneous: A flag, true if all the contractions are of the same size
  • Hausdorff_weights: A flag, true if this is a Hausdorff measure
  • barycentre: The barycentre of the measure
  • diameter: The diemater of the fractal support
  • measure: The measure of the whole fractal, usually set to one
  • weights: The probability weights describing the invariant measure
  • disjoint: Flag for if the fractal support is disjoint
  • connectedness: Matrix describing which subcomponents are connected
  • symmetry_group: Vector of AutomorphicMap, describing symmetries of the measure
source

Now let's try a more complicated example. The second (translation) argument of Similarity can also be a vector. For example:

ρ = 0.41
 IFS = [
     Similarity(ρ,[0,0])
     Similarity(ρ,[1-ρ,0])
@@ -35,7 +35,7 @@
     Similarity(ρ,[(1-ρ)/2,(1-ρ)/(2*sqrt(3))])
     ]
  Γ = InvariantMeasure(IFS)

We can plot an approximation attractors in $\mathbb{R}$ or $\mathbb{R}^2$, as follows:

using Plots
-plot(Γ,color = "black", markersize=0.75,label="My fractal")
Example block output

Preset fractals

Several preset fractals are available. These may be called with:

CantorSet()
+plot(Γ,color = "black", markersize=0.75,label="My fractal")
Example block output

Preset fractals

Several preset fractals are available. These may be called with:

CantorSet()
 CantorDust()
 Sierpinski() # triangle/gasket
 KochCurve() # Koch curve
@@ -49,18 +49,18 @@
 
 Γ_shift = 1.5*Γ + [-2,0.5]
 plot!(Γ_shift, markersize=0.75,
-    label="Sierpinski Triangle (translated and stretched)",aspect_ratio=1.0)
Example block output

SubInvariantMeasure

Consider an IFS with $M\in\mathbb{N}$ components. For a vector

\[\mathbf{m}=[m_1,\ldots,m_N]\in\{1,\ldots,M\}^N,\]

it is standard to denote a sub-component of the fractal by

\[\Gamma_{\mathbf{m}} := s_{m_1}\circ\ldots \circ s_{m_N}(\Gamma)\]

The following type also describes a measure whose support is a sub-component, in the above sense.

IFSintegrals.SubInvariantMeasure — Type
struct SubInvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
+    label="Sierpinski Triangle (translated and stretched)",aspect_ratio=1.0)
Example block output

SubInvariantMeasure

Consider an IFS with $M\in\mathbb{N}$ components. For a vector

\[\mathbf{m}=[m_1,\ldots,m_N]\in\{1,\ldots,M\}^N,\]

it is standard to denote a sub-component of the fractal by

\[\Gamma_{\mathbf{m}} := s_{m_1}\circ\ldots \circ s_{m_N}(\Gamma)\]

The following type also describes a measure whose support is a sub-component, in the above sense.

IFSintegrals.SubInvariantMeasure — Type
struct SubInvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
     parent_measure::InvariantMeasure
     IFS::Vector{Similarity{V,M}} # could be removed ?
     index::Vector{Int64}
     barycentre::V
     diameter::Float64
     measure::Float64
-end

Represents a fractal measure which has been derived from an InvariantMeasure.

Fields

  • parentmeasure: This measure is supported on a subet of the support of parentmeasure
  • IFS: The vector of similarities describing the fractal support of this measure
  • index: The vector index corresponding to the contractions applied to parent_measure
  • barycentre: The barycentre of this measure
  • diameter: The diameter of the fractal support
  • measure: The measure of the support
source

Using standard vector index syntax, a SubInvariantMeasure can be easily constructed:

Γ = Sierpinski()
+end

Represents a fractal measure which has been derived from an InvariantMeasure.

Fields

  • parentmeasure: This measure is supported on a subet of the support of parentmeasure
  • IFS: The vector of similarities describing the fractal support of this measure
  • index: The vector index corresponding to the contractions applied to parent_measure
  • barycentre: The barycentre of this measure
  • diameter: The diameter of the fractal support
  • measure: The measure of the support
source

Using standard vector index syntax, a SubInvariantMeasure can be easily constructed:

