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"1.4.1+1" diff --git a/v5.27.1/assets/Project.toml b/v5.27.1/assets/Project.toml new file mode 100644 index 000000000..9a18ffb63 --- /dev/null +++ b/v5.27.1/assets/Project.toml @@ -0,0 +1,19 @@ +[deps] +BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" +Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" +Latexify = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" +OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" +SymbolicUtils = "d1185830-fcd6-423d-90d6-eec64667417b" +Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" + +[compat] +BenchmarkTools = "1.3" +Documenter = "1" +Latexify = "0.15, 0.16" +OrdinaryDiffEq = "6.31" +Plots = "1.36" +StaticArrays = "1.5" +SymbolicUtils = "1" +Symbolics = "5" diff --git a/v5.27.1/assets/documenter.js b/v5.27.1/assets/documenter.js new file mode 100644 index 000000000..75719b2a9 --- /dev/null +++ b/v5.27.1/assets/documenter.js @@ -0,0 +1,1057 @@ +// 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', + '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', + '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" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +window.MathJax = { + "tex": { + "packages": [ + "base", + "ams", + "autoload", + "mathtools", + "require" + ], + "inlineMath": [ + [ + "$", + "$" + ], + [ + "\\(", + "\\)" + ] + ] + }, + "loader": { + "load": [ + "[tex]/require", + "[tex]/mathtools" + ] + }, + "options": { + "ignoreHtmlClass": "tex2jax_ignore", + "processHtmlClass": "tex2jax_process" + } +} +; + +(function () { + var script = document.createElement('script'); + script.src = 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-svg.js'; + script.async = true; + document.head.appendChild(script); +})(); + +}) +//////////////////////////////////////////////////////////////////////////////// +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@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${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")) { + 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/v5.27.1/assets/favicon.ico b/v5.27.1/assets/favicon.ico new file mode 100644 index 0000000000000000000000000000000000000000..cc690e2b6a551f080c577e96fd8438668d29e2f6 GIT binary patch literal 15406 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.button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-link.is-outlined{background-color:transparent;border-color:#1abc9c;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-outlined:hover,html.theme--documenter-dark .button.is-link.is-outlined.is-hovered,html.theme--documenter-dark .button.is-link.is-outlined:focus,html.theme--documenter-dark .button.is-link.is-outlined.is-focused{background-color:#1abc9c;border-color:#1abc9c;color:#fff}html.theme--documenter-dark .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1abc9c #1abc9c !important}html.theme--documenter-dark .button.is-link.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-link.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined{background-color:transparent;border-color:#1abc9c;box-shadow:none;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1abc9c #1abc9c !important}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-link.is-light{background-color:#edfdf9;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:hover,html.theme--documenter-dark .button.is-link.is-light.is-hovered{background-color:#e2fbf6;border-color:transparent;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:active,html.theme--documenter-dark .button.is-link.is-light.is-active{background-color:#d7f9f3;border-color:transparent;color:#15987e}html.theme--documenter-dark .button.is-info{background-color:#024c7d;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-info:hover,html.theme--documenter-dark .button.is-info.is-hovered{background-color:#024470;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-info:focus,html.theme--documenter-dark .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-info:focus:not(:active),html.theme--documenter-dark .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(2,76,125,0.25)}html.theme--documenter-dark .button.is-info:active,html.theme--documenter-dark .button.is-info.is-active{background-color:#023d64;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-info[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-info{background-color:#024c7d;border-color:#024c7d;box-shadow:none}html.theme--documenter-dark .button.is-info.is-inverted{background-color:#fff;color:#024c7d}html.theme--documenter-dark .button.is-info.is-inverted:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#024c7d}html.theme--documenter-dark .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-info.is-outlined{background-color:transparent;border-color:#024c7d;color:#024c7d}html.theme--documenter-dark .button.is-info.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-outlined.is-focused{background-color:#024c7d;border-color:#024c7d;color:#fff}html.theme--documenter-dark .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #024c7d #024c7d !important}html.theme--documenter-dark .button.is-info.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-info.is-outlined{background-color:transparent;border-color:#024c7d;box-shadow:none;color:#024c7d}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#024c7d}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #024c7d #024c7d !important}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-info.is-light{background-color:#ebf7ff;color:#0e9dfb}html.theme--documenter-dark .button.is-info.is-light:hover,html.theme--documenter-dark .button.is-info.is-light.is-hovered{background-color:#def2fe;border-color:transparent;color:#0e9dfb}html.theme--documenter-dark .button.is-info.is-light:active,html.theme--documenter-dark .button.is-info.is-light.is-active{background-color:#d2edfe;border-color:transparent;color:#0e9dfb}html.theme--documenter-dark .button.is-success{background-color:#008438;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-success:hover,html.theme--documenter-dark .button.is-success.is-hovered{background-color:#073;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-success:focus,html.theme--documenter-dark .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-success:focus:not(:active),html.theme--documenter-dark .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(0,132,56,0.25)}html.theme--documenter-dark .button.is-success:active,html.theme--documenter-dark .button.is-success.is-active{background-color:#006b2d;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-success[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success{background-color:#008438;border-color:#008438;box-shadow:none}html.theme--documenter-dark .button.is-success.is-inverted{background-color:#fff;color:#008438}html.theme--documenter-dark .button.is-success.is-inverted:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#008438}html.theme--documenter-dark .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#008438;color:#008438}html.theme--documenter-dark .button.is-success.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-outlined.is-focused{background-color:#008438;border-color:#008438;color:#fff}html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #008438 #008438 !important}html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#008438;box-shadow:none;color:#008438}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#008438}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #008438 #008438 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-success.is-light{background-color:#ebfff3;color:#00eb64}html.theme--documenter-dark .button.is-success.is-light:hover,html.theme--documenter-dark .button.is-success.is-light.is-hovered{background-color:#deffec;border-color:transparent;color:#00eb64}html.theme--documenter-dark .button.is-success.is-light:active,html.theme--documenter-dark .button.is-success.is-light.is-active{background-color:#d1ffe5;border-color:transparent;color:#00eb64}html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning:hover,html.theme--documenter-dark .button.is-warning.is-hovered{background-color:#a07700;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning:focus,html.theme--documenter-dark .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning:focus:not(:active),html.theme--documenter-dark .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(173,129,0,0.25)}html.theme--documenter-dark .button.is-warning:active,html.theme--documenter-dark .button.is-warning.is-active{background-color:#946e00;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:#ad8100;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-outlined.is-focused{background-color:#ad8100;border-color:#ad8100;color:#fff}html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#92190c;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(158,27,13,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#86170b;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:#9e1b0d;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#9e1b0d;border-color:#9e1b0d;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .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 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.button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}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 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.button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons 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.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 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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 .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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.docs-logo>img.is-5by4{padding-top:80%}html.theme--documenter-dark .image.is-4by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--documenter-dark .image.is-3by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--documenter-dark .image.is-5by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--documenter-dark .image.is-16by9,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--documenter-dark .image.is-2by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--documenter-dark .image.is-3by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by3{padding-top:300%}html.theme--documenter-dark .image.is-16x16,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v5.27.1/assets/themeswap.js b/v5.27.1/assets/themeswap.js new file mode 100644 index 000000000..9f5eebe6a --- /dev/null +++ b/v5.27.1/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. 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. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + 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; + // 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/v5.27.1/assets/warner.js b/v5.27.1/assets/warner.js new file mode 100644 index 000000000..3f6f5d008 --- /dev/null +++ b/v5.27.1/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. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v5.27.1/comparison/index.html b/v5.27.1/comparison/index.html new file mode 100644 index 000000000..2d8436b18 --- /dev/null +++ b/v5.27.1/comparison/index.html @@ -0,0 +1,2 @@ + +Comparison Against SymPy · Symbolics.jl
GitHub

Comparison of Julia's Symbolics.jl vs SymPy for Symbolic Computation

Symbolics.jl is a symbolic modeling language for Julia, built in Julia. Its goal is very different from Sympy: it was made to support symbolic-numerics, the combination of symbolic computing with numerical methods to allow for extreme performance computing that would not be possible without modifying the model. Because of this, Symbolics.jl excels in many areas due to purposeful design decisions:

  • Performance: Symbolics.jl is built in Julia, whereas SymPy was built in Python. Thus, the performance bar for Symbolics.jl is much higher. Symbolics.jl started because SymPy was far too slow and SymEngine was far too inflexible for the projects they were doing. Performance is key to Symbolics.jl. If you find any performance issues, please file an issue.
  • build_function: lambdify is “fine” for some people, but if you're building a super fast MPI-enabled Julia/C/Fortran simulation code, having a function that hits the Python interpreter is less than optimal. By default, build_function builds fast JIT-compiled functions due to being in Julia. However, it has support for things like static arrays, non-allocating functions via mutation, fast functions on sparse matrices and arrays of arrays, etc.: all core details of doing high performance computing.
  • Parallelism: Symbolics.jl has pervasive parallelism. The symbolic simplification via SymbolicUtils.jl has built-in parallelism, Symbolics.jl builds functions that parallelize across threads. Symbolics.jl is compatible with GPU libraries like CUDA.jl.
  • Extensible: Symbolics.jl and its underlying tools are written in pure Julia. Want to add new or better simplification rules? Add some Julia code! Need to add new derivatives? Add some Julia code! You get the picture. Breaking down these barriers makes it easier for the user to tailor the program to their needs and accelerates the development of the library.
  • Deep integration with the Julia ecosystem: Symbolics.jl's integration with neural networks is not the only thing that's deep. Symbolics.jl is built with the same philosophy as other SciML packages, eschewing “monorepos” for a distributed development approach that ties together the work of many developers. The differentiation parts utilize tools from automatic differentiation libraries, all linear algebra functionality comes from tracing Julia Base itself, symbolic rewriting (simplification and substitution) comes from SymbolicUtils.jl, parallelism comes from Julia Base libraries, Dagger.jl, etc. SciML Tools like DataDrivenDiffEq.jl can reconstruct symbolic expressions from neural networks and data, while NeuralPDE.jl can automatically solve partial differential equations from symbolic descriptions using physics-informed neural networks. The list keeps going. All told, by design Symbolics.jl's development moves fast because it's effectively using the work of hundreds of Julia developers, allowing it to grow fast.
  • While Symbolics.jl has many features missing from SymPy, it does not superset SymPy's functionality. For a list of missing features, see this issue.
diff --git a/v5.27.1/examples/perturbation/index.html b/v5.27.1/examples/perturbation/index.html new file mode 100644 index 000000000..0f6ddee90 --- /dev/null +++ b/v5.27.1/examples/perturbation/index.html @@ -0,0 +1,86 @@ + +Mixed Symbolic-Numeric Perturbation Theory · Symbolics.jl
GitHub

Mixed Symbolic-Numeric Perturbation Theory

Background

Symbolics.jl is a fast and modern Computer Algebra System (CAS) written in the Julia Programming Language. It is an integral part of the SciML ecosystem of differential equation solvers and scientific machine learning packages. While Symbolics.jl is primarily designed for modern scientific computing (e.g., auto-differentiation, machine learning), it is a powerful CAS that can also be useful for classic scientific computing. One such application is using the perturbation theory to solve algebraic and differential equations.

Perturbation methods are a collection of techniques to solve intractable problems that generally don't have a closed solution but depend on a tunable parameter and have closed or easy solutions for some values of the parameter. The main idea is to assume a solution as a power series in the tunable parameter (say $ϵ$), such that $ϵ = 0$ corresponds to an easy solution.

We will discuss the general steps of the perturbation methods to solve algebraic (this tutorial) and differential equations

The hallmark of the perturbation method is the generation of long and convoluted intermediate equations, which are subjected to algorithmic and mechanical manipulations. Therefore, these problems are well suited for CAS. In fact, CAS software packages have been used to help with the perturbation calculations since the early 1970s.

In this tutorial, our goal is to show how to use a mix of symbolic manipulations (Symbolics.jl) and numerical methods to solve simple perturbation problems.

Solving the Quintic

We start with the “hello world!” analog of the perturbation problems, solving the quintic (fifth-order) equations. We want to find a real valued $x$ such that $x^5 + x = 1$. According to the Abel's theorem, a general quintic equation does not have a closed form solution. Of course, we can easily solve this equation numerically; for example, by using the Newton's method. We use the following implementation of the Newton's method:

using Symbolics, SymbolicUtils
+
+function solve_newton(f, x, x₀; abstol=1e-8, maxiter=50)
+    xₙ = Float64(x₀)
+    fₙ₊₁ = x - f / Symbolics.derivative(f, x)
+
+    for i = 1:maxiter
+        xₙ₊₁ = substitute(fₙ₊₁, Dict(x => xₙ))
+        if abs(xₙ₊₁ - xₙ) < abstol
+            return xₙ₊₁
+        else
+            xₙ = xₙ₊₁
+        end
+    end
+    return xₙ₊₁
+end
solve_newton (generic function with 1 method)

In this code, Symbolics.derivative(eq, x) does exactly what it names implies: it calculates the symbolic derivative of eq (a Symbolics.jl expression) with respect to x (a Symbolics.jl variable). We use Symbolics.substitute(eq, D) to evaluate the update formula by substituting variables or sub-expressions (defined in a dictionary D) in eq. It should be noted that substitute is the workhorse of our code and will be used multiple times in the rest of these tutorials. solve_newton is written with simplicity and clarity in mind, and not performance.

Let's go back to our quintic. We can define a Symbolics variable as @variables x and then solve the equation solve_newton(x^5 + x - 1, x, 1.0) (here, x₀ = 1.0 is our first guess). The answer is 0.7549. Now, let's see how we can solve the same problem using the perturbation methods.

We introduce a tuning parameter $\epsilon$ into our equation: $x^5 + \epsilon x = 1$. If $\epsilon = 1$, we get our original problem. For $\epsilon = 0$, the problem transforms to an easy one: $x^5 = 1$ which has an exact real solution $x = 1$ (and four complex solutions which we ignore here). We expand $x$ as a power series on $\epsilon$:

\[ x(\epsilon) = a_0 + a_1 \epsilon + a_2 \epsilon^2 + O(\epsilon^3)\]

\[a_0\]

is the solution of the easy equation, therefore $a_0 = 1$. Substituting into the original problem,

\[ (a_0 + a_1 \epsilon + a_2 \epsilon^2)^5 + \epsilon (a_0 + a_1 \epsilon + a_2 \epsilon^2) - 1 = 0\]

Expanding the equations, we get

\[ \epsilon (1 + 5 a_1) + \epsilon^2 (a_1 + 5 a_2 + 10 a_1^2) + 𝑂(\epsilon^3) = 0\]

This equation should hold for each power of $\epsilon$. Therefore,

\[ 1 + 5 a_1 = 0\]

and

\[ a_1 + 5 a_2 + 10 a_1^2 = 0\]

This system of equations does not initially seem to be linear because of the presence of terms like $10 a_1^2$, but upon closer inspection is found to be linear (this is a feature of the perturbation methods). In addition, the system is in a triangular form, meaning the first equation depends only on $a_1$, the second one on $a_1$ and $a_2$, such that we can replace the result of $a_1$ from the first one into the second equation and remove the non-linear term. We solve the first equation to get $a_1 = -\frac{1}{5}$. Substituting in the second one and solve for $a_2$:

\[ a_2 = \frac{(-\frac{1}{5} + 10(-(\frac{1}{5})²)}{5} = -\frac{1}{25}\]

Finally,

\[ x(\epsilon) = 1 - \frac{\epsilon}{5} - \frac{\epsilon^2}{25} + O(\epsilon^3)\]

Solving the original problem, $x(1) = 0.76$, compared to 0.7548 calculated numerically. We can improve the accuracy by including more terms in the expansion of $x$. However, the calculations, while straightforward, become messy and intractable to do manually very quickly. This is why a CAS is very helpful to solve perturbation problems.

Now, let's see how we can do these calculations in Julia. Let $n$ be the order of the expansion. We start by defining the symbolic variables:

n = 2
+@variables ϵ a[1:n]
2-element Vector{Any}:
+ ϵ
+  a[1:2]

Then, we define

x = 1 + a[1]*ϵ + a[2]*ϵ^2

\[ \begin{equation} +1 + a_1 \epsilon + \epsilon^{2} a_2 +\end{equation} + \]

The next step is to substitute x in the problem equation

  eq = x^5 + ϵ*x - 1

\[ \begin{equation} +-1 + \left( 1 + a_1 \epsilon + \epsilon^{2} a_2 \right) \epsilon + \left( 1 + a_1 \epsilon + \epsilon^{2} a_2 \right)^{5} +\end{equation} + \]

The expanded form of eq is

expand(eq)

\[ \begin{equation} +\epsilon + 5 a_1 \epsilon + \epsilon^{2} a_1 + 5 \epsilon^{2} a_2 + 10 \epsilon^{2} a_1^{2} + \epsilon^{3} a_2 + 20 \epsilon^{3} a_1 a_2 + 10 \epsilon^{3} a_1^{3} + 10 \epsilon^{4} a_2^{2} + 30 \epsilon^{4} a_1^{2} a_2 + 5 \epsilon^{4} a_1^{4} + 30 \epsilon^{5} a_2^{2} a_1 + 20 \epsilon^{5} a_1^{3} a_2 + 10 \epsilon^{6} a_2^{3} + \epsilon^{5} a_1^{5} + 30 \epsilon^{6} a_2^{2} a_1^{2} + 5 \epsilon^{6} a_1^{4} a_2 + 20 \epsilon^{7} a_2^{3} a_1 + 10 \epsilon^{7} a_2^{2} a_1^{3} + 5 \epsilon^{8} a_2^{4} + 10 \epsilon^{8} a_2^{3} a_1^{2} + 5 \epsilon^{9} a_2^{4} a_1 + \epsilon^{10} a_2^{5} +\end{equation} + \]

We need a way to get the coefficients of different powers of ϵ. Function collect_powers(eq, x, ns) returns the powers of variable x in expression eq. Argument ns is the range of the powers.

function collect_powers(eq, x, ns; max_power=100)
+    eq = substitute(expand(eq), Dict(x^j => 0 for j=last(ns)+1:max_power))
+
+    eqs = []
+    for i in ns
+        powers = Dict(x^j => (i==j ? 1 : 0) for j=1:last(ns))
+        push!(eqs, substitute(eq, powers))
+    end
+    eqs
+end
collect_powers (generic function with 1 method)

To return the coefficients of $ϵ$ and $ϵ^2$ in eq, we can write

eqs = collect_powers(eq, ϵ, 1:2)
2-element Vector{Any}:
+                 1 + 5a[1]
+ a[1] + 5a[2] + 10(a[1]^2)

A few words on how collect_powers works, It uses substitute to find the coefficient of a given power of x by passing a Dict with all powers of x set to 0, except the target power which is set to 1. For example, the following expression returns the coefficient of ϵ^2 in eq,

substitute(expand(eq), Dict(
+  ϵ => 0,
+  ϵ^2 => 1,
+  ϵ^3 => 0,
+  ϵ^4 => 0,
+  ϵ^5 => 0,
+  ϵ^6 => 0,
+  ϵ^7 => 0,
+  ϵ^8 => 0)
+)

\[ \begin{equation} +a_1 + 5 a_2 + 10 a_1^{2} +\end{equation} + \]

Back to our problem. Having the coefficients of the powers of ϵ, we can set each equation in eqs to 0 (remember, we rearrange the problem such that eq is 0) and solve the system of linear equations to find the numerical values of the coefficients. Symbolics.jl has a function Symbolics.solve_for that can solve systems of linear equations. However, the presence of higher-order terms in eqs prevents Symbolics.solve_for(eqs .~ 0, a) from workings properly. Instead, we can exploit the fact that our system is in a triangular form and start by solving eqs[1] for a₁ and then substitute this in eqs[2] and solve for a₂, and so on. This cascading process is done by function solve_coef(eqs, ps):

function solve_coef(eqs, ps)
+    vals = Dict()
+
+    for i = 1:length(ps)
+        eq = substitute(eqs[i], vals)
+        vals[ps[i]] = Symbolics.solve_for(eq ~ 0, ps[i])
+    end
+    vals
+end
solve_coef (generic function with 1 method)

Here, eqs is an array of expressions (assumed to be equal to 0) and ps is an array of variables. The result is a dictionary of variable => value pairs. We apply solve_coef to eqs to get the numerical values of the parameters:

vals = solve_coef(eqs, a)
Dict{Any, Any} with 2 entries:
+  a[2] => -0.04
+  a[1] => -0.2

Finally, we substitute back the values of a in the definition of x as a function of 𝜀. Note that 𝜀 is a number (usually Float64), whereas ϵ is a symbolic variable.

X = 𝜀 -> 1 + vals[a[1]]*𝜀 + vals[a[2]]*𝜀^2
#7 (generic function with 1 method)

Therefore, the solution to our original problem becomes X(1), which is equal to 0.76. We can use larger values of n to improve the accuracy of estimations.

nx
10.8
20.76
30.752
40.752
50.7533
60.7543
70.7548
80.7550

Remember, the numerical value is 0.7549. The two functions collect_powers and solve_coef(eqs, a) are used in all the examples in this and the next tutorial.

Solving the Kepler's Equation

Historically, the perturbation methods were first invented to solve orbital calculations of the Moon and the planets. In homage to this history, our second example has a celestial theme. Our goal is solving the Kepler's equation:

\[ E - e\sin(E) = M\]

where $e$ is the eccentricity of the elliptical orbit, $M$ is the mean anomaly, and $E$ (unknown) is the eccentric anomaly (the angle between the position of a planet in an elliptical orbit and the point of periapsis). This equation is central to solving two-body Keplerian orbits.

Similar to the first example, it is easy to solve this problem using the Newton's method. For example, let $e = 0.01671$ (the eccentricity of the Earth) and $M = \pi/2$. We have solve_newton(x - e*sin(x) - M, x, M) equals to 1.5875 (compared to π/2 = 1.5708). Now, we try to solve the same problem using the perturbation techniques (see function test_kepler).

For $e = 0$, we get $E = M$. Therefore, we can use $e$ as our perturbation parameter. For consistency with other problems, we also rename $e$ to $\epsilon$ and $E$ to $x$.

From here on, we use the helper function def_taylor to define Taylor's series by calling it as x = def_taylor(ϵ, a, 1), where the arguments are, respectively, the perturbation variable, which is an array of coefficients (starting from the coefficient of $\epsilon^1$), and an optional constant term.

def_taylor(x, ps) = sum([a*x^i for (i,a) in enumerate(ps)])
+def_taylor(x, ps, p₀) = p₀ + def_taylor(x, ps)
def_taylor (generic function with 2 methods)

We start by defining the variables (assuming n = 3):

n = 3
+@variables ϵ M a[1:n]
+x = def_taylor(ϵ, a, M)

\[ \begin{equation} +M + a_1 \epsilon + \epsilon^{2} a_2 + \epsilon^{3} a_3 +\end{equation} + \]

We further simplify by substituting sin with its power series using the expand_sin helper function:

expand_sin(x, n) = sum([(isodd(k) ? -1 : 1)*(-x)^(2k-1)/factorial(2k-1) for k=1:n])
expand_sin (generic function with 1 method)

To test,

expand_sin(0.1, 10) ≈ sin(0.1)
true

The problem equation is

eq = x - ϵ * expand_sin(x, n) - M

\[ \begin{equation} +a_1 \epsilon + \epsilon^{2} a_2 + \epsilon^{3} a_3 - \left( M + a_1 \epsilon + \epsilon^{2} a_2 + \epsilon^{3} a_3 + \frac{1}{6} \left( - M - a_1 \epsilon - \epsilon^{2} a_2 - \epsilon^{3} a_3 \right)^{3} - \frac{1}{120} \left( - M - a_1 \epsilon - \epsilon^{2} a_2 - \epsilon^{3} a_3 \right)^{5} \right) \epsilon +\end{equation} + \]

We follow the same process as the first example. We collect the coefficients of the powers of ϵ

eqs = collect_powers(eq, ϵ, 1:n)
3-element Vector{Any}:
+                                                          -M + a[1] + (1//6)*(M^3) - (1//120)*(M^5)
+                                              -a[1] + a[2] + (1//2)*(M^2)*a[1] - (1//24)*(M^4)*a[1]
+ -a[2] + a[3] + (1//2)*(M^2)*a[2] + (1//2)*M*(a[1]^2) - (1//24)*(M^4)*a[2] - (1//12)*(M^3)*(a[1]^2)

and then solve for a:

vals = solve_coef(eqs, a)
Dict{Any, Any} with 3 entries:
+  a[2] => M - (1//6)*(M^3) + (1//120)*(M^5) - (1//2)*(M^2)*(M - (1//6)*(M^3) + …
+  a[3] => M - (1//6)*(M^3) + (1//120)*(M^5) - (1//2)*(M^2)*(M - (1//6)*(M^3) + …
+  a[1] => M - (1//6)*(M^3) + (1//120)*(M^5)

Finally, we substitute vals back in x:

x′ = substitute(x, vals)
+X = (𝜀, 𝑀) -> substitute(x′, Dict(ϵ => 𝜀, M => 𝑀))
+X(0.01671, π/2)

\[ \begin{equation} +1.5876 +\end{equation} + \]

The result is 1.5876, compared to the numerical value of 1.5875. It is customary to order X based on the powers of 𝑀 instead of 𝜀. We can calculate this series as collect_powers(sol, M, 0:3). The result (after manual cleanup) is

(1 + 𝜀 + 𝜀^2 + 𝜀^3)*𝑀
+- (𝜀 + 4*𝜀^2 + 10*𝜀^3)*𝑀^3/6
++ (𝜀 + 16*𝜀^2 + 91*𝜀^3)*𝑀^5/120

Comparing the formula to the one for 𝐸 in the Wikipedia article on the Kepler's equation:

\[ E = \frac{1}{1-\epsilon}M + -\frac{\epsilon}{(1-\epsilon)^4} \frac{M^3}{3!} + \frac{(9\epsilon^2 + + \epsilon)}{(1-\epsilon)^7}\frac{M^5}{5!}\cdots\]

The first deviation is in the coefficient of $\epsilon^3 M^5$.

diff --git a/v5.27.1/function.jl b/v5.27.1/function.jl new file mode 100644 index 000000000..c44f2d7e2 --- /dev/null +++ b/v5.27.1/function.jl @@ -0,0 +1,11 @@ +function (ˍ₋out, ˍ₋arg1) + begin + begin + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:422 =# @inbounds begin + ˍ₋out[1] = (+)(ˍ₋arg1[2], (^)(ˍ₋arg1[1], 2)) + ˍ₋out[2] = (+)(ˍ₋arg1[1], (^)(ˍ₋arg1[2], 2)) + nothing + end + end + end +end \ No newline at end of file diff --git a/v5.27.1/getting_started/index.html b/v5.27.1/getting_started/index.html new file mode 100644 index 000000000..1e1338fa4 --- /dev/null +++ b/v5.27.1/getting_started/index.html @@ -0,0 +1,228 @@ + +Getting Started with Symbolics.jl · Symbolics.jl
GitHub

Getting Started with Symbolics.jl

Symbolics.jl is a symbolic modeling language. In this tutorial, we will walk you through the process of getting Symbolics.jl up and running, and start doing our first symbolic calculations.

Installing Symbolics.jl

Installing Symbolics.jl is as simple as using the Julia package manager. This is done via the command ]add Symbolics. After precompilation is complete, do using Symbolics in the terminal (REPL) and when that command is completed, you're ready to start!

Building Symbolic Expressions

The way to define symbolic variables is via the @variables macro:

using Symbolics
+@variables x y

\[ \begin{equation} +\left[ +\begin{array}{c} +x \\ +y \\ +\end{array} +\right] +\end{equation} + \]

After defining variables as symbolic, symbolic expressions, which we call a istree object, can be generated by utilizing Julia expressions. For example:

z = x^2 + y

\[ \begin{equation} +y + x^{2} +\end{equation} + \]

Here, z is an expression tree for “square x and add y”. To make an array of symbolic expressions, simply make an array of symbolic expressions:

A = [x^2 + y 0 2x
+     0       0 2y
+     y^2 + x 0 0]

\[ \begin{equation} +\left[ +\begin{array}{ccc} +y + x^{2} & 0 & 2 x \\ +0 & 0 & 2 y \\ +x + y^{2} & 0 & 0 \\ +\end{array} +\right] +\end{equation} + \]

Note that by default, @variables returns Sym or istree objects wrapped in Num to make them behave like subtypes of Real. Any operation on these Num objects will return a new Num object, wrapping the result of computing symbolically on the underlying values.

If you are following this tutorial in the Julia REPL, A will not be shown with LaTeX equations. To get these equations, we can use Latexify.jl. Symbolics.jl comes with Latexify recipes, so it works automatically:

using Latexify
+latexify(A)

\begin{equation} \left[ \begin{array}{ccc} y + x^{2} & 0 & 2 x \ +0 & 0 & 2 y \ +x + y^{2} & 0 & 0 \ +\end{array} \right] \end{equation}

Normal Julia functions work on Symbolics expressions, so if we want to create the sparse version of A we would just call sparse:

using SparseArrays
+spA = sparse(A)
+latexify(A)

\begin{equation} \left[ \begin{array}{ccc} y + x^{2} & 0 & 2 x \ +0 & 0 & 2 y \ +x + y^{2} & 0 & 0 \ +\end{array} \right] \end{equation}

We can thus use normal Julia functions as generators for sparse expressions. For example, here we will define

function f(u)
+  [u[1] - u[3], u[1]^2 - u[2], u[3] + u[2]]
+end
+f([x, y, z]) # Recall that z = x^2 + y

\[ \begin{equation} +\left[ +\begin{array}{c} +x - y - x^{2} \\ + - y + x^{2} \\ +2 y + x^{2} \\ +\end{array} +\right] +\end{equation} + \]

Or we can build an array of variables and use it to trace the function:

u = Symbolics.variables(:u, 1:5)
+f(u)

\[ \begin{equation} +\left[ +\begin{array}{c} +u_1 - u_3 \\ + - u_2 + u_1^{2} \\ +u_2 + u_3 \\ +\end{array} +\right] +\end{equation} + \]

Note

Symbolics.variables(:u, 1:5) creates a Julia array of symbolic variables. This uses O(n) compute and memory but is a very general representation. Symbolics.jl also has the ability to represent symbolic arrays which gives an O(1) representation but is more limited in its functionality. For more information, see the Symbolic Arrays page.

Derivatives

One common thing to compute in a symbolic system is derivatives. In Symbolics.jl, derivatives are represented lazily via operations, just like any other function. To build a differential operator, use Differential like:

@variables t
+D = Differential(t)
(::Differential) (generic function with 3 methods)

This is the differential operator $D = \frac{\partial}{\partial t}$. We can compose the differential operator by *, e.g. Differential(t) * Differential(x) or Differential(t)^2. Now let's write down the derivative of some expression:

z = t + t^2
+D(z)

\[ \begin{equation} +\frac{\mathrm{d}}{\mathrm{d}t} \left( t + t^{2} \right) +\end{equation} + \]

Notice that this hasn't computed anything yet: D is a lazy operator because it lets us symbolically represent “The derivative of $z$ with respect to $t$”, which is useful for example when representing our favorite thing in the world, differential equations. However, if we want to expand the derivative operators, we'd use expand_derivatives:

expand_derivatives(D(z))

\[ \begin{equation} +1 + 2 t +\end{equation} + \]

The variable, that you are taking the derivative with respect to, is accessed with:

D.x

\[ \begin{equation} +t +\end{equation} + \]

We can also have simplified functions for multivariable calculus. For example, we can compute the Jacobian of an array of expressions like:

Symbolics.jacobian([x + x*y, x^2 + y], [x, y])

\[ \begin{equation} +\left[ +\begin{array}{cc} +1 + y & x \\ +2 x & 1 \\ +\end{array} +\right] +\end{equation} + \]

and similarly we can do Hessians, gradients, and define whatever other derivatives you want.

Simplification and Substitution

Symbolics interfaces with SymbolicUtils.jl to allow for simplifying symbolic expressions. This is done simply through the simplify command:

simplify(2x + 2y)

\[ \begin{equation} +2 \left( x + y \right) +\end{equation} + \]

This can be applied to arrays by using Julia's broadcast mechanism:

B = simplify.([t + t^2 + t + t^2  2t + 4t
+               x + y + y + 2t     x^2 - x^2 + y^2])

\[ \begin{equation} +\left[ +\begin{array}{cc} +2 \left( t + t^{2} \right) & 6 t \\ +2 \left( t + y \right) + x & y^{2} \\ +\end{array} +\right] +\end{equation} + \]

We can then use substitute to change values of an expression around:

simplify.(substitute.(B, (Dict(x => y^2),)))

\[ \begin{equation} +\left[ +\begin{array}{cc} +2 \left( t + t^{2} \right) & 6 t \\ +2 \left( t + y \right) + y^{2} & y^{2} \\ +\end{array} +\right] +\end{equation} + \]

and we can use this to interactively evaluate expressions without generating and compiling Julia functions:

V = substitute.(B, (Dict(x => 2.0, y => 3.0, t => 4.0),))

\[ \begin{equation} +\left[ +\begin{array}{cc} +40 & 24 \\ +16 & 9 \\ +\end{array} +\right] +\end{equation} + \]

Where we can reference the values via:

Symbolics.value.(V)
2×2 Matrix{Float64}:
+ 40.0  24.0
+ 16.0   9.0

Non-Interactive Development

Note that the macros are for the high-level case where you're doing symbolic computation on your own code. If you want to do symbolic computation on someone else's code, like in a macro, you may not want to do @variables x because you might want the name “x” to come from the user's code. For these cases, you can use the interpolation operator to interpolate the runtime value of x, i.e. @variables $x. Check the documentation of @variables for more details.

a, b, c = :runtime_symbol_value, :value_b, :value_c
(:runtime_symbol_value, :value_b, :value_c)
vars = @variables t $a $b(t) $c(t)[1:3]
4-element Vector{Any}:
+      t
+ runtime_symbol_value
+   value_b(t)
+       (value_c(t))[1:3]
(t, a, b, c)
(t, :runtime_symbol_value, :value_b, :value_c)

One could also use variable and variables. Read their documentation for more details.

If we need to use this to generate new Julia code, we can simply convert the output to an Expr:

Symbolics.toexpr(x + y^2)
:((+)(x, (^)(y, 2)))

Syms and callable Syms

In the definition

@variables t x(t) y(t)

\[ \begin{equation} +\left[ +\begin{array}{c} +t \\ +x\left( t \right) \\ +y\left( t \right) \\ +\end{array} +\right] +\end{equation} + \]

t is of type Sym{Real}, but the name x refers to an object that represents the Term x(t). The operation of this expression is itself the object Sym{FnType{Tuple{Real}, Real}}(:x). The type Sym{FnType{...}} represents a callable object. In this case specifically, it's a function that takes 1 Real argument (noted by Tuple{Real}) and returns a Real result. You can call such a callable Sym with either a number or a symbolic expression of a permissible type.

This expression also defines t as an independent variable, while x(t) and y(t) are dependent variables. This is accounted for in differentiation:

z = x + y*t
+expand_derivatives(D(z))

\[ \begin{equation} +y\left( t \right) + \frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} + t \frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} +\end{equation} + \]

Since x and y are time-dependent, they are not automatically eliminated from the expression, and thus the D(x) and D(y) operations still exist in the expanded derivatives for correctness.

We can also define unrestricted functions:

@variables g(..)
1-element Vector{Symbolics.CallWithMetadata{SymbolicUtils.FnType{Tuple, Real}, Base.ImmutableDict{DataType, Any}}}:
+ g⋆

Here, g is a variable that is a function of other variables. Any time that we reference g we have to utilize it as a function:

z = g(x) + g(y)

\[ \begin{equation} +g\left( y\left( t \right) \right) + g\left( x\left( t \right) \right) +\end{equation} + \]

Registering Functions

One of the benefits of a one-language Julia symbolic stack is that the primitives are all written in Julia, and therefore it's trivially extendible from Julia itself. By default, new functions are traced to the primitives and the symbolic expressions are written on the primitives. However, we can expand the allowed primitives by registering new functions. For example, let's register a new function h:

h(x, y) = x^2 + y
+@register_symbolic h(x, y)

When we use h(x, y), it is a symbolic expression and doesn't expand:

h(x, y) + y^2

\[ \begin{equation} +h\left( x\left( t \right), y\left( t \right) \right) + \left( y\left( t \right) \right)^{2} +\end{equation} + \]

To use it with the differentiation system, we need to register its derivatives. We would do it like this:

# Derivative w.r.t. the first argument
+Symbolics.derivative(::typeof(h), args::NTuple{2,Any}, ::Val{1}) = 2args[1]
+# Derivative w.r.t. the second argument
+Symbolics.derivative(::typeof(h), args::NTuple{2,Any}, ::Val{2}) = 1

and now it works with the rest of the system:

Symbolics.derivative(h(x, y) + y^2, x)

\[ \begin{equation} +2 x\left( t \right) +\end{equation} + \]

Symbolics.derivative(h(x, y) + y^2, y)

\[ \begin{equation} +1 + 2 y\left( t \right) +\end{equation} + \]

Note

@register_symbolic only allows for scalar outputs. If full array functions are needed, then see @register_array_symbolic for registering functions of symbolic arrays.

Building Functions

The function for building functions from symbolic expressions is the aptly-named build_function. The first argument is the symbolic expression or the array of symbolic expressions to compile, and the trailing arguments are the arguments for the function. For example:

to_compute = [x^2 + y, y^2 + x]
+f_expr = build_function(to_compute, [x, y])
+Base.remove_linenums!.(f_expr)
(:(function (ˍ₋arg1,)
+      begin
+          begin
+              (SymbolicUtils.Code.create_array)(typeof(ˍ₋arg1), nothing, Val{1}(), Val{(2,)}(), (+)(ˍ₋arg1[2], (^)(ˍ₋arg1[1], 2)), (+)(ˍ₋arg1[1], (^)(ˍ₋arg1[2], 2)))
+          end
+      end
+  end), :(function (ˍ₋out, ˍ₋arg1)
+      begin
+          begin
+              #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:422 =# @inbounds begin
+                      ˍ₋out[1] = (+)(ˍ₋arg1[2], (^)(ˍ₋arg1[1], 2))
+                      ˍ₋out[2] = (+)(ˍ₋arg1[1], (^)(ˍ₋arg1[2], 2))
+                      nothing
+                  end
+          end
+      end
+  end))

gives back two codes. The first is a function f([x, y]) that computes and builds an output vector [x^2 + y, y^2 + x]. Because this tool was made to be used by all the cool kids writing fast Julia codes, it is specialized to Julia and supports features like StaticArrays. For example:

using StaticArrays
+myf = eval(f_expr[1])
+myf(SA[2.0, 3.0])
2-element StaticArraysCore.SVector{2, Float64} with indices SOneTo(2):
+  7.0
+ 11.0

The second function is an in-place non-allocating mutating function which mutates its first value. Thus, we'd use it like the following:

myf! = eval(f_expr[2])
+out = zeros(2)
+myf!(out, [2.0, 3.0])
+out
2-element Vector{Float64}:
+  7.0
+ 11.0

To save the symbolic calculations for later, we can take this expression and save it out to a file:

write("function.jl", string(f_expr[2]))
373

Note that if we need to avoid eval, for example to avoid world-age issues, one could do expression = Val{false}:

Base.remove_linenums!(build_function(to_compute, [x, y], expression=Val{false})[1])
RuntimeGeneratedFunction(#=in Symbolics=#, #=using Symbolics=#, :((ˍ₋arg1,)->begin
+          #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:373 =#
+          #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:374 =#
+          #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:375 =#
+          begin
+              begin
+                  #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:468 =#
+                  (SymbolicUtils.Code.create_array)(typeof(ˍ₋arg1), nothing, Val{1}(), Val{(2,)}(), (+)(ˍ₋arg1[2], (^)(ˍ₋arg1[1], 2)), (+)(ˍ₋arg1[1], (^)(ˍ₋arg1[2], 2)))
+              end
+          end
+      end))

which will use RuntimeGeneratedFunctions.jl to build Julia functions which avoid world-age issues.

Building Non-Allocating Parallel Functions for Sparse Matrices

Now let's show off a little bit. build_function is kind of like if lambdify ate its spinach. To show this, let's build a non-allocating function that fills sparse matrices in a multithreaded manner. To do this, we just have to represent the operation that we're turning into a function via a sparse matrix. For example:

using LinearAlgebra
+N = 8
+A = sparse(Tridiagonal([x^i for i in 1:N-1], [x^i * y^(8-i) for i in 1:N], [y^i for i in 1:N-1]))
+show(A)
sparse([1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8], [1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8], Num[x(t)*(y(t)^7), x(t), y(t), (x(t)^2)*(y(t)^6), x(t)^2, y(t)^2, (x(t)^3)*(y(t)^5), x(t)^3, y(t)^3, (x(t)^4)*(y(t)^4), x(t)^4, y(t)^4, (x(t)^5)*(y(t)^3), x(t)^5, y(t)^5, (x(t)^6)*(y(t)^2), x(t)^6, y(t)^6, (x(t)^7)*y(t), x(t)^7, y(t)^7, x(t)^8], 8, 8)

Now we call build_function:

Base.remove_linenums!(build_function(A,[x,y],parallel=Symbolics.MultithreadedForm())[2])
:(function (ˍ₋out, ˍ₋arg1)
+      begin
+          begin
+              var"##out#242" = ˍ₋out.nzval
+              begin
+                  (Symbolics.var"#noop#429"())(map(fetch, (begin
+                                  let
+                                      task = Base.Task((Symbolics.Funcall)(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##out#242"), :ˍ₋arg1), Symbolics.var"#_RGF_ModTag", Symbolics.var"#_RGF_ModTag", (0xebac6064, 0x4dc584eb, 0xb0714722, 0x9618d69f, 0x28be9532), Nothing}(nothing), (var"##out#242", ˍ₋arg1)))
+                                      task.sticky = false
+                                      Base.schedule(task)
+                                      task
+                                  end
+                              end, begin
+                                  let
+                                      task = Base.Task((Symbolics.Funcall)(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##out#242"), :ˍ₋arg1), Symbolics.var"#_RGF_ModTag", Symbolics.var"#_RGF_ModTag", (0xcda6261e, 0xb9bc59a7, 0x66b0cbcc, 0x5e505e55, 0x0160f67b), Nothing}(nothing), (var"##out#242", ˍ₋arg1)))
+                                      task.sticky = false
+                                      Base.schedule(task)
+                                      task
+                                  end
+                              end))...)
+              end
+          end
+      end
+  end)

Let's unpack what that's doing. It's using A.nzval to linearly index through the sparse matrix, avoiding the A[i,j] form because that is a more expensive way to index a sparse matrix if you know exactly the order that the data is stored. Then, it's splitting up the calculation into Julia threads, so they can be operated on in parallel. It synchronizes after spawning all the tasks, so the computation is ensured to be complete before moving on. And it does this with all in-place operations to the original matrix instead of generating arrays. This is the fanciest way you could fill a sparse matrix, and Symbolics makes this dead simple.

(Note: this example may be slower with multithreading due to the thread spawning overhead, but the full version was not included in the documentation for brevity. It will be the faster version if N is sufficiently large!)

Importantly, when using the in-place functions generated by build_function, it is important that the mutating argument matches the sparse structure of the original function, as demonstrated below.

using Symbolics, SparseArrays, LinearAlgebra
+
+N = 10
+_S = sprand(N, N, 0.1)
+_Q = sprand(N, N, 0.1)
+
+F(z) = [ # some complicated sparse amenable function
+    _S * z
+    _Q * z.^2
+]
+
+Symbolics.@variables z[1:N]
+
+sj = Symbolics.sparsejacobian(F(z), z) # sparse jacobian
+
+f_expr = build_function(sj, z)
+
+rows, cols, _ = findnz(sj)
+out = sparse(rows, cols, zeros(length(cols)), size(sj)...) # pre-allocate, and correct structure
+myf = eval(last(f_expr))
+myf(out, rand(N)) # note that out matches the sparsity structure of sj
+out
20×10 SparseArrays.SparseMatrixCSC{Float64, Int64} with 19 stored entries:
+⎡⠤⠀⠀⠄⠐⎤
+⎢⡀⠀⠠⠀⡀⎥
+⎢⠀⠂⠐⠀⠘⎥
+⎢⠄⠀⠀⠈⠰⎥
+⎣⠀⠀⠈⠐⠆⎦
diff --git a/v5.27.1/index.html b/v5.27.1/index.html new file mode 100644 index 000000000..6fed7fe38 --- /dev/null +++ b/v5.27.1/index.html @@ -0,0 +1,342 @@ + +Home · Symbolics.jl
GitHub

Symbolics.jl

Symbolics.jl is a fast and modern Computer Algebra System (CAS) for a fast and modern programming language (Julia). The goal is to have a high-performance and parallelized symbolic algebra system that is directly extendable in the same language as that of the users.

Installation

To install Symbolics.jl, use the Julia package manager:

using Pkg
+Pkg.add("Symbolics")

Citation

If you use Symbolics.jl, please cite this paper

@article{gowda2021high,
+  title={High-performance symbolic-numerics via multiple dispatch},
+  author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher},
+  journal={arXiv preprint arXiv:2105.03949},
+  year={2021}
+}

Feature Summary

Because Symbolics.jl is built into the Julia language and works with its dispatches, generic functions in Base Julia will work with symbolic expressions! Make matrices of symbolic expressions and multiply them: it will just work. Take the LU-factorization. Etc. Thus, see the Julia Documentation for a large list of functionality available in Symbolics.jl.