Γ = Sierpinski()
 m = [1,3,2] # vector index
 Γₘ = Γ[m] # construct SubInvariantMeasure
 
 plot(Γ, markersize=0.75,
     label="Sierpinski Triangle (default)")
 plot!(Γₘ, markersize=0.75,
-    label="Subcomponent",aspect_ratio=1.0)
Example block output + label="Subcomponent",aspect_ratio=1.0)Example block output diff --git a/dev/objects.inv b/dev/objects.inv new file mode 100644 index 0000000..4ebec0e --- /dev/null +++ b/dev/objects.inv @@ -0,0 +1,7 @@ +# Sphinx inventory version 2 +# Project: IFSintegrals.jl +# Version: 0.1.0 +# The remainder of this file is compressed using zlib. +xRN0 +,T\wCML^@Aߓv햪tl9϶%q(TTE=ю,rI wHT2`lWRLԵ5GYa&5 |6 @AeZRJ*% +H!M,>u%GfNMQCV.8 x 10H$'Eͫx +&}wGxOmEçZ`iդ[&M:a)\HeJ>_Ksm#Hrw̸XZP \ No newline at end of file diff --git a/dev/plotting/index.html b/dev/plotting/index.html index 475884a..f8c9486 100644 --- a/dev/plotting/index.html +++ b/dev/plotting/index.html @@ -1,2 +1,2 @@ -Plotting · IFSintegrals.jl
GitHub
+Plotting · IFSintegrals.jl
GitHub
diff --git a/dev/quadrature/index.html b/dev/quadrature/index.html index b752ebc..2d026cc 100644 --- a/dev/quadrature/index.html +++ b/dev/quadrature/index.html @@ -1,2 +1,2 @@ -Approximating integrals · IFSintegrals.jl
GitHub
IFSintegrals.barycentre_rule — Function
x,w = barycentre_rule(Γ::Union{InvariantMeasure,SubInvariantMeasure},h::Real)

returns a vector of N weights wⱼ>0 and nodes xⱼ ∈ Rⁿ, for approximation of integrals defined on an IFS Γ⊂Rⁿ.

source
x,y,w = barycentre_rule(Γ₁::Union{InvariantMeasure,SubInvariantMeasure},Γ₂::Union{InvariantMeasure,SubInvariantMeasure},h::Real)

returns N weights wⱼ>0 and nodes x,y ∈ Rⁿ, for approximation of double integrals over Γ₁,Γ₂⊂Rⁿ. Uses Barycentre rule quadrature, the fractal Γ will be subdivided until each subcomponent has a diameter of less than h.

source
IFSintegrals.gauss_quad — Function
x,w = gauss_quad(Γ::SelfSimilarFractal{V,M}, N::Int64) where {V<:Real, M<:Real}

Returns N Gaussian weights w ∈ Rᴺ and nodes x ∈ Rᴺˣᴺ. Here Γ must be an SelfSimilarFractal in one spatial dimension. N is the order of the Gauss rule, i.e. number of weights and nodes.

source
IFSintegrals.chaos_quad — Function
chaos_quad(Γ::SelfSimilarFractal{V,M},N::Int64;x₀=Γ.barycentre:::AbstractVector) where {V<:AbstractVector, M<:Union{Real,AbstractMatrix}}

Returns a vector of N weights w>0 and nodes x ∈ Rⁿ, for approximation of integrals defined on an IFS Γ⊂Rⁿ. Using Chaos game quadrature. Optional third input is the initial guess, which is taken as barycentre by default.

source
+Approximating integrals · IFSintegrals.jl
GitHub
IFSintegrals.barycentre_rule — Function
x,w = barycentre_rule(Γ::Union{InvariantMeasure,SubInvariantMeasure},h::Real)

returns a vector of N weights wⱼ>0 and nodes xⱼ ∈ Rⁿ, for approximation of integrals defined on an IFS Γ⊂Rⁿ.

source
x,y,w = barycentre_rule(Γ₁::Union{InvariantMeasure,SubInvariantMeasure},Γ₂::Union{InvariantMeasure,SubInvariantMeasure},h::Real)

returns N weights wⱼ>0 and nodes x,y ∈ Rⁿ, for approximation of double integrals over Γ₁,Γ₂⊂Rⁿ. Uses Barycentre rule quadrature, the fractal Γ will be subdivided until each subcomponent has a diameter of less than h.

source
IFSintegrals.gauss_quad — Function
x,w = gauss_quad(Γ::SelfSimilarFractal{V,M}, N::Int64) where {V<:Real, M<:Real}

Returns N Gaussian weights w ∈ Rᴺ and nodes x ∈ Rᴺˣᴺ. Here Γ must be an SelfSimilarFractal in one spatial dimension. N is the order of the Gauss rule, i.e. number of weights and nodes.

source
IFSintegrals.chaos_quad — Function
chaos_quad(Γ::SelfSimilarFractal{V,M},N::Int64;x₀=Γ.barycentre:::AbstractVector) where {V<:AbstractVector, M<:Union{Real,AbstractMatrix}}

Returns a vector of N weights w>0 and nodes x ∈ Rⁿ, for approximation of integrals defined on an IFS Γ⊂Rⁿ. Using Chaos game quadrature. Optional third input is the initial guess, which is taken as barycentre by default.

source