A general list of the features is:

  • Symbolic arithmetic with type information and multiple dispatch
  • Symbolic polynomials and trigonometric functions
  • Pattern matching, simplification and substitution
  • Differentiation
  • Symbolic linear algebra (factorizations, inversion, determinants, eigencomputations, etc.)
  • Discrete math (representations of summations, products, binomial coefficients, etc.)
  • Logical and Boolean expressions
  • Symbolic equation solving and conversion to arbitrary precision
  • Support for non-standard algebras (non-commutative symbols and customizable rulesets)
  • Special functions (list provided by SpecialFunctions.jl)
  • Automatic conversion of Julia code to symbolic code
  • Generation of (high performance and parallel) functions from symbolic expressions
  • Fast automated sparsity detection and generation of sparse Jacobians and Hessians

and much more.

Extension Packages

Below is a list of known extension packages. If you would like your package to be listed here, feel free to open a pull request!

  • ModelingToolkit.jl: Symbolic representations of common numerical systems
    • Ordinary differential equations
    • Stochastic differential equations
    • Partial differential equations
    • Nonlinear systems
    • Optimization problems
    • Optimal Control
    • Causal and acausal modeling (Simulink/Modelica)
    • Automated model transformation, simplification, and composition
  • Catalyst.jl: Symbolic representations of chemical reactions
    • Symbolically build and represent large systems of chemical reactions
    • Generate code for ODEs, SDEs, continuous-time Markov Chains, and more
    • Simulate the models using the SciML ecosystem with O(1) Gillespie methods
  • DataDrivenDiffEq.jl: Automatic identification of equations from data
    • Automated construction of ODEs and DAEs from data
    • Representations of Koopman operators and Dynamic Mode Decomposition (DMD)
  • SymbolicRegression.jl: Distributed High-Performance symbolic regression
    • Parallelized generic algorithms for finding equations from data
    • Pareto frontier-based scoring
  • ReversePropagation.jl: Source-to-source reverse mode automatic differentiation
    • Automated tracing of code and construction of backpropagation equations
    • Composes with symbolic transformation and simplification functionality

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
Status `~/work/Symbolics.jl/Symbolics.jl/docs/Project.toml`
+  [6e4b80f9] BenchmarkTools v1.5.0
+  [e30172f5] Documenter v1.3.0
+  [23fbe1c1] Latexify v0.16.2
+  [1dea7af3] OrdinaryDiffEq v6.74.1
+  [91a5bcdd] Plots v1.40.2
+  [90137ffa] StaticArrays v1.9.3
+  [d1185830] SymbolicUtils v1.5.1
+  [0c5d862f] Symbolics v5.27.1 `~/work/Symbolics.jl/Symbolics.jl`
and using this machine and Julia version.
Julia Version 1.10.2
+Commit bd47eca2c8a (2024-03-01 10:14 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LIBM: libopenlibm
+  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  JULIA_DEBUG = Documenter
A more complete overview of all dependencies and their versions is also provided.
Status `~/work/Symbolics.jl/Symbolics.jl/docs/Manifest.toml`
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You can also download the manifest file and the project file.

diff --git a/v5.27.1/manual/arrays/index.html b/v5.27.1/manual/arrays/index.html new file mode 100644 index 000000000..b37749a65 --- /dev/null +++ b/v5.27.1/manual/arrays/index.html @@ -0,0 +1,67 @@ + +Symbolic Arrays · Symbolics.jl
GitHub

Symbolic Arrays

Symbolic Arrays vs Arrays of Symbolic Expressions

Symbolics.jl contains two forms for handling symbolic arrays:

  1. Arrays of symbolic expressions: these are Julia arrays with Symbolics.jl objects in them.
  2. Symbolic Arrays: these are symbolic (O(1)) representations of arrays.

Arrays of symbolic expressions are simply Symbolics.jl objects put into Julia arrays. For example:

using Symbolics
+@variables x y
+u = [x,y]

\[ \begin{equation} +\left[ +\begin{array}{c} +x \\ +y \\ +\end{array} +\right] +\end{equation} + \]

is a vector of two symbolic variables. As shorthand,

u2 = Symbolics.variables(:x, 1:3, 3:6)

\[ \begin{equation} +\left[ +\begin{array}{cccc} +x_{1}ˏ_3 & x_{1}ˏ_4 & x_{1}ˏ_5 & x_{1}ˏ_6 \\ +x_{2}ˏ_3 & x_{2}ˏ_4 & x_{2}ˏ_5 & x_{2}ˏ_6 \\ +x_{3}ˏ_3 & x_{3}ˏ_4 & x_{3}ˏ_5 & x_{3}ˏ_6 \\ +\end{array} +\right] +\end{equation} + \]

creates a Julia matrix of symbolic variables. Indexing u or u2 gives symbolic values which act as a normal scalar symbolic value. This form these uses Julia's array functionality and performs symbolic operations on the scalar values.

On the otherhand, Julia's symbolic array form is an O(1) representation of the whole array.

@variables A[1:5, 1:3]
1-element Vector{Symbolics.Arr{Num, 2}}:
+ A[1:5,1:3]

When using this form, A[1,1] is not a symbolic variable but a symbolic expression for indexing the variable A. This representation holds linear algebra expressions in a non-expanded form. For example:

@variables B[1:3, 1:3]
+A * B

\[ \begin{equation} +A B +\end{equation} + \]

in comparison to:

a = Symbolics.variables(:a, 1:5, 1:3)
+b = Symbolics.variables(:b, 1:3, 1:3)
+a * b

\[ \begin{equation} +\left[ +\begin{array}{ccc} +a_{1}ˏ_1 b_{1}ˏ_1 + a_{1}ˏ_2 b_{2}ˏ_1 + a_{1}ˏ_3 b_{3}ˏ_1 & a_{1}ˏ_1 b_{1}ˏ_2 + a_{1}ˏ_2 b_{2}ˏ_2 + a_{1}ˏ_3 b_{3}ˏ_2 & a_{1}ˏ_1 b_{1}ˏ_3 + a_{1}ˏ_2 b_{2}ˏ_3 + a_{1}ˏ_3 b_{3}ˏ_3 \\ +a_{2}ˏ_1 b_{1}ˏ_1 + a_{2}ˏ_2 b_{2}ˏ_1 + a_{2}ˏ_3 b_{3}ˏ_1 & a_{2}ˏ_1 b_{1}ˏ_2 + a_{2}ˏ_2 b_{2}ˏ_2 + a_{2}ˏ_3 b_{3}ˏ_2 & a_{2}ˏ_1 b_{1}ˏ_3 + a_{2}ˏ_2 b_{2}ˏ_3 + a_{2}ˏ_3 b_{3}ˏ_3 \\ +a_{3}ˏ_1 b_{1}ˏ_1 + a_{3}ˏ_2 b_{2}ˏ_1 + a_{3}ˏ_3 b_{3}ˏ_1 & a_{3}ˏ_1 b_{1}ˏ_2 + a_{3}ˏ_2 b_{2}ˏ_2 + a_{3}ˏ_3 b_{3}ˏ_2 & a_{3}ˏ_1 b_{1}ˏ_3 + a_{3}ˏ_2 b_{2}ˏ_3 + a_{3}ˏ_3 b_{3}ˏ_3 \\ +a_{4}ˏ_1 b_{1}ˏ_1 + a_{4}ˏ_2 b_{2}ˏ_1 + a_{4}ˏ_3 b_{3}ˏ_1 & a_{4}ˏ_1 b_{1}ˏ_2 + a_{4}ˏ_2 b_{2}ˏ_2 + a_{4}ˏ_3 b_{3}ˏ_2 & a_{4}ˏ_1 b_{1}ˏ_3 + a_{4}ˏ_2 b_{2}ˏ_3 + a_{4}ˏ_3 b_{3}ˏ_3 \\ +a_{5}ˏ_1 b_{1}ˏ_1 + a_{5}ˏ_2 b_{2}ˏ_1 + a_{5}ˏ_3 b_{3}ˏ_1 & a_{5}ˏ_1 b_{1}ˏ_2 + a_{5}ˏ_2 b_{2}ˏ_2 + a_{5}ˏ_3 b_{3}ˏ_2 & a_{5}ˏ_1 b_{1}ˏ_3 + a_{5}ˏ_2 b_{2}ˏ_3 + a_{5}ˏ_3 b_{3}ˏ_3 \\ +\end{array} +\right] +\end{equation} + \]

This makes the symbolic array form much more efficient, but requires that the expressions uses things with registered symbolic array functions which currently has much lower coverage. Also, there are many fallbacks for which arrays of symbolics which makes this approach more accessible but with larger expressions.

We recommend defaulting to arrays of symbolics unless you need the expression symplifications of the symbolic array approach.

Using Symbolic Arrays

Symbolic array-valued expressions (symbolic arrays) are supported by Symbolics. Symbolic array expressions propagate useful metadata that depends on input arrays: array dimension, element type and shape.

You can create a symbolic array variable with the following syntax:

using Symbolics
+@variables A[1:5, 1:3] b[1:3]
2-element Vector{Symbolics.Arr{Num}}:
+ A[1:5,1:3]
+ b[1:3]

Here, A is a symbolic matrix of size (5, 3) and b is a symbolic vector of length 3.

size(A)
(5, 3)
size(b)
(3,)
ndims(A)
2
ndims(b)
1
eltype(A)
Real
eltype(b)
Real

Array operations

Operations on symbolic arrays return symbolic array expressions:

c = A * b

\[ \begin{equation} +A b +\end{equation} + \]

size(c)
(5,)
eltype(c)
Real

Adjoints, matrix-matrix, and matrix-vector multiplications are supported. Dot product returns a scalar-valued expression:

b'b

\[ \begin{equation} +adjoint(b) * b_1 +\end{equation} + \]

size(b'b)
()

Outer product returns a matrix:

b * b'

\[ \begin{equation} +b \mathrm{adjoint}\left( b \right) +\end{equation} + \]

size(b*b')
(3, 3)

Broadcast, map and reduce

A .* b'

\[ \begin{equation} +\mathrm{broadcast}\left( *, A, \mathrm{adjoint}\left( b \right) \right) +\end{equation} + \]

map(asin, (A*b))

\[ \begin{equation} +\mathrm{map}\left( \arcsin, A b \right) +\end{equation} + \]

#sum(A) #latexify not working
typeof(sum(A))
Num
typeof(sum(A, dims=2))
Symbolics.Arr{Num, 2}

Indexing and delayed computation

Indexing array expressions is fairly flexible in Symbolics. Let's go through all the possible ways to index arrays.

Scalar indexing and scalarization

AAt = A*A'

\[ \begin{equation} +A \mathrm{adjoint}\left( A \right) +\end{equation} + \]

AAt[2,3]

\[ \begin{equation} +A * adjoint(A)_{2}ˏ_3 +\end{equation} + \]

Here we indexed for the element (2,3), but we got back a symbolic indexing expression. You may want to force the element to be computed in terms of the elements of A. This can be done, using the scalarize function.

Symbolics.scalarize(AAt[2,3])

\[ \begin{equation} +A_{2}ˏ_1 A_{3}ˏ_1 + A_{2}ˏ_2 A_{3}ˏ_2 + A_{2}ˏ_3 A_{3}ˏ_3 +\end{equation} + \]

@syms i::Int j::Int
+Symbolics.scalarize(AAt[i,j])

In general, any scalar expression which is derived from array expressions can be scalarized.

#sum(A[:,1]) + sum(A[2,:])#latexify not working
Symbolics.scalarize(sum(A[:,1]) + sum(A[2,:]))
diff --git a/v5.27.1/manual/build_function/index.html b/v5.27.1/manual/build_function/index.html new file mode 100644 index 000000000..affb4df08 --- /dev/null +++ b/v5.27.1/manual/build_function/index.html @@ -0,0 +1,39 @@ + +Function Building and Compilation (build_function) · Symbolics.jl
GitHub

Function Building and Compilation (build_function)

At any time, callable functions can be generated from Symbolics IR by using Symbolics.toexpr. This performs some cleaning to return an expression without extraneous pieces that commonly matches expressions one would write in functions like those for differential equation solvers and optimization libraries. These functions can be automatically parallelized and specialize on Julia types like static arrays and sparse matrices.

The core compilation process of Symbolics IR is build_function. build_function takes an operation or an AbstractArray of operations and generates a compilable version of the model for numerical solvers. The form of this output is dependent on the target. By default, the target outputs Julia code, but other formats, such as C, Stan, and MATLAB are available. These can be generated as expressions which can then be evaluated into a callable function, or the compilers for the respective targets can be invoked to directly give back the function handle.

build_function

Symbolics.build_function — Function

build_function

Generates a numerically-usable function from a Symbolics Num.

build_function(ex, args...;
+               expression = Val{true},
+               target = JuliaTarget(),
+               parallel=nothing,
+               kwargs...)

Arguments:

  • ex: The Num to compile
  • args: The arguments of the function
  • expression: Whether to generate code or whether to generate the compiled form. By default, expression = Val{true}, which means that the code for the function is returned. If Val{false}, then the returned value is compiled.

Keyword Arguments:

  • target: The output target of the compilation process. Possible options are:
    • JuliaTarget: Generates a Julia function
    • CTarget: Generates a C function
    • StanTarget: Generates a function for compiling with the Stan probabilistic programming language
    • MATLABTarget: Generates an anonymous function for use in MATLAB and Octave environments
  • parallel: The kind of parallelism to use in the generated function. Defaults to SerialForm(), i.e. no parallelism, if ex is a single expression or an array containing <= 1500 non-zero expressions. If ex is an array of > 1500 non-zero expressions, then ShardedForm(80, 4) is used. See below for more on ShardedForm. Note that the parallel forms are not exported and thus need to be chosen like Symbolics.SerialForm(). The choices are:
    • SerialForm(): Serial execution.
    • ShardedForm(cutoff, ncalls): splits the output function into sub-functions which contain at most cutoff number of output rhss. These sub-functions are called by the top-level function that buildfunction returns. This helps in reducing the compile time of the generated function.
    • MultithreadedForm(): Multithreaded execution with a static split, evenly splitting the number of expressions per thread.
  • fname: Used by some targets for the name of the function in the target space.

Note that not all build targets support the full compilation interface. Check the individual target documentation for details.

source

Target-Specific Definitions

Symbolics._build_function — Method
_build_function(target::JuliaTarget, rhss::AbstractArray, args...;
+                   conv=toexpr,
+                   expression = Val{true},
+                   expression_module = @__MODULE__(),
+                   checkbounds = false,
+                   postprocess_fbody=ex -> ex,
+                   linenumbers = false,
+                   outputidxs=nothing,
+                   skipzeros = false,
+                   force_SA = false,
+                   wrap_code = (nothing, nothing),
+                   fillzeros = skipzeros && !(rhss isa SparseMatrixCSC),
+                   states = LazyState(),
+                   iip_config = (true, true),
+                   parallel=nothing, cse = false, kwargs...)

Build function target: JuliaTarget

function _build_function(target::JuliaTarget, rhss, args...;
+                         conv = toexpr, 
+                         expression = Val{true},
+                         checkbounds = false,
+                         linenumbers = false,
+                         headerfun = addheader, outputidxs=nothing,
+                         convert_oop = true, force_SA = false,
+                         skipzeros = outputidxs===nothing,
+                         fillzeros = skipzeros && !(typeof(rhss)<:SparseMatrixCSC),
+                         parallel=SerialForm(), kwargs...)

Generates a Julia function which can then be utilized for further evaluations. If expression=Val{false}, the return is a Julia function which utilizes RuntimeGeneratedFunctions.jl to be free of world-age issues.

If the rhss is a scalar, the generated function is a function with a scalar output. Otherwise, if it's an AbstractArray, the output is two functions, one for out-of-place AbstractArray output and a second which is a mutating function. The outputted functions match the given argument order, i.e., f(u,p,args...) for the out-of-place and scalar functions and f!(du,u,p,args..) for the in-place version.

Special Keyword Arguments:

  • parallel: The kind of parallelism to use in the generated function. Defaults to SerialForm(), i.e. no parallelism. Note that the parallel forms are not exported and thus need to be chosen like Symbolics.SerialForm(). The choices are:
    • SerialForm(): Serial execution.
    • ShardedForm(cutoff, ncalls): splits the output function into sub-functions which contain at most cutoff number of output rhss. These sub-functions are called by the top-level function that buildfunction returns.
    • MultithreadedForm(): Multithreaded execution with a static split, evenly splitting the number of expressions per thread.
  • conv: The conversion function of symbolic types to Expr. By default, this uses the toexpr function.
  • checkbounds: For whether to enable bounds checking inside the generated function. Defaults to false, meaning that @inbounds is applied.
  • linenumbers: Determines whether the generated function expression retains the line numbers. Defaults to true.
  • convert_oop: Determines whether the OOP version should try to convert the output to match the type of the first input. This is useful for cases like LabelledArrays or other array types that carry extra information. Defaults to true.
  • force_SA: Forces the output of the OOP version to be a StaticArray. Defaults to false, and outputs a static array when the first argument is a static array.
  • skipzeros: Whether to skip filling zeros in the in-place version if the filling function is 0.
  • fillzeros: Whether to perform fill(out,0) before the calculations to ensure safety with skipzeros.
source
Symbolics._build_function — Method

Build function target: CTarget

function _build_function(target::CTarget, eqs::Array{<:Equation}, args...;
+                         conv = toexpr, expression = Val{true},
+                         fname = :diffeqf,
+                         lhsname=:du,rhsnames=[Symbol("RHS$i") for i in 1:length(args)],
+                         libpath=tempname(),compiler=:gcc)

This builds an in-place C function. Only works on arrays of equations. If expression == Val{false}, then this builds a function in C, compiles it, and returns a lambda to that compiled function. These special keyword arguments control the compilation:

  • libpath: the path to store the binary. Defaults to a temporary path.
  • compiler: which C compiler to use. Defaults to :gcc, which is currently the only available option.
source
Symbolics._build_function — Method

Build function target: StanTarget

function _build_function(target::StanTarget, eqs::Array{<:Equation}, vs, ps, iv;
+                         conv = toexpr, expression = Val{true},
+                         fname = :diffeqf, lhsname=:internal_var___du,
+                         rhsnames=[:internal_var___u,:internal_var___p,:internal_var___t])

This builds an in-place Stan function compatible with the Stan differential equation solvers. Unlike other build targets, this one requires (vs, ps, iv) as the function arguments. Only allowed on arrays of equations.

source
Symbolics._build_function — Method

Build function target: MATLABTarget

function _build_function(target::MATLABTarget, eqs::Array{<:Equation}, args...;
+                         conv = toexpr, expression = Val{true},
+                         lhsname=:internal_var___du,
+                         rhsnames=[:internal_var___u,:internal_var___p,:internal_var___t])

This builds an out of place anonymous function @(t,rhsnames[1]) to be used in MATLAB. Compatible with the MATLAB differential equation solvers. Only allowed on expressions, and arrays of expressions.

source

Limitations

build_function

diff --git a/v5.27.1/manual/derivatives/index.html b/v5.27.1/manual/derivatives/index.html new file mode 100644 index 000000000..1a922106a --- /dev/null +++ b/v5.27.1/manual/derivatives/index.html @@ -0,0 +1,39 @@ + +Derivatives and Differentials · Symbolics.jl
GitHub

Derivatives and Differentials

A Differential(op) is a partial derivative with respect to op, which can then be applied to some other operations. For example, D=Differential(t) is what would commonly be referred to as d/dt, which can then be applied to other operations using its function call, so D(x+y) is d(x+y)/dt.

By default, the derivatives are left unexpanded to capture the symbolic representation of the differential equation. If the user would like to expand out all the differentials, the expand_derivatives function eliminates all the differentials down to basic one-variable expressions.

Symbolics.Differential — Type
struct Differential <: Symbolics.Operator

Represents a differential operator.

Fields

  • x: The variable or expression to differentiate with respect to.

Examples

julia> using Symbolics
+
+julia> @variables x y;
+
+julia> D = Differential(x)
+(D'~x)
+
+julia> D(y) # Differentiate y wrt. x
+(D'~x)(y)
+
+julia> Dx = Differential(x) * Differential(y) # d^2/dxy operator
+(D'~x(t)) ∘ (D'~y(t))
+
+julia> D3 = Differential(x)^3 # 3rd order differential operator
+(D'~x(t)) ∘ (D'~x(t)) ∘ (D'~x(t))
source
Symbolics.expand_derivatives — Function
expand_derivatives(O; ...)
+expand_derivatives(O, simplify; occurrences)
+

TODO

Examples


+julia> @variables x y z k;
+
+julia> f=k*(abs(x-y)/y-z)^2
+k*((abs(x - y) / y - z)^2)
+
+julia> Dx=Differential(x) # Differentiate wrt x
+(::Differential) (generic function with 2 methods)
+
+julia> dfx=expand_derivatives(Dx(f))
+(k*((2abs(x - y)) / y - 2z)*IfElse.ifelse(signbit(x - y), -1, 1)) / y
source
Missing docstring.

Missing docstring for is_derivative. Check Documenter's build log for details.

Note

For symbolic differentiation, all registered functions in the symbolic expression need a registered derivative. For more information, see the function registration page.

High-Level Differentiation Functions

The following functions are not exported and thus must be accessed in a namespaced way, i.e. Symbolics.jacobian.

Symbolics.derivative — Function
derivative(O, var; simplify)
+

A helper function for computing the derivative of the expression O with respect to var.

source
Symbolics.jacobian — Function
jacobian(ops, vars; simplify, scalarize)
+

A helper function for computing the Jacobian of an array of expressions with respect to an array of variable expressions.

source
Symbolics.sparsejacobian — Function
sparsejacobian(ops, vars; simplify)
+

A helper function for computing the sparse Jacobian of an array of expressions with respect to an array of variable expressions.

source
Symbolics.sparsejacobian_vals — Function
sparsejacobian_vals(ops, vars, I, J; simplify)
+

A helper function for computing the values of the sparse Jacobian of an array of expressions with respect to an array of variable expressions given the sparsity structure.

source
Symbolics.gradient — Function
gradient(O, vars; simplify)
+

A helper function for computing the gradient of the expression O with respect to an array of variable expressions.

source
Symbolics.hessian — Function
hessian(O, vars; simplify)
+

A helper function for computing the Hessian of the expression O with respect to an array of variable expressions.

source
Symbolics.sparsehessian — Function
sparsehessian(op, vars; simplify, full)
+

A helper function for computing the sparse Hessian of an expression with respect to an array of variable expressions.

source
Symbolics.sparsehessian_vals — Function
sparsehessian_vals(op, vars, I, J; simplify)
+

A helper function for computing the values of the sparse Hessian of an expression with respect to an array of variable expressions given the sparsity structure.

source

Adding Analytical Derivatives

There are many derivatives pre-defined by DiffRules.jl. For example,

using Symbolics
+@variables x y z
+f(x,y,z) = x^2 + sin(x+y) - z
f (generic function with 1 method)

f automatically has the derivatives defined via the tracing mechanism. It will do this by directly building the internals of your function and differentiating that.

However, often you may want to define your own derivatives so that way automatic Jacobian etc. calculations can utilize this information. This can allow for more succinct versions of the derivatives to be calculated to scale to larger systems. You can define derivatives for your function via the dispatch:

# `N` arguments are accepted by the relevant method of `my_function`
+Symbolics.derivative(::typeof(my_function), args::NTuple{N,Any}, ::Val{i})

where i means that it's the derivative with respect to the ith argument. args is the array of arguments, so, for example, if your function is f(x,t), then args = [x,t]. You should return an Term for the derivative of your function.

For example, sin(t)'s derivative (by t) is given by the following:

Symbolics.derivative(::typeof(sin), args::NTuple{1,Any}, ::Val{1}) = cos(args[1])
diff --git a/v5.27.1/manual/expression_manipulation/index.html b/v5.27.1/manual/expression_manipulation/index.html new file mode 100644 index 000000000..00b08b6b2 --- /dev/null +++ b/v5.27.1/manual/expression_manipulation/index.html @@ -0,0 +1,80 @@ + +Expression Manipulation · Symbolics.jl
GitHub

Expression Manipulation

Symbolics.jl provides functionality for easily manipulating expressions. Most of the functionality comes by the expression objects obeying the standard mathematical semantics. For example, if one has A a matrix of symbolic expressions wrapped in Num, then A^2 calculates the expressions for the squared matrix. It is thus encouraged to use standard Julia for performing many of the manipulation on the IR. For example, calculating the sparse form of the matrix via sparse(A) is valid, legible, and easily understandable to all Julia programmers.

Functionality Inherited From SymbolicUtils.jl

SymbolicUtils.substitute — Function
substitute(expr, s)

Performs the substitution on expr according to rule(s) s.

Examples

julia> @variables t x y z(t)
+4-element Vector{Num}:
+    t
+    x
+    y
+ z(t)
+julia> ex = x + y + sin(z)
+(x + y) + sin(z(t))
+julia> substitute(ex, Dict([x => z, sin(z) => z^2]))
+(z(t) + y) + (z(t) ^ 2)
source
SymbolicUtils.simplify — Function
simplify(x; expand=false,
+            threaded=false,
+            thread_subtree_cutoff=100,
+            rewriter=nothing)

Simplify an expression (x) by applying rewriter until there are no changes. expand=true applies expand in the beginning of each fixpoint iteration.

By default, simplify will assume denominators are not zero and allow cancellation in fractions. Pass simplify_fractions=false to prevent this.

source

Documentation for rewriter can be found here, using the @rule macro or the @acrule macro from SymbolicUtils.jl.

Additional Manipulation Functions

Other additional manipulation functions are given below.

Symbolics.get_variables — Function
get_variables(O) -> Vector{BasicSymbolic}

Returns the variables in the expression. Note that the returned variables are not wrapped in the Num type.

Examples

julia> @variables t x y z(t)
+4-element Vector{Num}:
+    t
+    x
+    y
+ z(t)
+
+julia> ex = x + y + sin(z)
+(x + y) + sin(z(t))
+
+julia> Symbolics.get_variables(ex)
+3-element Vector{Any}:
+ x
+ y
+ z(t)
source
Symbolics.tosymbol — Function
tosymbol(x::Union{Num,Symbolic}; states=nothing, escape=true) -> Symbol

Convert x to a symbol. states are the states of a system, and escape means if the target has escapes like val"y(t)". If escape is false, then it will only output y instead of y(t).

Examples

julia> @variables t z(t)
+2-element Vector{Num}:
+    t
+ z(t)
+
+julia> Symbolics.tosymbol(z)
+Symbol("z(t)")
+
+julia> Symbolics.tosymbol(z; escape=false)
+:z
source
Symbolics.diff2term — Function
diff2term(x) -> Symbolic

Convert a differential variable to a Term. Note that it only takes a Term not a Num.

julia> @variables x t u(x, t) z(t)[1:2]; Dt = Differential(t); Dx = Differential(x);
+
+julia> Symbolics.diff2term(Symbolics.value(Dx(Dt(u))))
+uˍtx(x, t)
+
+julia> Symbolics.diff2term(Symbolics.value(Dt(z[1])))
+var"z(t)[1]ˍt"
source
Symbolics.solve_for — Function
solve_for(eq, var; simplify, check) -> Any
+

Solve equation(s) eqs for a set of variables vars.

Assumes length(eqs) == length(vars)

Currently only works if all equations are linear. check if the expr is linear w.r.t vars.

Examples

julia> @variables x y
+2-element Vector{Num}:
+ x
+ y
+
+julia> Symbolics.solve_for(x + y ~ 0, x)
+-y
+
+julia> Symbolics.solve_for([x + y ~ 0, x - y ~ 2], [x, y])
+2-element Vector{Float64}:
+  1.0
+ -1.0
source
Symbolics.degree — Function
degree(p, sym=nothing)

Extract the degree of p with respect to sym.

Examples

julia> @variables x;
+
+julia> Symbolics.degree(x^0)
+0
+
+julia> Symbolics.degree(x)
+1
+
+julia> Symbolics.degree(x^2)
+2
source
Symbolics.coeff — Function
coeff(p, sym=nothing)

Extract the coefficient of p with respect to sym. Note that p might need to be expanded and/or simplified with expand and/or simplify.

Examples

julia> @variables a x y;
+
+julia> Symbolics.coeff(2a, x)
+0
+
+julia> Symbolics.coeff(3x + 2y, y)
+2
+
+julia> Symbolics.coeff(x^2 + y, x^2)
+1
source
Base.replace — Function
replace(expr::Symbolic, rules...)

Walk the expression and replace subexpressions according to rules. rules could be rules constructed with @rule, a function, or a pair where the left hand side is matched with equality (using isequal) and is replaced by the right hand side.

Rules will be applied left-to-right simultaneously, so only one pattern will be applied to any subexpression, and the patterns will only be applied to the input text, not the replacements.

Set fixpoint = true to repeatedly apply rules until no change to the expression remains to be made.

source
Base.occursin — Function
occursin(c, x)

Returns true if any part of x fufills the condition given in c. c can be a function or an expression. If it is a function, returns true if x is true for any part of x. If c is an expression, returns true if x contains c.

Examples:

@syms x y
+Symbolics.occursin(x, log(x) + x + 1) # returns `true`.
+Symbolics.occursin(x, log(y) + y + 1) # returns `false`.
@variables t X(t)
+D = Differential(t)
+Symbolics.occursin(Symbolics.is_derivative, X + D(X) + D(X^2)) # returns `true`.
source
occursin(needle::Symbolic, haystack::Symbolic)

Determine whether the second argument contains the first argument. Note that this function doesn't handle associativity, commutativity, or distributivity.

source
Symbolics.filterchildren — Function

filterchildren(c, x) Returns all parts of x that fufills the condition given in c. c can be a function or an expression. If it is a function, returns everything for which the function is true. If c is an expression, returns all expressions that matches it.

Examples:

@syms x
+Symbolics.filterchildren(x, log(x) + x + 1)

returns [x, x]

@variables t X(t)
+D = Differential(t)
+Symbolics.filterchildren(Symbolics.is_derivative, X + D(X) + D(X^2))

returns [Differential(t)(X(t)^2), Differential(t)(X(t))]

source
Symbolics.fixpoint_sub — Function
fixpoint_sub(expr, dict; operator = Nothing)

Given a symbolic expression, equation or inequality expr perform the substitutions in dict recursively until the expression does not change. Substitutions that depend on one another will thus be recursively expanded. For example, fixpoint_sub(x, Dict(x => y, y => 3)) will return 3. The operator keyword can be specified to prevent substitution of expressions inside operators of the given type.

See also: fast_substitute.

source
Symbolics.fast_substitute — Function
fast_substitute(expr, dict; operator = Nothing)

Given a symbolic expression, equation or inequality expr perform the substitutions in dict. This only performs the substitutions once. For example, fast_substitute(x, Dict(x => y, y => 3)) will return y. The operator keyword can be specified to prevent substitution of expressions inside operators of the given type.

See also: fixpoint_sub.

source
diff --git a/v5.27.1/manual/faq/index.html b/v5.27.1/manual/faq/index.html new file mode 100644 index 000000000..8659f34b6 --- /dev/null +++ b/v5.27.1/manual/faq/index.html @@ -0,0 +1,19 @@ + +Frequently Asked Questions · Symbolics.jl
GitHub

Frequently Asked Questions

Limits of Symbolic Computation

Transforming my function to a symbolic equation has failed. What do I do?

If you see the error:

ERROR: TypeError: non-boolean (Num) used in boolean context

this is likely coming from an algorithm which cannot be traced into a purely symbolic algorithm. Many numerical solvers, for instance, have this property. It shows up when you're doing something like if x < tol. If x is a number, then this is true or false. If x is a symbol, then it's x < tol, so Julia just cannot know how many iterations to do and throws an error.

This shows up in adaptive algorithms, for example:

function factorial(x)
+  out = x
+  while x > 1
+    x -= 1
+    out *= x
+  end
+  out
+end
factorial (generic function with 1 method)

The number of iterations this algorithm runs for is dependent on the value of x, and so there is no static representation of the algorithm. If x is 5, then it's out = x*(x-1)*(x-2)*(x-3)*(x-4), while if x is 3, then it's out = x*(x-1)*(x-2). It should thus be no surprise that:

using Symbolics
+@variables x
+try
+    factorial(x)
+catch e
+    e
+end
TypeError(:if, "", Bool, x > 1)

fails. It's not that there is anything wrong with this code, but it's not going to work because fundamentally this is not a symbolically-representable algorithm.

The space of algorithms which can be turned into symbolic algorithms is what we call quasi-static, that is, there is a way to represent the algorithm as static. Loops are allowed, but the amount of loop iterations should not require that you know the value of the symbol x. If the algorithm is quasi-static, then Symbolics.jl tracing will produce the static form of the code, unrolling the operations, and generating a flat representation of the algorithm.

What can be done?

If you need to represent this function f symbolically, then you'll need to make sure it's not traced and instead is directly represented in the underlying computational graph. Just like how sqrt(x) symbolically does not try to represent the underlying algorithm, this must be done to your f. This is done by doing @register_symbolic f(x). If you have to define things like derivatives to f, then the function registration documentation.

Equality and set membership tests

Comparing symbols with == produces a symbolic equality, not a Bool. To produce a Bool, call isequal.

To test if a symbol is part of a collection of symbols, i.e., a vector, either create a Set and use in, e.g.

try
+    x in [x]
+catch e
+    e
+end
TypeError(:if, "", Bool, x == x)
x in Set([x])
true
any(isequal(x), [x])
true

If == is used instead, you will receive TypeError: non-boolean (Num) used in boolean context. What this error is telling you is that the symbolic x == y expression is being used where a Bool is required, such as if x == y, and since the symbolic expression is held lazily this will error because the appropriate branch cannot be selected (since x == y is unknown for arbitrary symbolic values!). This is why the check isequal(x,y) is required, since this is a non-lazy check of whether the symbol x is always equal to the symbol y, rather than an expression of whether x and y currently have the same value.

diff --git a/v5.27.1/manual/function.jl b/v5.27.1/manual/function.jl new file mode 100644 index 000000000..e44d555fa --- /dev/null +++ b/v5.27.1/manual/function.jl @@ -0,0 +1,16 @@ +function (ˍ₋out, u) + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:373 =# + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:374 =# + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:375 =# + begin + #= /home/runner/work/Symbolics.jl/Symbolics.jl/src/build_function.jl:546 =# + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:422 =# @inbounds begin + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:418 =# + ˍ₋out[1] = (+)((getindex)(u, 1), (*)(-1, (getindex)(u, 3))) + ˍ₋out[2] = (+)((*)(-1, (getindex)(u, 2)), (^)((getindex)(u, 1), 2)) + ˍ₋out[3] = (+)((getindex)(u, 2), (getindex)(u, 3)) + #= /home/runner/.julia/packages/SymbolicUtils/c0xQb/src/code.jl:420 =# + nothing + end + end +end \ No newline at end of file diff --git a/v5.27.1/manual/functions/index.html b/v5.27.1/manual/functions/index.html new file mode 100644 index 000000000..b071e0e95 --- /dev/null +++ b/v5.27.1/manual/functions/index.html @@ -0,0 +1,41 @@ + +Function Registration and Tracing · Symbolics.jl
GitHub

Function Registration and Tracing

Direct Tracing

Because Symbolics expressions respect Julia semantics, one way to generate symbolic expressions is to simply place Symbolics variables as inputs into existing Julia code. For example, the following uses the standard Julia function for the Lorenz equations to generate the symbolic expression for the Lorenz equations:

using Symbolics
+function lorenz(du,u,p,t)
+ du[1] = 10.0(u[2]-u[1])
+ du[2] = u[1]*(28.0-u[3]) - u[2]
+ du[3] = u[1]*u[2] - (8/3)*u[3]
+end
+@variables t p[1:3] u(t)[1:3]
+du = Array{Any}(undef, 3)
+lorenz(du,u,p,t)
+du
3-element Vector{Any}:
+                         10.0(-(u(t))[1] + (u(t))[2])
+ -(u(t))[2] + (u(t))[1]*(28.0 - (u(t))[3])
+                         -2.6666666666666665(u(t))[3] + (u(t))[1]*(u(t))[2]

Or similarly:

@variables t x(t) y(t) z(t) dx(t) dy(t) dz(t) σ ρ β
+du = [dx,dy,dz]
+u = [x,y,z]
+p = [σ,ρ,β]
+lorenz(du,u,p,t)
+du

\[ \begin{equation} +\left[ +\begin{array}{c} +10 \left( - x\left( t \right) + y\left( t \right) \right) \\ + - y\left( t \right) + x\left( t \right) \left( 28 - z\left( t \right) \right) \\ + - 2.6667 z\left( t \right) + x\left( t \right) y\left( t \right) \\ +\end{array} +\right] +\end{equation} + \]

Registering Functions

The Symbolics graph only allows registered Julia functions within its type. All other functions are automatically traced down to registered functions. By default, Symbolics.jl pre-registers the common functions utilized in SymbolicUtils.jl and pre-defines their derivatives. However, the user can utilize the @register_symbolic macro to add their function to allowed functions of the computation graph.

Additionally, @register_array_symbolic can be used to define array functions. For size propagation it's required that a computation of how the sizes are computed is also supplied.

Defining Derivatives of Registered Functions

In order for symbolic differentiation to work, an overload of Symbolics.derivative is required. The syntax is derivative(typeof(f), args::NTuple{i,Any}, ::Val{j}) where i is the number of arguments to the function and j is which argument is being differentiated. So for example:

function derivative(::typeof(min), args::NTuple{2,Any}, ::Val{1})
+    x, y = args
+    IfElse.ifelse(x < y, one(x), zero(x))
+end

is the partial derivative of the Julia min(x,y) function with respect to x.

Note

Downstream symbolic derivative functionality only work if every partial derivative that is required in the derivative expression is defined. Note that you only need to define the partial derivatives which are actually computed.

Registration API

Symbolics.@register_symbolic — Macro
@register_symbolic(expr, define_promotion = true, Ts = [Real])

Overload appropriate methods so that Symbolics can stop tracing into the registered function. If define_promotion is true, then a promotion method in the form of

SymbolicUtils.promote_symtype(::typeof(f_registered), args...) = Real # or the annotated return type

is defined for the register function. Note that when defining multiple register overloads for one function, all the rest of the registers must set define_promotion to false except for the first one, to avoid method overwriting.

Examples

@register_symbolic foo(x, y)
+@register_symbolic foo(x, y::Bool) false # do not overload a duplicate promotion rule
+@register_symbolic goo(x, y::Int) # `y` is not overloaded to take symbolic objects
+@register_symbolic hoo(x, y)::Int # `hoo` returns `Int`

See @register_array_symbolic to register functions which return arrays.

source
Symbolics.@register_array_symbolic — Macro
@register_array_symbolic(expr)

Example:

# Let's say vandermonde takes an n-vector and returns an n x n matrix
+@register_array_symbolic vandermonde(x::AbstractVector) begin
+    size=(length(x), length(x))
+    eltype=eltype(x) # optional, will default to the promoted eltypes of x
+end

You can also register calls on callable structs:

@register_array_symbolic (c::Conv)(x::AbstractMatrix) begin
+    size=size(x) .- size(c.kernel) .+ 1
+    eltype=promote_type(eltype(x), eltype(c))
+end
source
diff --git a/v5.27.1/manual/groebner/index.html b/v5.27.1/manual/groebner/index.html new file mode 100644 index 000000000..77b377384 --- /dev/null +++ b/v5.27.1/manual/groebner/index.html @@ -0,0 +1,2 @@ + +Groebner bases · Symbolics.jl
GitHub

Groebner bases

Groebner bases use the implementation of the F4 algorithm from Groebner.jl package as its backend. We refer to the documentation of Groebner.jl, which lists some implementations details and possible use-cases of Groebner bases.

Symbolics.groebner_basis — Function
groebner_basis(polynomials)

Computes a Groebner basis of the ideal generated by the given polynomials.

This function requires a Groebner bases backend (such as Groebner.jl) to be loaded.

source
diff --git a/v5.27.1/manual/io/index.html b/v5.27.1/manual/io/index.html new file mode 100644 index 000000000..ea6471046 --- /dev/null +++ b/v5.27.1/manual/io/index.html @@ -0,0 +1,9 @@ + +I/O, Saving, and Latex · Symbolics.jl
GitHub

I/O, Saving, and Latex

Note that Julia's standard I/O functionality can be used to save Symbolics expressions out to files. For example, here we will generate an in-place version of f and save the anonymous function to a .jl file:

using Symbolics
+@variables u[1:3]
+function f(u)
+  [u[1]-u[3],u[1]^2-u[2],u[3]+u[2]]
+end
+ex1, ex2 = build_function(f(u),u)
+write("function.jl", string(ex2))
898

Now we can do something like:

g = include("function.jl")
#1 (generic function with 1 method)

and that will load the function back in. Note that this can be done to save the transformation results of Symbolics.jl so that they can be stored and used in a precompiled Julia package.

Latexification

Symbolics.jl's expressions support Latexify.jl, and thus

using Latexify
+latexify(ex)

will produce LaTeX output from Symbolics models and expressions. This works on basics like Term all the way to higher primitives like ODESystem and ReactionSystem.

diff --git a/v5.27.1/manual/limits/index.html b/v5.27.1/manual/limits/index.html new file mode 100644 index 000000000..1837ed12c --- /dev/null +++ b/v5.27.1/manual/limits/index.html @@ -0,0 +1,2 @@ + +Symbolic Limits · Symbolics.jl
GitHub

Symbolic Limits

Experimental symbolic limit support is provided by the limit function, documented below. See SymbolicLimits.jl for more information and implementation details.

Symbolics.limit — Function
limit(expr, var, h[, side::Symbol])

Compute the limit of expr as var approaches h.

side indicates the direction from which var approaches h. It may be one of :left, :right, or :both. If side is :both and the two sides do not align, an error is thrown. Side defaults to :both for finite h, :left for h = Inf, and :right for h = -Inf.

expr must be compoesed of log, exp, constants, and the rational opperators +, -, *, and /. This limitation may eventually be relaxed.

Warning

Because symbolic limit computation is undecidable, this function necessarily employs heuristics and may occasionally return wrong answers. Nevertheless, please report wrong answers as issues as we aim to have heuristics that produce correct answers in all practical cases.

source
diff --git a/v5.27.1/manual/parsing/index.html b/v5.27.1/manual/parsing/index.html new file mode 100644 index 000000000..39470e502 --- /dev/null +++ b/v5.27.1/manual/parsing/index.html @@ -0,0 +1,18 @@ + +Parsing Julia Expressions to Symbolic Expressions · Symbolics.jl
GitHub

Parsing Julia Expressions to Symbolic Expressions

Julia expressions such as :(y - x) are fundamentally different from symbolic expressions, as they do not have an algebra defined on them. Thus, it can be very helpful when building domain-specific languages (DSLs) and parsing files to convert from Julia expressions to Symbolics.jl expressions for further manipulation. Towards this end is the parse_expr_to_symbolic which performs the parsing.

Warn

Take the limitations mentioned in the parse_expr_to_symbolic docstrings seriously! Because Julia expressions contain no symbolic metadata, there is limited information and thus the parsing requires heuristics to work.

Symbolics.parse_expr_to_symbolic — Function
parse_expr_to_symbolic(ex, mod::Module)

Applies the parse_expr_to_symbolic function in the current module, i.e. parse_expr_to_symbolic(ex, mod) where mod is the module of the function caller.

Arguments

  • ex: the expression to parse
  • mod: the module to apply the parsing in. See the limitations section for details.

Example

ex = :(y(t) ~ x(t))
+parse_expr_to_symbolic(ex,Main) # gives the symbolic expression `y(t) ~ x(t)` in empty Main
+
+# Now do a whole system
+
+ex = [:(y ~ x)
+      :(y ~ -2x + 3 / z)
+      :(z ~ 2)]
+eqs = parse_expr_to_symbolic.(ex, (Main,))
+
+@variables x y z
+ex = [y ~ x
+      y ~ -2x + 3 / z
+      z ~ 2]
+all(isequal.(eqs,ex)) # true

Limitations

Symbolic-ness Tied to Environment Definitions

The parsing to a symbolic expression has to be able to recognize the difference between functions, numbers, and globals defined within one's Julia environment and those that are to be made symbolic. The way this functionality handles this problem is that it does not define anything as symbolic that is already defined in the chosen mod module. For example, f(x,y) will have f as non-symbolic if the function f (named f) is defined in mod, i.e. if isdefined(mod,:f) is true. When the symbol is defined, it will be replaced by its value. Notably, this means that the parsing behavior changes depending on the environment that it is applied.

For example:

parse_expr_to_symbolic(:(x - y),@__MODULE__) # x - y
+x = 2.0
+parse_expr_to_symbolic(:(x - y),@__MODULE__) # 2.0 - y

This is required to detect that standard functions like - are functions instead of symbolic symbols. For safety, one should create anonymous modules or other sub-environments to ensure no stray variables are defined.

Metadata is Blank

Because all the variables defined by the expressions are not defined with the standard @variables, there is no metadata that is or can be associated with any of the generated variables. Instead, they all have blank metadata, but are defined in the Real domain. Thus, the variables which come out of this parsing may not evaluate as equal to a symbolic variable defined elsewhere.

source
Missing docstring.

Missing docstring for @parse_expr_to_symbolic. Check Documenter's build log for details.

diff --git a/v5.27.1/manual/sparsity_detection/index.html b/v5.27.1/manual/sparsity_detection/index.html new file mode 100644 index 000000000..49bb36582 --- /dev/null +++ b/v5.27.1/manual/sparsity_detection/index.html @@ -0,0 +1,71 @@ + +Structure and Sparsity Detection · Symbolics.jl
GitHub

Structure and Sparsity Detection

Using the tracing system provided by Symbolics.jl expressions, Symbolics.jl can automatically detect the sparsity patterns of Julia functions efficiently. This functionality is described in more detail in the paper:

@article{gowda2019sparsity,
+  title={Sparsity Programming: Automated Sparsity-Aware Optimizations in Differentiable Programming},
+  author={Gowda, Shashi and Ma, Yingbo and Churavy, Valentin and Edelman, Alan and Rackauckas, Christopher},
+  year={2019}
+}

Please cite this work if the functionality is used.

Sparsity Detection

Symbolics.jacobian_sparsity — Function
jacobian_sparsity(
+    exprs::AbstractArray,
+    vars::AbstractArray
+) -> SparseArrays.SparseMatrixCSC{Bool, Int64}
+

Return the sparsity pattern of the Jacobian of an array of expressions with respect to an array of variable expressions.

Arguments

  • exprs: an array of symbolic expressions.
  • vars: an array of symbolic variables.

Examples

julia> using Symbolics
+
+julia> vars = @variables x₁ x₂;
+
+julia> exprs = [2x₁, 3x₂, 4x₁ * x₂];
+
+julia> Symbolics.jacobian_sparsity(exprs, vars)
+3×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 4 stored entries:
+ 1  ⋅
+ ⋅  1
+ 1  1
source
jacobian_sparsity(
+    f!::Function,
+    output::AbstractArray,
+    input::AbstractArray,
+    args...;
+    kwargs...
+) -> SparseArrays.SparseMatrixCSC{Bool, Int64}
+

Return the sparsity pattern of the Jacobian of the mutating function f!.

Arguments

  • f!: an in-place function f!(output, input, args...; kwargs...).
  • output: output array.
  • input: input array.

The eltype of output and input can be either symbolic or primitive.

Examples

julia> using Symbolics
+
+julia> f!(y, x) = y .= [x[2], 2x[1], 3x[1] * x[2]];
+
+julia> output = Vector{Float64}(undef, 3);
+
+julia> input = Vector{Float64}(undef, 2);
+
+julia> Symbolics.jacobian_sparsity(f!, output, input)
+3×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 4 stored entries:
+ ⋅  1
+ 1  ⋅
+ 1  1
source
Symbolics.hessian_sparsity — Function
hessian_sparsity(
+    expr,
+    vars::AbstractVector;
+    full
+) -> Union{SparseArrays.SparseMatrixCSC{Bool, Int64}, SparseArrays.SparseMatrixCSC{Int64, Int64}}
+

Return the sparsity pattern of the Hessian of an expression with respect to an array of variable expressions.

Arguments

  • expr: a symbolic expression.
  • vars: a vector of symbolic variables.

Examples

julia> using Symbolics
+
+julia> vars = @variables x₁ x₂;
+
+julia> expr = 3x₁^2 + 4x₁ * x₂;
+
+julia> Symbolics.hessian_sparsity(expr, vars)
+2×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 3 stored entries:
+ 1  1
+ 1  ⋅
source
hessian_sparsity(
+    f::Function,
+    input::AbstractVector,
+    args...;
+    full,
+    kwargs...
+) -> Union{SparseArrays.SparseMatrixCSC{Bool, Int64}, SparseArrays.SparseMatrixCSC{Int64, Int64}}
+

Return the sparsity pattern of the Hessian of the given function f.

Arguments

  • f: an out-of-place function f(input, args...; kwargs...).
  • input: a vector of input values whose eltype can be either symbolic or primitive.

Examples

julia> using Symbolics
+
+julia> f(x) = 4x[1] * x[2] - 5x[2]^2;
+
+julia> input = Vector{Float64}(undef, 2);
+
+julia> Symbolics.hessian_sparsity(f, input)
+2×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 3 stored entries:
+ ⋅  1
+ 1  1
source

Structure Detection

Symbolics.islinear — Function
islinear(ex, u)
+

Check if an expression is linear with respect to a list of variable expressions.

source
Symbolics.isaffine — Function
isaffine(ex, u)
+

Check if an expression is affine with respect to a list of variable expressions.

source
diff --git a/v5.27.1/manual/types/index.html b/v5.27.1/manual/types/index.html new file mode 100644 index 000000000..f4f34d5f6 --- /dev/null +++ b/v5.27.1/manual/types/index.html @@ -0,0 +1,8 @@ + +Supported types and dispatch in Symbolics · Symbolics.jl
GitHub

Supported types and dispatch in Symbolics

There is a tension between types as a representation of expression trees and types that are a subtype of types already present in Julia.

We want to be able to deal with expression trees in a unified way and not constrain expression trees themselves to be under an abstract type in Julia's type hierarchy. (For example, if we said that all expression trees are subtype of Real, then we couldn't represent array operations using the same expression tree.). But we also want to be able to pass in expression trees into places in existing code that accept Real values.

We accomplish this by wrapping expression trees in a simple wrapper type which is a subtype of our desired abstract type. For example, we wrap expression trees in the type Num which is a subtype of Real to make it behave like a Real number.

The methods on Num objects are forwarded to the wrapped expression tree. And care is taken so that an expression tree never internally contains Num – this is both for performance and separation of concerns.

User-facing APIs in Symbolics always take wrapped objects like Num, they are then internally unwrapped for expression tree manipulation.

Due to it requiring such wrappers, we only fully support a limited number of types as both the types of expression trees and the type as Julia sees them.

These types are

  • Real numbers (wrapped using Num)
  • complex numbers (stored as Complex{Num} where Complex is from Base Julia)
  • arrays of Real and complex numbers (wrapped using Arr, so Arr{Num} or Arr{Complex{Num}})

@variables and types

Use the syntax @variables x::T to create a symbol named x of symbolic type T. If T is a subtype of any of the above listed types which support a wrapper, the resulting variable will be wrapped in that type. As seen in the examples below, x,z,X,Z all have a suitable wrapper type. Hence, their types are shown. However, s being of symbolic type String does not have a corresponding wrapper supported by Symbolics, and hence, it returns a Sym{String} object. This is the trivial expression tree of a single variable without a wrapper, and is not a subtype of String or AbstractString.

using Symbolics
+@variables x::Real z::Complex{Real} (X::Real)[1:10, 1:10] (Z::Complex{Real})[1:10] s::String
5-element Vector{Any}:
+ x
+ z
+  X[1:10,1:10]
+  Z[1:10]
+  s
typeof(x)
Num
typeof(z)
Complex{Num}
typeof(X)
Symbolics.Arr{Num, 2}
typeof(Z)
Symbolics.Arr{Complex{Num}, 1}
typeof(s)
SymbolicUtils.BasicSymbolic{String}
diff --git a/v5.27.1/manual/variables/index.html b/v5.27.1/manual/variables/index.html new file mode 100644 index 000000000..7e5ed689e --- /dev/null +++ b/v5.27.1/manual/variables/index.html @@ -0,0 +1,71 @@ + +Variable and Equation Types · Symbolics.jl
GitHub

Variable and Equation Types

Symbolics IR mirrors the Julia AST but allows for easy mathematical manipulation by itself following mathematical semantics. The base of the IR is the Sym type, which defines a symbolic variable. Registered (mathematical) functions on Syms (or istree objects) return an expression that istree. For example, op1 = x+y is one symbolic object and op2 = 2z is another, and so op1*op2 is another tree object. Then, at the top, an Equation, normally written as op1 ~ op2, defines the symbolic equality between two operations.

Types

Sym, Term, and FnType are from SymbolicUtils.jl. Note that in Symbolics, we always use Sym{Real}, Term{Real}, and FnType{Tuple{Any}, Real}. To get the arguments of an istree object, use arguments(t::Term), and to get the operation, use operation(t::Term). However, note that one should never dispatch on Term or test isa Term. Instead, one needs to use SymbolicUtils.istree to check if arguments and operation is defined.

Symbolics.@variables — Macro

Define one or more unknown variables.

@variables t α σ(..) β[1:2]
+@variables w(..) x(t) y z(t, α, x)
+
+expr = β[1]* x + y^α + σ(3) * (z - t) - β[2] * w(t - 1)

(..) signifies that the value should be left uncalled.

Symbolics supports creating variables that denote an array of some size.

julia> @variables x[1:3]
+1-element Vector{Symbolics.Arr{Num, 1}}:
+ x[1:3]
+
+julia> @variables y[1:3, 1:6] # support for  tensors
+1-element Vector{Symbolics.Arr{Num, 2}}:
+ y[1:3,1:6]
+
+julia> @variables t z(t)[1:3] # also works for dependent variables
+2-element Vector{Any}:
+ t
+  (z(t))[1:3]

A symbol or expression that represents an array can be turned into an array of symbols or expressions using the scalarize function.

julia> Symbolics.scalarize(z)
+3-element Vector{Num}:
+ (z(t))[1]
+ (z(t))[2]
+ (z(t))[3]

Note that @variables returns a vector of all the defined variables.

@variables can also take runtime symbol values by the $ interpolation operator, and in this case, @variables doesn't automatically assign the value, instead, it only returns a vector of symbolic variables. All the rest of the syntax also applies here.

julia> a, b, c = :runtime_symbol_value, :value_b, :value_c
+(:runtime_symbol_value, :value_b, :value_c)
+
+julia> vars = @variables t $a $b(t) $c(t)[1:3]
+4-element Vector{Any}:
+      t
+ runtime_symbol_value
+   value_b(t)
+       (value_c(t))[1:3]
+
+julia> (t, a, b, c)
+(t, :runtime_symbol_value, :value_b, :value_c)
source
Symbolics.variable — Function
variable(name::Symbol, idx::Integer...; T=Real)

Create a variable with the given name along with subscripted indices with the symtype=T. When T=FnType, it creates a symbolic function.

julia> Symbolics.variable(:x, 4, 2, 0)
+x₄ˏ₂ˏ₀
+
+julia> Symbolics.variable(:x, 4, 2, 0, T=Symbolics.FnType)
+x₄ˏ₂ˏ₀⋆

Also see variables.

source
Symbolics.variables — Function
variables(name::Symbol, indices...)

Create a multi-dimensional array of individual variables named with subscript notation. Use @variables instead to create symbolic array variables (as opposed to array of variables). See variable to create one variable with subscripts.

julia> Symbolics.variables(:x, 1:3, 3:6)
+3×4 Matrix{Num}:
+ x₁ˏ₃  x₁ˏ₄  x₁ˏ₅  x₁ˏ₆
+ x₂ˏ₃  x₂ˏ₄  x₂ˏ₅  x₂ˏ₆
+ x₃ˏ₃  x₃ˏ₄  x₃ˏ₅  x₃ˏ₆
source
Symbolics.Equation — Type
struct Equation

An equality relationship between two expressions.

Fields

  • lhs: The expression on the left-hand side of the equation.

  • rhs: The expression on the right-hand side of the equation.

source
Base.:~ — Method
~(lhs, rhs) -> Any
+

Create an Equation out of two Num instances, or an Num and a Number.

Examples

julia> using Symbolics
+
+julia> @variables x y;
+
+julia> @variables A[1:3, 1:3] B[1:3, 1:3];
+
+julia> x ~ y
+x ~ y
+
+julia> x - y ~ 0
+x - y ~ 0
+
+julia> A ~ B
+(broadcast(~, A, B))[1:3,1:3]
+
+julia> A .~ 3x
+(broadcast(~, A, 3x))[1:3,1:3]
source

A note about functions restricted to Numbers

Sym and Term objects are NOT subtypes of Number. Symbolics provides a simple wrapper type called Num which is a subtype of Real. Num wraps either a Sym or a Term or any other object, defines the same set of operations as symbolic expressions and forwards those to the values it wraps. You can use Symbolics.value function to unwrap a Num.

By default, the @variables macros return Num-wrapped objects to allow calling functions which are restricted to Number or Real.

using Symbolics
+@variables t x y z(t);
+Symbolics.operation(Symbolics.value(x + y))
+ (generic function with 482 methods)
Symbolics.operation(Symbolics.value(z))

\[ \begin{equation} +z +\end{equation} + \]

Symbolics.arguments(Symbolics.value(x + y))
2-element Vector{Any}:
+ x
+ y

Note that Julia converts irrationals — like π and ℯ — to Float64 whenever they are involved in arithmetic with other numbers, including integers. An expression like 2π will be converted to a float immediately, so an expression like 2π * x will leave the symbolic x multiplied by a Float64. It may be preferable to have a symbolic representation of π also, which can be achieved with Num(π). For generic programming, it may be helpful to simply redefine the variable π to be of the same type as some other argument, as in

function f(x)
+    let π=oftype(x, π)
+        1 + (2//3 + 4π/5) * x
+    end
+end
+f(t)

\[ \begin{equation} +1 + t \left( \frac{2}{3} + \frac{4}{5} \mathrm{identity}\left( \pi \right) \right) +\end{equation} + \]

This will work for any floating-point input, as well as symbolic input.

Symbolic Control Flow

Control flow can be expressed in Symbolics.jl in the following ways:

  • IfElse.ifelse(cond,x,y): this is a dispatch-able version of the ifelse function provided by IfElse.jl which allows for encoding conditionals in the symbolic branches.

Inspection Functions

SymbolicUtils.istree — Function

istree(x)

Returns true if x is a term. If true, operation, arguments must also be defined for x appropriately.

source
SymbolicUtils.operation — Function

operation(x)

If x is a term as defined by istree(x), operation(x) returns the head of the term if x represents a function call, for example, the head is the function being called.

source
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What do I do?","text":"","category":"section"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"If you see the error:","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"ERROR: TypeError: non-boolean (Num) used in boolean context","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"this is likely coming from an algorithm which cannot be traced into a purely symbolic algorithm. Many numerical solvers, for instance, have this property. It shows up when you're doing something like if x < tol. If x is a number, then this is true or false. If x is a symbol, then it's x < tol, so Julia just cannot know how many iterations to do and throws an error.","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"This shows up in adaptive algorithms, for example:","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"function factorial(x)\n out = x\n while x > 1\n x -= 1\n out *= x\n end\n out\nend","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"The number of iterations this algorithm runs for is dependent on the value of x, and so there is no static representation of the algorithm. If x is 5, then it's out = x*(x-1)*(x-2)*(x-3)*(x-4), while if x is 3, then it's out = x*(x-1)*(x-2). It should thus be no surprise that:","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"using Symbolics\n@variables x\ntry\n factorial(x)\ncatch e\n e\nend","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"fails. It's not that there is anything wrong with this code, but it's not going to work because fundamentally this is not a symbolically-representable algorithm.","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"The space of algorithms which can be turned into symbolic algorithms is what we call quasi-static, that is, there is a way to represent the algorithm as static. Loops are allowed, but the amount of loop iterations should not require that you know the value of the symbol x. If the algorithm is quasi-static, then Symbolics.jl tracing will produce the static form of the code, unrolling the operations, and generating a flat representation of the algorithm.","category":"page"},{"location":"manual/faq/#What-can-be-done?","page":"Frequently Asked Questions","title":"What can be done?","text":"","category":"section"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"If you need to represent this function f symbolically, then you'll need to make sure it's not traced and instead is directly represented in the underlying computational graph. Just like how sqrt(x) symbolically does not try to represent the underlying algorithm, this must be done to your f. This is done by doing @register_symbolic f(x). 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To produce a Bool, call isequal.","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"To test if a symbol is part of a collection of symbols, i.e., a vector, either create a Set and use in, e.g.","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"try \n x in [x]\ncatch e\n e\nend","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"x in Set([x])","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"any(isequal(x), [x])","category":"page"},{"location":"manual/faq/","page":"Frequently Asked Questions","title":"Frequently Asked Questions","text":"If == is used instead, you will receive TypeError: non-boolean (Num) used in boolean context. 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Its goal is very different from Sympy: it was made to support symbolic-numerics, the combination of symbolic computing with numerical methods to allow for extreme performance computing that would not be possible without modifying the model. Because of this, Symbolics.jl excels in many areas due to purposeful design decisions:","category":"page"},{"location":"comparison/","page":"Comparison Against SymPy","title":"Comparison Against SymPy","text":"Performance: Symbolics.jl is built in Julia, whereas SymPy was built in Python. Thus, the performance bar for Symbolics.jl is much higher. Symbolics.jl started because SymPy was far too slow and SymEngine was far too inflexible for the projects they were doing. Performance is key to Symbolics.jl. If you find any performance issues, please file an issue.\nbuild_function: lambdify is “fine” for some people, but if you're building a super fast MPI-enabled Julia/C/Fortran simulation code, having a function that hits the Python interpreter is less than optimal. By default, build_function builds fast JIT-compiled functions due to being in Julia. However, it has support for things like static arrays, non-allocating functions via mutation, fast functions on sparse matrices and arrays of arrays, etc.: all core details of doing high performance computing.\nParallelism: Symbolics.jl has pervasive parallelism. The symbolic simplification via SymbolicUtils.jl has built-in parallelism, Symbolics.jl builds functions that parallelize across threads. Symbolics.jl is compatible with GPU libraries like CUDA.jl.\nExtensible: Symbolics.jl and its underlying tools are written in pure Julia. Want to add new or better simplification rules? Add some Julia code! Need to add new derivatives? Add some Julia code! You get the picture. Breaking down these barriers makes it easier for the user to tailor the program to their needs and accelerates the development of the library.\nDeep integration with the Julia ecosystem: Symbolics.jl's integration with neural networks is not the only thing that's deep. Symbolics.jl is built with the same philosophy as other SciML packages, eschewing “monorepos” for a distributed development approach that ties together the work of many developers. The differentiation parts utilize tools from automatic differentiation libraries, all linear algebra functionality comes from tracing Julia Base itself, symbolic rewriting (simplification and substitution) comes from SymbolicUtils.jl, parallelism comes from Julia Base libraries, Dagger.jl, etc. SciML Tools like DataDrivenDiffEq.jl can reconstruct symbolic expressions from neural networks and data, while NeuralPDE.jl can automatically solve partial differential equations from symbolic descriptions using physics-informed neural networks. The list keeps going. All told, by design Symbolics.jl's development moves fast because it's effectively using the work of hundreds of Julia developers, allowing it to grow fast.\nWhile Symbolics.jl has many features missing from SymPy, it does not superset SymPy's functionality. For a list of missing features, see this issue.","category":"page"},{"location":"manual/groebner/#Groebner-bases","page":"Groebner bases","title":"Groebner bases","text":"","category":"section"},{"location":"manual/groebner/","page":"Groebner bases","title":"Groebner bases","text":"Groebner bases use the implementation of the F4 algorithm from Groebner.jl package as its backend. We refer to the documentation of Groebner.jl, which lists some implementations details and possible use-cases of Groebner bases.","category":"page"},{"location":"manual/groebner/","page":"Groebner bases","title":"Groebner bases","text":"groebner_basis","category":"page"},{"location":"manual/groebner/#Symbolics.groebner_basis","page":"Groebner bases","title":"Symbolics.groebner_basis","text":"groebner_basis(polynomials)\n\nComputes a Groebner basis of the ideal generated by the given polynomials.\n\nThis function requires a Groebner bases backend (such as Groebner.jl) to be loaded.\n\n\n\n\n\n","category":"function"},{"location":"manual/types/#Supported-types-and-dispatch-in-Symbolics","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"","category":"section"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"There is a tension between types as a representation of expression trees and types that are a subtype of types already present in Julia.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"We want to be able to deal with expression trees in a unified way and not constrain expression trees themselves to be under an abstract type in Julia's type hierarchy. (For example, if we said that all expression trees are subtype of Real, then we couldn't represent array operations using the same expression tree.). But we also want to be able to pass in expression trees into places in existing code that accept Real values.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"We accomplish this by wrapping expression trees in a simple wrapper type which is a subtype of our desired abstract type. For example, we wrap expression trees in the type Num which is a subtype of Real to make it behave like a Real number.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"The methods on Num objects are forwarded to the wrapped expression tree. And care is taken so that an expression tree never internally contains Num – this is both for performance and separation of concerns.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"User-facing APIs in Symbolics always take wrapped objects like Num, they are then internally unwrapped for expression tree manipulation.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"Due to it requiring such wrappers, we only fully support a limited number of types as both the types of expression trees and the type as Julia sees them.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"These types are","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"Real numbers (wrapped using Num)\ncomplex numbers (stored as Complex{Num} where Complex is from Base Julia)\narrays of Real and complex numbers (wrapped using Arr, so Arr{Num} or Arr{Complex{Num}})","category":"page"},{"location":"manual/types/#@variables-and-types","page":"Supported types and dispatch in Symbolics","title":"@variables and types","text":"","category":"section"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"Use the syntax @variables x::T to create a symbol named x of symbolic type T. If T is a subtype of any of the above listed types which support a wrapper, the resulting variable will be wrapped in that type. As seen in the examples below, x,z,X,Z all have a suitable wrapper type. Hence, their types are shown. However, s being of symbolic type String does not have a corresponding wrapper supported by Symbolics, and hence, it returns a Sym{String} object. This is the trivial expression tree of a single variable without a wrapper, and is not a subtype of String or AbstractString.","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"using Symbolics\n@variables x::Real z::Complex{Real} (X::Real)[1:10, 1:10] (Z::Complex{Real})[1:10] s::String","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"typeof(x)","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"typeof(z)","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"typeof(X)","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"typeof(Z)","category":"page"},{"location":"manual/types/","page":"Supported types and dispatch in Symbolics","title":"Supported types and dispatch in Symbolics","text":"typeof(s)","category":"page"},{"location":"manual/parsing/#Parsing-Julia-Expressions-to-Symbolic-Expressions","page":"Parsing Julia Expressions to Symbolic Expressions","title":"Parsing Julia Expressions to Symbolic Expressions","text":"","category":"section"},{"location":"manual/parsing/","page":"Parsing Julia Expressions to Symbolic Expressions","title":"Parsing Julia Expressions to Symbolic Expressions","text":"Julia expressions such as :(y - x) are fundamentally different from symbolic expressions, as they do not have an algebra defined on them. Thus, it can be very helpful when building domain-specific languages (DSLs) and parsing files to convert from Julia expressions to Symbolics.jl expressions for further manipulation. Towards this end is the parse_expr_to_symbolic which performs the parsing.","category":"page"},{"location":"manual/parsing/","page":"Parsing Julia Expressions to Symbolic Expressions","title":"Parsing Julia Expressions to Symbolic Expressions","text":"warn: Warn\nTake the limitations mentioned in the parse_expr_to_symbolic docstrings seriously! Because Julia expressions contain no symbolic metadata, there is limited information and thus the parsing requires heuristics to work. ","category":"page"},{"location":"manual/parsing/","page":"Parsing Julia Expressions to Symbolic Expressions","title":"Parsing Julia Expressions to Symbolic Expressions","text":"parse_expr_to_symbolic\n@parse_expr_to_symbolic","category":"page"},{"location":"manual/parsing/#Symbolics.parse_expr_to_symbolic","page":"Parsing Julia Expressions to Symbolic Expressions","title":"Symbolics.parse_expr_to_symbolic","text":"parse_expr_to_symbolic(ex, mod::Module)\n\nApplies the parse_expr_to_symbolic function in the current module, i.e. parse_expr_to_symbolic(ex, mod) where mod is the module of the function caller.\n\nArguments\n\nex: the expression to parse\nmod: the module to apply the parsing in. See the limitations section for details.\n\nExample\n\nex = :(y(t) ~ x(t))\nparse_expr_to_symbolic(ex,Main) # gives the symbolic expression `y(t) ~ x(t)` in empty Main\n\n# Now do a whole system\n\nex = [:(y ~ x)\n :(y ~ -2x + 3 / z)\n :(z ~ 2)]\neqs = parse_expr_to_symbolic.(ex, (Main,))\n\n@variables x y z\nex = [y ~ x\n y ~ -2x + 3 / z\n z ~ 2]\nall(isequal.(eqs,ex)) # true\n\nLimitations\n\nSymbolic-ness Tied to Environment Definitions\n\nThe parsing to a symbolic expression has to be able to recognize the difference between functions, numbers, and globals defined within one's Julia environment and those that are to be made symbolic. The way this functionality handles this problem is that it does not define anything as symbolic that is already defined in the chosen mod module. For example, f(x,y) will have f as non-symbolic if the function f (named f) is defined in mod, i.e. if isdefined(mod,:f) is true. When the symbol is defined, it will be replaced by its value. Notably, this means that the parsing behavior changes depending on the environment that it is applied.\n\nFor example:\n\nparse_expr_to_symbolic(:(x - y),@__MODULE__) # x - y\nx = 2.0\nparse_expr_to_symbolic(:(x - y),@__MODULE__) # 2.0 - y\n\nThis is required to detect that standard functions like - are functions instead of symbolic symbols. For safety, one should create anonymous modules or other sub-environments to ensure no stray variables are defined.\n\nMetadata is Blank\n\nBecause all the variables defined by the expressions are not defined with the standard @variables, there is no metadata that is or can be associated with any of the generated variables. Instead, they all have blank metadata, but are defined in the Real domain. Thus, the variables which come out of this parsing may not evaluate as equal to a symbolic variable defined elsewhere.\n\n\n\n\n\n","category":"function"},{"location":"manual/variables/#Variable-and-Equation-Types","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"","category":"section"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Symbolics IR mirrors the Julia AST but allows for easy mathematical manipulation by itself following mathematical semantics. The base of the IR is the Sym type, which defines a symbolic variable. Registered (mathematical) functions on Syms (or istree objects) return an expression that istree. For example, op1 = x+y is one symbolic object and op2 = 2z is another, and so op1*op2 is another tree object. Then, at the top, an Equation, normally written as op1 ~ op2, defines the symbolic equality between two operations.","category":"page"},{"location":"manual/variables/#Types","page":"Variable and Equation Types","title":"Types","text":"","category":"section"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Sym, Term, and FnType are from SymbolicUtils.jl. Note that in Symbolics, we always use Sym{Real}, Term{Real}, and FnType{Tuple{Any}, Real}. To get the arguments of an istree object, use arguments(t::Term), and to get the operation, use operation(t::Term). However, note that one should never dispatch on Term or test isa Term. Instead, one needs to use SymbolicUtils.istree to check if arguments and operation is defined.","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"@variables\nSymbolics.variable\nSymbolics.variables\nEquation\nBase.:~(::Num, ::Num)","category":"page"},{"location":"manual/variables/#Symbolics.@variables","page":"Variable and Equation Types","title":"Symbolics.@variables","text":"Define one or more unknown variables.\n\n@variables t α σ(..) β[1:2]\n@variables w(..) x(t) y z(t, α, x)\n\nexpr = β[1]* x + y^α + σ(3) * (z - t) - β[2] * w(t - 1)\n\n(..) signifies that the value should be left uncalled.\n\nSymbolics supports creating variables that denote an array of some size.\n\njulia> @variables x[1:3]\n1-element Vector{Symbolics.Arr{Num, 1}}:\n x[1:3]\n\njulia> @variables y[1:3, 1:6] # support for tensors\n1-element Vector{Symbolics.Arr{Num, 2}}:\n y[1:3,1:6]\n\njulia> @variables t z(t)[1:3] # also works for dependent variables\n2-element Vector{Any}:\n t\n (z(t))[1:3]\n\nA symbol or expression that represents an array can be turned into an array of symbols or expressions using the scalarize function.\n\njulia> Symbolics.scalarize(z)\n3-element Vector{Num}:\n (z(t))[1]\n (z(t))[2]\n (z(t))[3]\n\nNote that @variables returns a vector of all the defined variables.\n\n@variables can also take runtime symbol values by the $ interpolation operator, and in this case, @variables doesn't automatically assign the value, instead, it only returns a vector of symbolic variables. All the rest of the syntax also applies here.\n\njulia> a, b, c = :runtime_symbol_value, :value_b, :value_c\n(:runtime_symbol_value, :value_b, :value_c)\n\njulia> vars = @variables t $a $b(t) $c(t)[1:3]\n4-element Vector{Any}:\n t\n runtime_symbol_value\n value_b(t)\n (value_c(t))[1:3]\n\njulia> (t, a, b, c)\n(t, :runtime_symbol_value, :value_b, :value_c)\n\n\n\n\n\n","category":"macro"},{"location":"manual/variables/#Symbolics.variable","page":"Variable and Equation Types","title":"Symbolics.variable","text":"variable(name::Symbol, idx::Integer...; T=Real)\n\nCreate a variable with the given name along with subscripted indices with the symtype=T. When T=FnType, it creates a symbolic function.\n\njulia> Symbolics.variable(:x, 4, 2, 0)\nx₄ˏ₂ˏ₀\n\njulia> Symbolics.variable(:x, 4, 2, 0, T=Symbolics.FnType)\nx₄ˏ₂ˏ₀⋆\n\nAlso see variables.\n\n\n\n\n\n","category":"function"},{"location":"manual/variables/#Symbolics.variables","page":"Variable and Equation Types","title":"Symbolics.variables","text":"variables(name::Symbol, indices...)\n\nCreate a multi-dimensional array of individual variables named with subscript notation. Use @variables instead to create symbolic array variables (as opposed to array of variables). See variable to create one variable with subscripts.\n\njulia> Symbolics.variables(:x, 1:3, 3:6)\n3×4 Matrix{Num}:\n x₁ˏ₃ x₁ˏ₄ x₁ˏ₅ x₁ˏ₆\n x₂ˏ₃ x₂ˏ₄ x₂ˏ₅ x₂ˏ₆\n x₃ˏ₃ x₃ˏ₄ x₃ˏ₅ x₃ˏ₆\n\n\n\n\n\n","category":"function"},{"location":"manual/variables/#Symbolics.Equation","page":"Variable and Equation Types","title":"Symbolics.Equation","text":"struct Equation\n\nAn equality relationship between two expressions.\n\nFields\n\nlhs: The expression on the left-hand side of the equation.\nrhs: The expression on the right-hand side of the equation.\n\n\n\n\n\n","category":"type"},{"location":"manual/variables/#Base.:~-Tuple{Num, Num}","page":"Variable and Equation Types","title":"Base.:~","text":"~(lhs, rhs) -> Any\n\n\nCreate an Equation out of two Num instances, or an Num and a Number.\n\nExamples\n\njulia> using Symbolics\n\njulia> @variables x y;\n\njulia> @variables A[1:3, 1:3] B[1:3, 1:3];\n\njulia> x ~ y\nx ~ y\n\njulia> x - y ~ 0\nx - y ~ 0\n\njulia> A ~ B\n(broadcast(~, A, B))[1:3,1:3]\n\njulia> A .~ 3x\n(broadcast(~, A, 3x))[1:3,1:3]\n\n\n\n\n\n","category":"method"},{"location":"manual/variables/#A-note-about-functions-restricted-to-Numbers","page":"Variable and Equation Types","title":"A note about functions restricted to Numbers","text":"","category":"section"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Sym and Term objects are NOT subtypes of Number. Symbolics provides a simple wrapper type called Num which is a subtype of Real. Num wraps either a Sym or a Term or any other object, defines the same set of operations as symbolic expressions and forwards those to the values it wraps. You can use Symbolics.value function to unwrap a Num.","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"By default, the @variables macros return Num-wrapped objects to allow calling functions which are restricted to Number or Real.","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"using Symbolics\n@variables t x y z(t);\nSymbolics.operation(Symbolics.value(x + y))","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Symbolics.operation(Symbolics.value(z))","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Symbolics.arguments(Symbolics.value(x + y))","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Note that Julia converts irrationals — like π and ℯ — to Float64 whenever they are involved in arithmetic with other numbers, including integers. An expression like 2π will be converted to a float immediately, so an expression like 2π * x will leave the symbolic x multiplied by a Float64. It may be preferable to have a symbolic representation of π also, which can be achieved with Num(π). For generic programming, it may be helpful to simply redefine the variable π to be of the same type as some other argument, as in","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"function f(x)\n let π=oftype(x, π)\n 1 + (2//3 + 4π/5) * x\n end\nend\nf(t)","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"This will work for any floating-point input, as well as symbolic input.","category":"page"},{"location":"manual/variables/#Symbolic-Control-Flow","page":"Variable and Equation Types","title":"Symbolic Control Flow","text":"","category":"section"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"Control flow can be expressed in Symbolics.jl in the following ways:","category":"page"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"IfElse.ifelse(cond,x,y): this is a dispatch-able version of the ifelse function provided by IfElse.jl which allows for encoding conditionals in the symbolic branches.","category":"page"},{"location":"manual/variables/#Inspection-Functions","page":"Variable and Equation Types","title":"Inspection Functions","text":"","category":"section"},{"location":"manual/variables/","page":"Variable and Equation Types","title":"Variable and Equation Types","text":"SymbolicUtils.istree\nSymbolicUtils.operation\nSymbolicUtils.arguments","category":"page"},{"location":"manual/variables/#SymbolicUtils.istree","page":"Variable and Equation Types","title":"SymbolicUtils.istree","text":"istree(x)\n\nReturns true if x is a term. If true, operation, arguments must also be defined for x appropriately.\n\n\n\n\n\n","category":"function"},{"location":"manual/variables/#SymbolicUtils.operation","page":"Variable and Equation Types","title":"SymbolicUtils.operation","text":"operation(x)\n\nIf x is a term as defined by istree(x), operation(x) returns the head of the term if x represents a function call, for example, the head is the function being called.\n\n\n\n\n\n","category":"function"},{"location":"manual/variables/#SymbolicUtils.arguments","page":"Variable and Equation Types","title":"SymbolicUtils.arguments","text":"arguments(x)\n\nGet the arguments of x, must be defined if istree(x) is true.\n\n\n\n\n\n","category":"function"},{"location":"manual/sparsity_detection/#Structure-and-Sparsity-Detection","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"","category":"section"},{"location":"manual/sparsity_detection/","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"Using the tracing system provided by Symbolics.jl expressions, Symbolics.jl can automatically detect the sparsity patterns of Julia functions efficiently. This functionality is described in more detail in the paper:","category":"page"},{"location":"manual/sparsity_detection/","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"@article{gowda2019sparsity,\n title={Sparsity Programming: Automated Sparsity-Aware Optimizations in Differentiable Programming},\n author={Gowda, Shashi and Ma, Yingbo and Churavy, Valentin and Edelman, Alan and Rackauckas, Christopher},\n year={2019}\n}","category":"page"},{"location":"manual/sparsity_detection/","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"Please cite this work if the functionality is used.","category":"page"},{"location":"manual/sparsity_detection/#Sparsity-Detection","page":"Structure and Sparsity Detection","title":"Sparsity Detection","text":"","category":"section"},{"location":"manual/sparsity_detection/","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"Symbolics.jacobian_sparsity\nSymbolics.hessian_sparsity","category":"page"},{"location":"manual/sparsity_detection/#Symbolics.jacobian_sparsity","page":"Structure and Sparsity Detection","title":"Symbolics.jacobian_sparsity","text":"jacobian_sparsity(\n exprs::AbstractArray,\n vars::AbstractArray\n) -> SparseArrays.SparseMatrixCSC{Bool, Int64}\n\n\nReturn the sparsity pattern of the Jacobian of an array of expressions with respect to an array of variable expressions.\n\nArguments\n\nexprs: an array of symbolic expressions.\nvars: an array of symbolic variables.\n\nExamples\n\njulia> using Symbolics\n\njulia> vars = @variables x₁ x₂;\n\njulia> exprs = [2x₁, 3x₂, 4x₁ * x₂];\n\njulia> Symbolics.jacobian_sparsity(exprs, vars)\n3×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 4 stored entries:\n 1 ⋅\n ⋅ 1\n 1 1\n\n\n\n\n\njacobian_sparsity(\n f!::Function,\n output::AbstractArray,\n input::AbstractArray,\n args...;\n kwargs...\n) -> SparseArrays.SparseMatrixCSC{Bool, Int64}\n\n\nReturn the sparsity pattern of the Jacobian of the mutating function f!.\n\nArguments\n\nf!: an in-place function f!(output, input, args...; kwargs...).\noutput: output array.\ninput: input array.\n\nThe eltype of output and input can be either symbolic or primitive.\n\nExamples\n\njulia> using Symbolics\n\njulia> f!(y, x) = y .= [x[2], 2x[1], 3x[1] * x[2]];\n\njulia> output = Vector{Float64}(undef, 3);\n\njulia> input = Vector{Float64}(undef, 2);\n\njulia> Symbolics.jacobian_sparsity(f!, output, input)\n3×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 4 stored entries:\n ⋅ 1\n 1 ⋅\n 1 1\n\n\n\n\n\n","category":"function"},{"location":"manual/sparsity_detection/#Symbolics.hessian_sparsity","page":"Structure and Sparsity Detection","title":"Symbolics.hessian_sparsity","text":"hessian_sparsity(\n expr,\n vars::AbstractVector;\n full\n) -> Union{SparseArrays.SparseMatrixCSC{Bool, Int64}, SparseArrays.SparseMatrixCSC{Int64, Int64}}\n\n\nReturn the sparsity pattern of the Hessian of an expression with respect to an array of variable expressions.\n\nArguments\n\nexpr: a symbolic expression.\nvars: a vector of symbolic variables.\n\nExamples\n\njulia> using Symbolics\n\njulia> vars = @variables x₁ x₂;\n\njulia> expr = 3x₁^2 + 4x₁ * x₂;\n\njulia> Symbolics.hessian_sparsity(expr, vars)\n2×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 3 stored entries:\n 1 1\n 1 ⋅\n\n\n\n\n\nhessian_sparsity(\n f::Function,\n input::AbstractVector,\n args...;\n full,\n kwargs...\n) -> Union{SparseArrays.SparseMatrixCSC{Bool, Int64}, SparseArrays.SparseMatrixCSC{Int64, Int64}}\n\n\nReturn the sparsity pattern of the Hessian of the given function f.\n\nArguments\n\nf: an out-of-place function f(input, args...; kwargs...).\ninput: a vector of input values whose eltype can be either symbolic or primitive.\n\nExamples\n\njulia> using Symbolics\n\njulia> f(x) = 4x[1] * x[2] - 5x[2]^2;\n\njulia> input = Vector{Float64}(undef, 2);\n\njulia> Symbolics.hessian_sparsity(f, input)\n2×2 SparseArrays.SparseMatrixCSC{Bool, Int64} with 3 stored entries:\n ⋅ 1\n 1 1\n\n\n\n\n\n","category":"function"},{"location":"manual/sparsity_detection/#Structure-Detection","page":"Structure and Sparsity Detection","title":"Structure Detection","text":"","category":"section"},{"location":"manual/sparsity_detection/","page":"Structure and Sparsity Detection","title":"Structure and Sparsity Detection","text":"Symbolics.islinear\nSymbolics.isaffine","category":"page"},{"location":"manual/sparsity_detection/#Symbolics.islinear","page":"Structure and Sparsity Detection","title":"Symbolics.islinear","text":"islinear(ex, u)\n\n\nCheck if an expression is linear with respect to a list of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/sparsity_detection/#Symbolics.isaffine","page":"Structure and Sparsity Detection","title":"Symbolics.isaffine","text":"isaffine(ex, u)\n\n\nCheck if an expression is affine with respect to a list of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"tutorials/auto_parallel/#Automated-Sparse-Parallelism-of-Julia-Functions-via-Tracing","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"","category":"section"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"Because the Symbolics.jl expressions obey Julia semantics, one can directly transform existing Julia functions into Symbolics.jl symbolic representations of the function by simply inputting the symbolic values into the function and using what is returned. For example, let's take the following numerical PDE discretization:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"using Symbolics, LinearAlgebra, SparseArrays, Plots\n\n# Define the constants for the PDE\nconst α₂ = 1.0\nconst α₃ = 1.0\nconst β₁ = 1.0\nconst β₂ = 1.0\nconst β₃ = 1.0\nconst r₁ = 1.0\nconst r₂ = 1.0\nconst _DD = 100.0\nconst γ₁ = 0.1\nconst γ₂ = 0.1\nconst γ₃ = 0.1\nconst N = 32\nconst X = reshape([i for i in 1:N for j in 1:N], N, N)\nconst Y = reshape([j for i in 1:N for j in 1:N], N, N)\nconst α₁ = 1.0 .* (X .>= 4*N/5)\n\nconst Mx = Array(Tridiagonal([1.0 for i in 1:N-1], [-2.0 for i in 1:N], [1.0 for i in 1:N-1]))\nconst My = copy(Mx)\nMx[2, 1] = 2.0\nMx[end-1,end] = 2.0\nMy[1, 2] = 2.0\nMy[end,end-1] = 2.0\n\n# Define the discretized PDE as an ODE function\nfunction f(u, p, t)\n A = u[:,:,1]\n B = u[:,:,2]\n C = u[:,:,3]\n MyA = My*A\n AMx = A*Mx\n DA = @. _DD*(MyA + AMx)\n dA = @. DA + α₁ - β₁*A - r₁*A*B + r₂*C\n dB = @. α₂ - β₂*B - r₁*A*B + r₂*C\n dC = @. α₃ - β₃*C + r₁*A*B - r₂*C\n cat(dA, dB, dC, dims=3)\nend","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"We can build the Symbolics version of this model by tracing the model function:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"# Define the initial condition as normal arrays\n@variables u[1:N, 1:N, 1:3]\ndu = simplify.(f(collect(u), nothing, 0.0))\nvec(du)[1:10]","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"The output, here the in-place modified du, is a symbolic representation of each output of the function. We can then utilize this in the Symbolics functionality. For example, let's build a parallel version of f first:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"fastf = eval(Symbolics.build_function(du,u,\n parallel=Symbolics.MultithreadedForm())[2])","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"Now let's compute the sparse Jacobian function and compile a fast multithreaded version:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"jac = Symbolics.sparsejacobian(vec(du), vec(u))\nrow,col,val = findnz(jac)\nscatter(row,col,legend=false,ms=1,c=:black)","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"fjac = eval(Symbolics.build_function(jac,u,\n parallel=Symbolics.MultithreadedForm())[2])","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"It takes awhile for this to generate, but the results will be worth it! Now let's set up the parabolic PDE to be solved by DifferentialEquations.jl. We will set up the vanilla version and the sparse multithreaded version:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"using OrdinaryDiffEq\nu0 = zeros(N, N, 3)\nMyA = zeros(N, N);\nAMx = zeros(N, N);\nDA = zeros(N, N);\nprob = ODEProblem(f, u0, (0.0, 10.0))\nfastprob = ODEProblem(ODEFunction((du, u, p, t) -> fastf(du, u),\n jac = (du, u, p, t) -> fjac(du, u),\n jac_prototype = similar(jac, Float64)),\n u0, (0.0, 10.0))","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"Let's see the timing difference:","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"using BenchmarkTools\n#@btime solve(prob, TRBDF2()); # 33.073 s (895404 allocations: 23.87 GiB)\n#warning the following solve takes a long time to compile, but afterwards is very fast.\n#@btime solve(fastprob, TRBDF2()); # 209.670 ms (8208 allocations: 109.25 MiB)","category":"page"},{"location":"tutorials/auto_parallel/","page":"Automated Sparse Parallelism of Julia Functions via Tracing","title":"Automated Sparse Parallelism of Julia Functions via Tracing","text":"Boom, an automatic 157x acceleration that grows as the size of the problem increases!","category":"page"},{"location":"manual/io/#I/O,-Saving,-and-Latex","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"","category":"section"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"Note that Julia's standard I/O functionality can be used to save Symbolics expressions out to files. For example, here we will generate an in-place version of f and save the anonymous function to a .jl file:","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"using Symbolics\n@variables u[1:3]\nfunction f(u)\n [u[1]-u[3],u[1]^2-u[2],u[3]+u[2]]\nend\nex1, ex2 = build_function(f(u),u)\nwrite(\"function.jl\", string(ex2))","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"Now we can do something like:","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"g = include(\"function.jl\")","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"and that will load the function back in. Note that this can be done to save the transformation results of Symbolics.jl so that they can be stored and used in a precompiled Julia package.","category":"page"},{"location":"manual/io/#Latexification","page":"I/O, Saving, and Latex","title":"Latexification","text":"","category":"section"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"Symbolics.jl's expressions support Latexify.jl, and thus","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"using Latexify\nlatexify(ex)","category":"page"},{"location":"manual/io/","page":"I/O, Saving, and Latex","title":"I/O, Saving, and Latex","text":"will produce LaTeX output from Symbolics models and expressions. This works on basics like Term all the way to higher primitives like ODESystem and ReactionSystem.","category":"page"},{"location":"tutorials/converting_to_C/#Automatic-Conversion-of-Julia-Code-to-C-Functions","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"","category":"section"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"Since Symbolics.jl can trace Julia code into Symbolics IR that can be built and compiled via build_function to C, this gives us a nifty way to automatically generate C functions from Julia code! To see this in action, let's start with the Lotka-Volterra equations:","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"using Symbolics\nfunction lotka_volterra!(du, u, p, t)\n x, y = u\n α, β, δ, γ = p\n du[1] = dx = α*x - β*x*y\n du[2] = dy = -δ*y + γ*x*y\nend","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"Now we trace this into Symbolics:","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"@variables t du[1:2] u[1:2] p[1:4]\ndu = collect(du)\nlotka_volterra!(du, u, p, t)\ndu","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"and then we build the function:","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"build_function(du, u, p, t, target=Symbolics.CTarget())","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"If we want to compile this, we do expression=Val{false}:","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"f = build_function(du, u, p, t, target=Symbolics.CTarget(), expression=Val{false})","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"now we check it computes the same thing:","category":"page"},{"location":"tutorials/converting_to_C/","page":"Automatic Conversion of Julia Code to C Functions","title":"Automatic Conversion of Julia Code to C Functions","text":"du = rand(2); du2 = rand(2)\nu = rand(2)\np = rand(4)\nt = rand()\nf(du, u, p, t)\nlotka_volterra!(du2, u, p, t)\ndu == du2 # true!","category":"page"},{"location":"manual/build_function/#Function-Building-and-Compilation-(build_function)","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"","category":"section"},{"location":"manual/build_function/","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"At any time, callable functions can be generated from Symbolics IR by using Symbolics.toexpr. This performs some cleaning to return an expression without extraneous pieces that commonly matches expressions one would write in functions like those for differential equation solvers and optimization libraries. These functions can be automatically parallelized and specialize on Julia types like static arrays and sparse matrices.","category":"page"},{"location":"manual/build_function/","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"The core compilation process of Symbolics IR is build_function. build_function takes an operation or an AbstractArray of operations and generates a compilable version of the model for numerical solvers. The form of this output is dependent on the target. By default, the target outputs Julia code, but other formats, such as C, Stan, and MATLAB are available. These can be generated as expressions which can then be evaluated into a callable function, or the compilers for the respective targets can be invoked to directly give back the function handle.","category":"page"},{"location":"manual/build_function/#build_function","page":"Function Building and Compilation (build_function)","title":"build_function","text":"","category":"section"},{"location":"manual/build_function/","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"build_function","category":"page"},{"location":"manual/build_function/#Symbolics.build_function","page":"Function Building and Compilation (build_function)","title":"Symbolics.build_function","text":"build_function\n\nGenerates a numerically-usable function from a Symbolics Num.\n\nbuild_function(ex, args...;\n expression = Val{true},\n target = JuliaTarget(),\n parallel=nothing,\n kwargs...)\n\nArguments:\n\nex: The Num to compile\nargs: The arguments of the function\nexpression: Whether to generate code or whether to generate the compiled form. By default, expression = Val{true}, which means that the code for the function is returned. If Val{false}, then the returned value is compiled.\n\nKeyword Arguments:\n\ntarget: The output target of the compilation process. Possible options are:\nJuliaTarget: Generates a Julia function\nCTarget: Generates a C function\nStanTarget: Generates a function for compiling with the Stan probabilistic programming language\nMATLABTarget: Generates an anonymous function for use in MATLAB and Octave environments\nparallel: The kind of parallelism to use in the generated function. Defaults to SerialForm(), i.e. no parallelism, if ex is a single expression or an array containing <= 1500 non-zero expressions. If ex is an array of > 1500 non-zero expressions, then ShardedForm(80, 4) is used. See below for more on ShardedForm. Note that the parallel forms are not exported and thus need to be chosen like Symbolics.SerialForm(). The choices are:\nSerialForm(): Serial execution.\nShardedForm(cutoff, ncalls): splits the output function into sub-functions which contain at most cutoff number of output rhss. These sub-functions are called by the top-level function that buildfunction returns. This helps in reducing the compile time of the generated function.\nMultithreadedForm(): Multithreaded execution with a static split, evenly splitting the number of expressions per thread.\nfname: Used by some targets for the name of the function in the target space.\n\nNote that not all build targets support the full compilation interface. Check the individual target documentation for details.\n\n\n\n\n\n","category":"function"},{"location":"manual/build_function/#Target-Specific-Definitions","page":"Function Building and Compilation (build_function)","title":"Target-Specific Definitions","text":"","category":"section"},{"location":"manual/build_function/","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"Symbolics._build_function(target::Symbolics.JuliaTarget,rhss::AbstractArray,args...;kwargs...)\nSymbolics._build_function(target::Symbolics.CTarget,eqs::Array{<:Equation},args...;kwargs...)\nSymbolics._build_function(target::Symbolics.StanTarget,eqs::Array{<:Equation}, vs, ps, iv;kwargs...)\nSymbolics._build_function(target::Symbolics.MATLABTarget,eqs::Array{<:Equation},args...;kwargs...)","category":"page"},{"location":"manual/build_function/#Symbolics._build_function-Tuple{Symbolics.JuliaTarget, AbstractArray, Vararg{Any}}","page":"Function Building and Compilation (build_function)","title":"Symbolics._build_function","text":"_build_function(target::JuliaTarget, rhss::AbstractArray, args...;\n conv=toexpr,\n expression = Val{true},\n expression_module = @__MODULE__(),\n checkbounds = false,\n postprocess_fbody=ex -> ex,\n linenumbers = false,\n outputidxs=nothing,\n skipzeros = false,\n force_SA = false,\n wrap_code = (nothing, nothing),\n fillzeros = skipzeros && !(rhss isa SparseMatrixCSC),\n states = LazyState(),\n iip_config = (true, true),\n parallel=nothing, cse = false, kwargs...)\n\nBuild function target: JuliaTarget\n\nfunction _build_function(target::JuliaTarget, rhss, args...;\n conv = toexpr, \n expression = Val{true},\n checkbounds = false,\n linenumbers = false,\n headerfun = addheader, outputidxs=nothing,\n convert_oop = true, force_SA = false,\n skipzeros = outputidxs===nothing,\n fillzeros = skipzeros && !(typeof(rhss)<:SparseMatrixCSC),\n parallel=SerialForm(), kwargs...)\n\nGenerates a Julia function which can then be utilized for further evaluations. If expression=Val{false}, the return is a Julia function which utilizes RuntimeGeneratedFunctions.jl to be free of world-age issues.\n\nIf the rhss is a scalar, the generated function is a function with a scalar output. Otherwise, if it's an AbstractArray, the output is two functions, one for out-of-place AbstractArray output and a second which is a mutating function. The outputted functions match the given argument order, i.e., f(u,p,args...) for the out-of-place and scalar functions and f!(du,u,p,args..) for the in-place version.\n\nSpecial Keyword Arguments:\n\nparallel: The kind of parallelism to use in the generated function. Defaults to SerialForm(), i.e. no parallelism. Note that the parallel forms are not exported and thus need to be chosen like Symbolics.SerialForm(). The choices are:\nSerialForm(): Serial execution.\nShardedForm(cutoff, ncalls): splits the output function into sub-functions which contain at most cutoff number of output rhss. These sub-functions are called by the top-level function that buildfunction returns.\nMultithreadedForm(): Multithreaded execution with a static split, evenly splitting the number of expressions per thread.\nconv: The conversion function of symbolic types to Expr. By default, this uses the toexpr function.\ncheckbounds: For whether to enable bounds checking inside the generated function. Defaults to false, meaning that @inbounds is applied.\nlinenumbers: Determines whether the generated function expression retains the line numbers. Defaults to true.\nconvert_oop: Determines whether the OOP version should try to convert the output to match the type of the first input. This is useful for cases like LabelledArrays or other array types that carry extra information. Defaults to true.\nforce_SA: Forces the output of the OOP version to be a StaticArray. Defaults to false, and outputs a static array when the first argument is a static array.\nskipzeros: Whether to skip filling zeros in the in-place version if the filling function is 0.\nfillzeros: Whether to perform fill(out,0) before the calculations to ensure safety with skipzeros.\n\n\n\n\n\n","category":"method"},{"location":"manual/build_function/#Symbolics._build_function-Tuple{Symbolics.CTarget, Array{<:Equation}, Vararg{Any}}","page":"Function Building and Compilation (build_function)","title":"Symbolics._build_function","text":"Build function target: CTarget\n\nfunction _build_function(target::CTarget, eqs::Array{<:Equation}, args...;\n conv = toexpr, expression = Val{true},\n fname = :diffeqf,\n lhsname=:du,rhsnames=[Symbol(\"RHS$i\") for i in 1:length(args)],\n libpath=tempname(),compiler=:gcc)\n\nThis builds an in-place C function. Only works on arrays of equations. If expression == Val{false}, then this builds a function in C, compiles it, and returns a lambda to that compiled function. These special keyword arguments control the compilation:\n\nlibpath: the path to store the binary. Defaults to a temporary path.\ncompiler: which C compiler to use. Defaults to :gcc, which is currently the only available option.\n\n\n\n\n\n","category":"method"},{"location":"manual/build_function/#Symbolics._build_function-Tuple{Symbolics.StanTarget, Array{<:Equation}, Any, Any, Any}","page":"Function Building and Compilation (build_function)","title":"Symbolics._build_function","text":"Build function target: StanTarget\n\nfunction _build_function(target::StanTarget, eqs::Array{<:Equation}, vs, ps, iv;\n conv = toexpr, expression = Val{true},\n fname = :diffeqf, lhsname=:internal_var___du,\n rhsnames=[:internal_var___u,:internal_var___p,:internal_var___t])\n\nThis builds an in-place Stan function compatible with the Stan differential equation solvers. Unlike other build targets, this one requires (vs, ps, iv) as the function arguments. Only allowed on arrays of equations.\n\n\n\n\n\n","category":"method"},{"location":"manual/build_function/#Symbolics._build_function-Tuple{Symbolics.MATLABTarget, Array{<:Equation}, Vararg{Any}}","page":"Function Building and Compilation (build_function)","title":"Symbolics._build_function","text":"Build function target: MATLABTarget\n\nfunction _build_function(target::MATLABTarget, eqs::Array{<:Equation}, args...;\n conv = toexpr, expression = Val{true},\n lhsname=:internal_var___du,\n rhsnames=[:internal_var___u,:internal_var___p,:internal_var___t])\n\nThis builds an out of place anonymous function @(t,rhsnames[1]) to be used in MATLAB. Compatible with the MATLAB differential equation solvers. Only allowed on expressions, and arrays of expressions.\n\n\n\n\n\n","category":"method"},{"location":"manual/build_function/#Limitations","page":"Function Building and Compilation (build_function)","title":"Limitations","text":"","category":"section"},{"location":"manual/build_function/","page":"Function Building and Compilation (build_function)","title":"Function Building and Compilation (build_function)","text":"build_function ","category":"page"},{"location":"manual/arrays/#symbolic_arrays","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"","category":"section"},{"location":"manual/arrays/#Symbolic-Arrays-vs-Arrays-of-Symbolic-Expressions","page":"Symbolic Arrays","title":"Symbolic Arrays vs Arrays of Symbolic Expressions","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Symbolics.jl contains two forms for handling symbolic arrays:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Arrays of symbolic expressions: these are Julia arrays with Symbolics.jl objects in them.\nSymbolic Arrays: these are symbolic (O(1)) representations of arrays.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Arrays of symbolic expressions are simply Symbolics.jl objects put into Julia arrays. For example:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"using Symbolics\n@variables x y\nu = [x,y]","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"is a vector of two symbolic variables. As shorthand,","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"u2 = Symbolics.variables(:x, 1:3, 3:6)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"creates a Julia matrix of symbolic variables. Indexing u or u2 gives symbolic values which act as a normal scalar symbolic value. This form these uses Julia's array functionality and performs symbolic operations on the scalar values.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"On the otherhand, Julia's symbolic array form is an O(1) representation of the whole array.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"@variables A[1:5, 1:3]","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"When using this form, A[1,1] is not a symbolic variable but a symbolic expression for indexing the variable A. This representation holds linear algebra expressions in a non-expanded form. For example:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"@variables B[1:3, 1:3]\nA * B","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"in comparison to:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"a = Symbolics.variables(:a, 1:5, 1:3)\nb = Symbolics.variables(:b, 1:3, 1:3)\na * b","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"This makes the symbolic array form much more efficient, but requires that the expressions uses things with registered symbolic array functions which currently has much lower coverage. Also, there are many fallbacks for which arrays of symbolics which makes this approach more accessible but with larger expressions.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"We recommend defaulting to arrays of symbolics unless you need the expression symplifications of the symbolic array approach.","category":"page"},{"location":"manual/arrays/#Using-Symbolic-Arrays","page":"Symbolic Arrays","title":"Using Symbolic Arrays","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Symbolic array-valued expressions (symbolic arrays) are supported by Symbolics. Symbolic array expressions propagate useful metadata that depends on input arrays: array dimension, element type and shape.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"You can create a symbolic array variable with the following syntax:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"using Symbolics\n@variables A[1:5, 1:3] b[1:3]","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Here, A is a symbolic matrix of size (5, 3) and b is a symbolic vector of length 3.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"size(A)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"size(b)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"ndims(A)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"ndims(b)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"eltype(A)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"eltype(b)","category":"page"},{"location":"manual/arrays/#Array-operations","page":"Symbolic Arrays","title":"Array operations","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Operations on symbolic arrays return symbolic array expressions:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"c = A * b","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"size(c)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"eltype(c)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Adjoints, matrix-matrix, and matrix-vector multiplications are supported. Dot product returns a scalar-valued expression:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"b'b","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"size(b'b)","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Outer product returns a matrix:","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"b * b'","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"size(b*b')","category":"page"},{"location":"manual/arrays/#Broadcast,-map-and-reduce","page":"Symbolic Arrays","title":"Broadcast, map and reduce","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"A .* b'","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"map(asin, (A*b))","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"#sum(A) #latexify not working","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"typeof(sum(A))","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"typeof(sum(A, dims=2))","category":"page"},{"location":"manual/arrays/#Indexing-and-delayed-computation","page":"Symbolic Arrays","title":"Indexing and delayed computation","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Indexing array expressions is fairly flexible in Symbolics. Let's go through all the possible ways to index arrays.","category":"page"},{"location":"manual/arrays/#Scalar-indexing-and-scalarization","page":"Symbolic Arrays","title":"Scalar indexing and scalarization","text":"","category":"section"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"AAt = A*A'","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"AAt[2,3]","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Here we indexed for the element (2,3), but we got back a symbolic indexing expression. You may want to force the element to be computed in terms of the elements of A. This can be done, using the scalarize function.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Symbolics.scalarize(AAt[2,3])","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"@syms i::Int j::Int\nSymbolics.scalarize(AAt[i,j])","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"In general, any scalar expression which is derived from array expressions can be scalarized.","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"#sum(A[:,1]) + sum(A[2,:])#latexify not working","category":"page"},{"location":"manual/arrays/","page":"Symbolic Arrays","title":"Symbolic Arrays","text":"Symbolics.scalarize(sum(A[:,1]) + sum(A[2,:]))","category":"page"},{"location":"manual/limits/#Symbolic-Limits","page":"Symbolic Limits","title":"Symbolic Limits","text":"","category":"section"},{"location":"manual/limits/","page":"Symbolic Limits","title":"Symbolic Limits","text":"Experimental symbolic limit support is provided by the limit function, documented below. See SymbolicLimits.jl for more information and implementation details.","category":"page"},{"location":"manual/limits/","page":"Symbolic Limits","title":"Symbolic Limits","text":"limit","category":"page"},{"location":"manual/limits/#Symbolics.limit","page":"Symbolic Limits","title":"Symbolics.limit","text":"limit(expr, var, h[, side::Symbol])\n\nCompute the limit of expr as var approaches h.\n\nside indicates the direction from which var approaches h. It may be one of :left, :right, or :both. If side is :both and the two sides do not align, an error is thrown. Side defaults to :both for finite h, :left for h = Inf, and :right for h = -Inf.\n\nexpr must be compoesed of log, exp, constants, and the rational opperators +, -, *, and /. This limitation may eventually be relaxed.\n\nwarning: Warning\nBecause symbolic limit computation is undecidable, this function necessarily employs heuristics and may occasionally return wrong answers. Nevertheless, please report wrong answers as issues as we aim to have heuristics that produce correct answers in all practical cases.\n\n\n\n\n\n","category":"function"},{"location":"manual/functions/#function_registration","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"","category":"section"},{"location":"manual/functions/#Direct-Tracing","page":"Function Registration and Tracing","title":"Direct Tracing","text":"","category":"section"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"Because Symbolics expressions respect Julia semantics, one way to generate symbolic expressions is to simply place Symbolics variables as inputs into existing Julia code. For example, the following uses the standard Julia function for the Lorenz equations to generate the symbolic expression for the Lorenz equations:","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"using Symbolics\nfunction lorenz(du,u,p,t)\n du[1] = 10.0(u[2]-u[1])\n du[2] = u[1]*(28.0-u[3]) - u[2]\n du[3] = u[1]*u[2] - (8/3)*u[3]\nend\n@variables t p[1:3] u(t)[1:3]\ndu = Array{Any}(undef, 3)\nlorenz(du,u,p,t)\ndu","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"Or similarly:","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"@variables t x(t) y(t) z(t) dx(t) dy(t) dz(t) σ ρ β\ndu = [dx,dy,dz]\nu = [x,y,z]\np = [σ,ρ,β]\nlorenz(du,u,p,t)\ndu","category":"page"},{"location":"manual/functions/#Registering-Functions","page":"Function Registration and Tracing","title":"Registering Functions","text":"","category":"section"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"The Symbolics graph only allows registered Julia functions within its type. All other functions are automatically traced down to registered functions. By default, Symbolics.jl pre-registers the common functions utilized in SymbolicUtils.jl and pre-defines their derivatives. However, the user can utilize the @register_symbolic macro to add their function to allowed functions of the computation graph.","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"Additionally, @register_array_symbolic can be used to define array functions. For size propagation it's required that a computation of how the sizes are computed is also supplied.","category":"page"},{"location":"manual/functions/#Defining-Derivatives-of-Registered-Functions","page":"Function Registration and Tracing","title":"Defining Derivatives of Registered Functions","text":"","category":"section"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"In order for symbolic differentiation to work, an overload of Symbolics.derivative is required. The syntax is derivative(typeof(f), args::NTuple{i,Any}, ::Val{j}) where i is the number of arguments to the function and j is which argument is being differentiated. So for example:","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"function derivative(::typeof(min), args::NTuple{2,Any}, ::Val{1})\n x, y = args\n IfElse.ifelse(x < y, one(x), zero(x))\nend","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"is the partial derivative of the Julia min(x,y) function with respect to x.","category":"page"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"note: Note\nDownstream symbolic derivative functionality only work if every partial derivative that is required in the derivative expression is defined. Note that you only need to define the partial derivatives which are actually computed.","category":"page"},{"location":"manual/functions/#Registration-API","page":"Function Registration and Tracing","title":"Registration API","text":"","category":"section"},{"location":"manual/functions/","page":"Function Registration and Tracing","title":"Function Registration and Tracing","text":"@register_symbolic\n@register_array_symbolic","category":"page"},{"location":"manual/functions/#Symbolics.@register_symbolic","page":"Function Registration and Tracing","title":"Symbolics.@register_symbolic","text":"@register_symbolic(expr, define_promotion = true, Ts = [Real])\n\nOverload appropriate methods so that Symbolics can stop tracing into the registered function. If define_promotion is true, then a promotion method in the form of\n\nSymbolicUtils.promote_symtype(::typeof(f_registered), args...) = Real # or the annotated return type\n\nis defined for the register function. Note that when defining multiple register overloads for one function, all the rest of the registers must set define_promotion to false except for the first one, to avoid method overwriting.\n\nExamples\n\n@register_symbolic foo(x, y)\n@register_symbolic foo(x, y::Bool) false # do not overload a duplicate promotion rule\n@register_symbolic goo(x, y::Int) # `y` is not overloaded to take symbolic objects\n@register_symbolic hoo(x, y)::Int # `hoo` returns `Int`\n\nSee @register_array_symbolic to register functions which return arrays.\n\n\n\n\n\n","category":"macro"},{"location":"manual/functions/#Symbolics.@register_array_symbolic","page":"Function Registration and Tracing","title":"Symbolics.@register_array_symbolic","text":"@register_array_symbolic(expr)\n\nExample:\n\n# Let's say vandermonde takes an n-vector and returns an n x n matrix\n@register_array_symbolic vandermonde(x::AbstractVector) begin\n size=(length(x), length(x))\n eltype=eltype(x) # optional, will default to the promoted eltypes of x\nend\n\nYou can also register calls on callable structs:\n\n@register_array_symbolic (c::Conv)(x::AbstractMatrix) begin\n size=size(x) .- size(c.kernel) .+ 1\n eltype=promote_type(eltype(x), eltype(c))\nend\n\n\n\n\n\n","category":"macro"},{"location":"getting_started/#Getting-Started-with-Symbolics.jl","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.jl is a symbolic modeling language. In this tutorial, we will walk you through the process of getting Symbolics.jl up and running, and start doing our first symbolic calculations.","category":"page"},{"location":"getting_started/#Installing-Symbolics.jl","page":"Getting Started with Symbolics.jl","title":"Installing Symbolics.jl","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Installing Symbolics.jl is as simple as using the Julia package manager. This is done via the command ]add Symbolics. After precompilation is complete, do using Symbolics in the terminal (REPL) and when that command is completed, you're ready to start!","category":"page"},{"location":"getting_started/#Building-Symbolic-Expressions","page":"Getting Started with Symbolics.jl","title":"Building Symbolic Expressions","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"The way to define symbolic variables is via the @variables macro:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using Symbolics\n@variables x y","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"After defining variables as symbolic, symbolic expressions, which we call a istree object, can be generated by utilizing Julia expressions. For example:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"z = x^2 + y","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Here, z is an expression tree for “square x and add y”. To make an array of symbolic expressions, simply make an array of symbolic expressions:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"A = [x^2 + y 0 2x\n 0 0 2y\n y^2 + x 0 0]","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Note that by default, @variables returns Sym or istree objects wrapped in Num to make them behave like subtypes of Real. Any operation on these Num objects will return a new Num object, wrapping the result of computing symbolically on the underlying values.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"If you are following this tutorial in the Julia REPL, A will not be shown with LaTeX equations. To get these equations, we can use Latexify.jl. Symbolics.jl comes with Latexify recipes, so it works automatically:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using Latexify\nlatexify(A)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Normal Julia functions work on Symbolics expressions, so if we want to create the sparse version of A we would just call sparse:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using SparseArrays\nspA = sparse(A)\nlatexify(A)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"We can thus use normal Julia functions as generators for sparse expressions. For example, here we will define","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"function f(u)\n [u[1] - u[3], u[1]^2 - u[2], u[3] + u[2]]\nend\nf([x, y, z]) # Recall that z = x^2 + y","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Or we can build an array of variables and use it to trace the function:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"u = Symbolics.variables(:u, 1:5)\nf(u)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"note: Note\nSymbolics.variables(:u, 1:5) creates a Julia array of symbolic variables. This uses O(n) compute and memory but is a very general representation. Symbolics.jl also has the ability to represent symbolic arrays which gives an O(1) representation but is more limited in its functionality. For more information, see the Symbolic Arrays page.","category":"page"},{"location":"getting_started/#Derivatives","page":"Getting Started with Symbolics.jl","title":"Derivatives","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"One common thing to compute in a symbolic system is derivatives. In Symbolics.jl, derivatives are represented lazily via operations, just like any other function. To build a differential operator, use Differential like:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"@variables t\nD = Differential(t)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"This is the differential operator D = fracpartialpartial t. We can compose the differential operator by *, e.g. Differential(t) * Differential(x) or Differential(t)^2. Now let's write down the derivative of some expression:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"z = t + t^2\nD(z)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Notice that this hasn't computed anything yet: D is a lazy operator because it lets us symbolically represent “The derivative of z with respect to t”, which is useful for example when representing our favorite thing in the world, differential equations. However, if we want to expand the derivative operators, we'd use expand_derivatives:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"expand_derivatives(D(z))","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"The variable, that you are taking the derivative with respect to, is accessed with:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"D.x","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"We can also have simplified functions for multivariable calculus. For example, we can compute the Jacobian of an array of expressions like:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.jacobian([x + x*y, x^2 + y], [x, y])","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"and similarly we can do Hessians, gradients, and define whatever other derivatives you want.","category":"page"},{"location":"getting_started/#Simplification-and-Substitution","page":"Getting Started with Symbolics.jl","title":"Simplification and Substitution","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics interfaces with SymbolicUtils.jl to allow for simplifying symbolic expressions. This is done simply through the simplify command:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"simplify(2x + 2y)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"This can be applied to arrays by using Julia's broadcast mechanism:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"B = simplify.([t + t^2 + t + t^2 2t + 4t\n x + y + y + 2t x^2 - x^2 + y^2])","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"We can then use substitute to change values of an expression around:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"simplify.(substitute.(B, (Dict(x => y^2),)))","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"and we can use this to interactively evaluate expressions without generating and compiling Julia functions:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"V = substitute.(B, (Dict(x => 2.0, y => 3.0, t => 4.0),))","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Where we can reference the values via:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.value.(V)","category":"page"},{"location":"getting_started/#Non-Interactive-Development","page":"Getting Started with Symbolics.jl","title":"Non-Interactive Development","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Note that the macros are for the high-level case where you're doing symbolic computation on your own code. If you want to do symbolic computation on someone else's code, like in a macro, you may not want to do @variables x because you might want the name “x” to come from the user's code. For these cases, you can use the interpolation operator to interpolate the runtime value of x, i.e. @variables $x. Check the documentation of @variables for more details.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"a, b, c = :runtime_symbol_value, :value_b, :value_c","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"vars = @variables t $a $b(t) $c(t)[1:3]","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"(t, a, b, c)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"One could also use variable and variables. Read their documentation for more details.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"If we need to use this to generate new Julia code, we can simply convert the output to an Expr:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.toexpr(x + y^2)","category":"page"},{"location":"getting_started/#Syms-and-callable-Syms","page":"Getting Started with Symbolics.jl","title":"Syms and callable Syms","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"In the definition","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"@variables t x(t) y(t)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"t is of type Sym{Real}, but the name x refers to an object that represents the Term x(t). The operation of this expression is itself the object Sym{FnType{Tuple{Real}, Real}}(:x). The type Sym{FnType{...}} represents a callable object. In this case specifically, it's a function that takes 1 Real argument (noted by Tuple{Real}) and returns a Real result. You can call such a callable Sym with either a number or a symbolic expression of a permissible type.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"This expression also defines t as an independent variable, while x(t) and y(t) are dependent variables. This is accounted for in differentiation:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"z = x + y*t\nexpand_derivatives(D(z))","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Since x and y are time-dependent, they are not automatically eliminated from the expression, and thus the D(x) and D(y) operations still exist in the expanded derivatives for correctness.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"We can also define unrestricted functions:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"@variables g(..)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Here, g is a variable that is a function of other variables. Any time that we reference g we have to utilize it as a function:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"z = g(x) + g(y)","category":"page"},{"location":"getting_started/#Registering-Functions","page":"Getting Started with Symbolics.jl","title":"Registering Functions","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"One of the benefits of a one-language Julia symbolic stack is that the primitives are all written in Julia, and therefore it's trivially extendible from Julia itself. By default, new functions are traced to the primitives and the symbolic expressions are written on the primitives. However, we can expand the allowed primitives by registering new functions. For example, let's register a new function h:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"h(x, y) = x^2 + y\n@register_symbolic h(x, y)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"When we use h(x, y), it is a symbolic expression and doesn't expand:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"h(x, y) + y^2","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"To use it with the differentiation system, we need to register its derivatives. We would do it like this:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"# Derivative w.r.t. the first argument\nSymbolics.derivative(::typeof(h), args::NTuple{2,Any}, ::Val{1}) = 2args[1]\n# Derivative w.r.t. the second argument\nSymbolics.derivative(::typeof(h), args::NTuple{2,Any}, ::Val{2}) = 1","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"and now it works with the rest of the system:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.derivative(h(x, y) + y^2, x)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Symbolics.derivative(h(x, y) + y^2, y)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"note: Note\n@register_symbolic only allows for scalar outputs. If full array functions are needed, then see @register_array_symbolic for registering functions of symbolic arrays.","category":"page"},{"location":"getting_started/#Building-Functions","page":"Getting Started with Symbolics.jl","title":"Building Functions","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"The function for building functions from symbolic expressions is the aptly-named build_function. The first argument is the symbolic expression or the array of symbolic expressions to compile, and the trailing arguments are the arguments for the function. For example:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"to_compute = [x^2 + y, y^2 + x]\nf_expr = build_function(to_compute, [x, y])\nBase.remove_linenums!.(f_expr)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"gives back two codes. The first is a function f([x, y]) that computes and builds an output vector [x^2 + y, y^2 + x]. Because this tool was made to be used by all the cool kids writing fast Julia codes, it is specialized to Julia and supports features like StaticArrays. For example:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using StaticArrays\nmyf = eval(f_expr[1])\nmyf(SA[2.0, 3.0])","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"The second function is an in-place non-allocating mutating function which mutates its first value. Thus, we'd use it like the following:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"myf! = eval(f_expr[2])\nout = zeros(2)\nmyf!(out, [2.0, 3.0])\nout","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"To save the symbolic calculations for later, we can take this expression and save it out to a file:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"write(\"function.jl\", string(f_expr[2]))","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Note that if we need to avoid eval, for example to avoid world-age issues, one could do expression = Val{false}:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Base.remove_linenums!(build_function(to_compute, [x, y], expression=Val{false})[1])","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"which will use RuntimeGeneratedFunctions.jl to build Julia functions which avoid world-age issues.","category":"page"},{"location":"getting_started/#Building-Non-Allocating-Parallel-Functions-for-Sparse-Matrices","page":"Getting Started with Symbolics.jl","title":"Building Non-Allocating Parallel Functions for Sparse Matrices","text":"","category":"section"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Now let's show off a little bit. build_function is kind of like if lambdify ate its spinach. To show this, let's build a non-allocating function that fills sparse matrices in a multithreaded manner. To do this, we just have to represent the operation that we're turning into a function via a sparse matrix. For example:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using LinearAlgebra\nN = 8\nA = sparse(Tridiagonal([x^i for i in 1:N-1], [x^i * y^(8-i) for i in 1:N], [y^i for i in 1:N-1]))\nshow(A)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Now we call build_function:","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Base.remove_linenums!(build_function(A,[x,y],parallel=Symbolics.MultithreadedForm())[2])","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Let's unpack what that's doing. It's using A.nzval to linearly index through the sparse matrix, avoiding the A[i,j] form because that is a more expensive way to index a sparse matrix if you know exactly the order that the data is stored. Then, it's splitting up the calculation into Julia threads, so they can be operated on in parallel. It synchronizes after spawning all the tasks, so the computation is ensured to be complete before moving on. And it does this with all in-place operations to the original matrix instead of generating arrays. This is the fanciest way you could fill a sparse matrix, and Symbolics makes this dead simple.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"(Note: this example may be slower with multithreading due to the thread spawning overhead, but the full version was not included in the documentation for brevity. It will be the faster version if N is sufficiently large!)","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"Importantly, when using the in-place functions generated by build_function, it is important that the mutating argument matches the sparse structure of the original function, as demonstrated below.","category":"page"},{"location":"getting_started/","page":"Getting Started with Symbolics.jl","title":"Getting Started with Symbolics.jl","text":"using Symbolics, SparseArrays, LinearAlgebra\n\nN = 10\n_S = sprand(N, N, 0.1)\n_Q = sprand(N, N, 0.1)\n\nF(z) = [ # some complicated sparse amenable function\n _S * z\n _Q * z.^2\n]\n\nSymbolics.@variables z[1:N]\n\nsj = Symbolics.sparsejacobian(F(z), z) # sparse jacobian\n\nf_expr = build_function(sj, z)\n\nrows, cols, _ = findnz(sj)\nout = sparse(rows, cols, zeros(length(cols)), size(sj)...) # pre-allocate, and correct structure\nmyf = eval(last(f_expr))\nmyf(out, rand(N)) # note that out matches the sparsity structure of sj\nout","category":"page"},{"location":"manual/expression_manipulation/#Expression-Manipulation","page":"Expression Manipulation","title":"Expression Manipulation","text":"","category":"section"},{"location":"manual/expression_manipulation/","page":"Expression Manipulation","title":"Expression Manipulation","text":"Symbolics.jl provides functionality for easily manipulating expressions. Most of the functionality comes by the expression objects obeying the standard mathematical semantics. For example, if one has A a matrix of symbolic expressions wrapped in Num, then A^2 calculates the expressions for the squared matrix. It is thus encouraged to use standard Julia for performing many of the manipulation on the IR. For example, calculating the sparse form of the matrix via sparse(A) is valid, legible, and easily understandable to all Julia programmers.","category":"page"},{"location":"manual/expression_manipulation/#Functionality-Inherited-From-SymbolicUtils.jl","page":"Expression Manipulation","title":"Functionality Inherited From SymbolicUtils.jl","text":"","category":"section"},{"location":"manual/expression_manipulation/","page":"Expression Manipulation","title":"Expression Manipulation","text":"SymbolicUtils.substitute\nSymbolicUtils.simplify","category":"page"},{"location":"manual/expression_manipulation/#SymbolicUtils.substitute","page":"Expression Manipulation","title":"SymbolicUtils.substitute","text":"substitute(expr, s)\n\nPerforms the substitution on expr according to rule(s) s.\n\nExamples\n\njulia> @variables t x y z(t)\n4-element Vector{Num}:\n t\n x\n y\n z(t)\njulia> ex = x + y + sin(z)\n(x + y) + sin(z(t))\njulia> substitute(ex, Dict([x => z, sin(z) => z^2]))\n(z(t) + y) + (z(t) ^ 2)\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#SymbolicUtils.simplify","page":"Expression Manipulation","title":"SymbolicUtils.simplify","text":"simplify(x; expand=false,\n threaded=false,\n thread_subtree_cutoff=100,\n rewriter=nothing)\n\nSimplify an expression (x) by applying rewriter until there are no changes. expand=true applies expand in the beginning of each fixpoint iteration.\n\nBy default, simplify will assume denominators are not zero and allow cancellation in fractions. Pass simplify_fractions=false to prevent this.\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/","page":"Expression Manipulation","title":"Expression Manipulation","text":"Documentation for rewriter can be found here, using the @rule macro or the @acrule macro from SymbolicUtils.jl.","category":"page"},{"location":"manual/expression_manipulation/#Additional-Manipulation-Functions","page":"Expression Manipulation","title":"Additional Manipulation Functions","text":"","category":"section"},{"location":"manual/expression_manipulation/","page":"Expression Manipulation","title":"Expression Manipulation","text":"Other additional manipulation functions are given below.","category":"page"},{"location":"manual/expression_manipulation/","page":"Expression Manipulation","title":"Expression Manipulation","text":"Symbolics.get_variables\nSymbolics.tosymbol\nSymbolics.diff2term\nSymbolics.solve_for\nSymbolics.degree\nSymbolics.coeff\nSymbolics.replace\nSymbolics.occursin\nSymbolics.filterchildren\nSymbolics.fixpoint_sub\nSymbolics.fast_substitute","category":"page"},{"location":"manual/expression_manipulation/#Symbolics.get_variables","page":"Expression Manipulation","title":"Symbolics.get_variables","text":"get_variables(O) -> Vector{BasicSymbolic}\n\nReturns the variables in the expression. Note that the returned variables are not wrapped in the Num type.\n\nExamples\n\njulia> @variables t x y z(t)\n4-element Vector{Num}:\n t\n x\n y\n z(t)\n\njulia> ex = x + y + sin(z)\n(x + y) + sin(z(t))\n\njulia> Symbolics.get_variables(ex)\n3-element Vector{Any}:\n x\n y\n z(t)\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.tosymbol","page":"Expression Manipulation","title":"Symbolics.tosymbol","text":"tosymbol(x::Union{Num,Symbolic}; states=nothing, escape=true) -> Symbol\n\nConvert x to a symbol. states are the states of a system, and escape means if the target has escapes like val\"y(t)\". If escape is false, then it will only output y instead of y(t).\n\nExamples\n\njulia> @variables t z(t)\n2-element Vector{Num}:\n t\n z(t)\n\njulia> Symbolics.tosymbol(z)\nSymbol(\"z(t)\")\n\njulia> Symbolics.tosymbol(z; escape=false)\n:z\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.diff2term","page":"Expression Manipulation","title":"Symbolics.diff2term","text":"diff2term(x) -> Symbolic\n\nConvert a differential variable to a Term. Note that it only takes a Term not a Num.\n\njulia> @variables x t u(x, t) z(t)[1:2]; Dt = Differential(t); Dx = Differential(x);\n\njulia> Symbolics.diff2term(Symbolics.value(Dx(Dt(u))))\nuˍtx(x, t)\n\njulia> Symbolics.diff2term(Symbolics.value(Dt(z[1])))\nvar\"z(t)[1]ˍt\"\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.solve_for","page":"Expression Manipulation","title":"Symbolics.solve_for","text":"solve_for(eq, var; simplify, check) -> Any\n\n\nSolve equation(s) eqs for a set of variables vars.\n\nAssumes length(eqs) == length(vars)\n\nCurrently only works if all equations are linear. check if the expr is linear w.r.t vars.\n\nExamples\n\njulia> @variables x y\n2-element Vector{Num}:\n x\n y\n\njulia> Symbolics.solve_for(x + y ~ 0, x)\n-y\n\njulia> Symbolics.solve_for([x + y ~ 0, x - y ~ 2], [x, y])\n2-element Vector{Float64}:\n 1.0\n -1.0\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.degree","page":"Expression Manipulation","title":"Symbolics.degree","text":"degree(p, sym=nothing)\n\nExtract the degree of p with respect to sym.\n\nExamples\n\njulia> @variables x;\n\njulia> Symbolics.degree(x^0)\n0\n\njulia> Symbolics.degree(x)\n1\n\njulia> Symbolics.degree(x^2)\n2\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.coeff","page":"Expression Manipulation","title":"Symbolics.coeff","text":"coeff(p, sym=nothing)\n\nExtract the coefficient of p with respect to sym. Note that p might need to be expanded and/or simplified with expand and/or simplify.\n\nExamples\n\njulia> @variables a x y;\n\njulia> Symbolics.coeff(2a, x)\n0\n\njulia> Symbolics.coeff(3x + 2y, y)\n2\n\njulia> Symbolics.coeff(x^2 + y, x^2)\n1\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Base.replace","page":"Expression Manipulation","title":"Base.replace","text":"replace(expr::Symbolic, rules...)\n\nWalk the expression and replace subexpressions according to rules. rules could be rules constructed with @rule, a function, or a pair where the left hand side is matched with equality (using isequal) and is replaced by the right hand side.\n\nRules will be applied left-to-right simultaneously, so only one pattern will be applied to any subexpression, and the patterns will only be applied to the input text, not the replacements.\n\nSet fixpoint = true to repeatedly apply rules until no change to the expression remains to be made.\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Base.occursin","page":"Expression Manipulation","title":"Base.occursin","text":"occursin(c, x)\n\nReturns true if any part of x fufills the condition given in c. c can be a function or an expression. If it is a function, returns true if x is true for any part of x. If c is an expression, returns true if x contains c.\n\nExamples:\n\n@syms x y\nSymbolics.occursin(x, log(x) + x + 1) # returns `true`.\nSymbolics.occursin(x, log(y) + y + 1) # returns `false`.\n\n@variables t X(t)\nD = Differential(t)\nSymbolics.occursin(Symbolics.is_derivative, X + D(X) + D(X^2)) # returns `true`.\n\n\n\n\n\noccursin(needle::Symbolic, haystack::Symbolic)\n\nDetermine whether the second argument contains the first argument. Note that this function doesn't handle associativity, commutativity, or distributivity.\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.filterchildren","page":"Expression Manipulation","title":"Symbolics.filterchildren","text":"filterchildren(c, x) Returns all parts of x that fufills the condition given in c. c can be a function or an expression. If it is a function, returns everything for which the function is true. If c is an expression, returns all expressions that matches it.\n\nExamples:\n\n@syms x\nSymbolics.filterchildren(x, log(x) + x + 1)\n\nreturns [x, x]\n\n@variables t X(t)\nD = Differential(t)\nSymbolics.filterchildren(Symbolics.is_derivative, X + D(X) + D(X^2))\n\nreturns [Differential(t)(X(t)^2), Differential(t)(X(t))]\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.fixpoint_sub","page":"Expression Manipulation","title":"Symbolics.fixpoint_sub","text":"fixpoint_sub(expr, dict; operator = Nothing)\n\nGiven a symbolic expression, equation or inequality expr perform the substitutions in dict recursively until the expression does not change. Substitutions that depend on one another will thus be recursively expanded. For example, fixpoint_sub(x, Dict(x => y, y => 3)) will return 3. The operator keyword can be specified to prevent substitution of expressions inside operators of the given type.\n\nSee also: fast_substitute.\n\n\n\n\n\n","category":"function"},{"location":"manual/expression_manipulation/#Symbolics.fast_substitute","page":"Expression Manipulation","title":"Symbolics.fast_substitute","text":"fast_substitute(expr, dict; operator = Nothing)\n\nGiven a symbolic expression, equation or inequality expr perform the substitutions in dict. This only performs the substitutions once. For example, fast_substitute(x, Dict(x => y, y => 3)) will return y. The operator keyword can be specified to prevent substitution of expressions inside operators of the given type.\n\nSee also: fixpoint_sub.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Derivatives-and-Differentials","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"","category":"section"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"A Differential(op) is a partial derivative with respect to op, which can then be applied to some other operations. For example, D=Differential(t) is what would commonly be referred to as d/dt, which can then be applied to other operations using its function call, so D(x+y) is d(x+y)/dt.","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"By default, the derivatives are left unexpanded to capture the symbolic representation of the differential equation. If the user would like to expand out all the differentials, the expand_derivatives function eliminates all the differentials down to basic one-variable expressions.","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"Differential\nexpand_derivatives\nis_derivative","category":"page"},{"location":"manual/derivatives/#Symbolics.Differential","page":"Derivatives and Differentials","title":"Symbolics.Differential","text":"struct Differential <: Symbolics.Operator\n\nRepresents a differential operator.\n\nFields\n\nx: The variable or expression to differentiate with respect to.\n\nExamples\n\njulia> using Symbolics\n\njulia> @variables x y;\n\njulia> D = Differential(x)\n(D'~x)\n\njulia> D(y) # Differentiate y wrt. x\n(D'~x)(y)\n\njulia> Dx = Differential(x) * Differential(y) # d^2/dxy operator\n(D'~x(t)) ∘ (D'~y(t))\n\njulia> D3 = Differential(x)^3 # 3rd order differential operator\n(D'~x(t)) ∘ (D'~x(t)) ∘ (D'~x(t))\n\n\n\n\n\n","category":"type"},{"location":"manual/derivatives/#Symbolics.expand_derivatives","page":"Derivatives and Differentials","title":"Symbolics.expand_derivatives","text":"expand_derivatives(O; ...)\nexpand_derivatives(O, simplify; occurrences)\n\n\nTODO\n\nExamples\n\n\njulia> @variables x y z k;\n\njulia> f=k*(abs(x-y)/y-z)^2\nk*((abs(x - y) / y - z)^2)\n\njulia> Dx=Differential(x) # Differentiate wrt x\n(::Differential) (generic function with 2 methods)\n\njulia> dfx=expand_derivatives(Dx(f))\n(k*((2abs(x - y)) / y - 2z)*IfElse.ifelse(signbit(x - y), -1, 1)) / y\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"note: Note\nFor symbolic differentiation, all registered functions in the symbolic expression need a registered derivative. For more information, see the function registration page.","category":"page"},{"location":"manual/derivatives/#High-Level-Differentiation-Functions","page":"Derivatives and Differentials","title":"High-Level Differentiation Functions","text":"","category":"section"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"The following functions are not exported and thus must be accessed in a namespaced way, i.e. Symbolics.jacobian.","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"Symbolics.derivative\nSymbolics.jacobian\nSymbolics.sparsejacobian\nSymbolics.sparsejacobian_vals\nSymbolics.gradient\nSymbolics.hessian\nSymbolics.sparsehessian\nSymbolics.sparsehessian_vals","category":"page"},{"location":"manual/derivatives/#Symbolics.derivative","page":"Derivatives and Differentials","title":"Symbolics.derivative","text":"derivative(O, var; simplify)\n\n\nA helper function for computing the derivative of the expression O with respect to var.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.jacobian","page":"Derivatives and Differentials","title":"Symbolics.jacobian","text":"jacobian(ops, vars; simplify, scalarize)\n\n\nA helper function for computing the Jacobian of an array of expressions with respect to an array of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.sparsejacobian","page":"Derivatives and Differentials","title":"Symbolics.sparsejacobian","text":"sparsejacobian(ops, vars; simplify)\n\n\nA helper function for computing the sparse Jacobian of an array of expressions with respect to an array of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.sparsejacobian_vals","page":"Derivatives and Differentials","title":"Symbolics.sparsejacobian_vals","text":"sparsejacobian_vals(ops, vars, I, J; simplify)\n\n\nA helper function for computing the values of the sparse Jacobian of an array of expressions with respect to an array of variable expressions given the sparsity structure.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.gradient","page":"Derivatives and Differentials","title":"Symbolics.gradient","text":"gradient(O, vars; simplify)\n\n\nA helper function for computing the gradient of the expression O with respect to an array of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.hessian","page":"Derivatives and Differentials","title":"Symbolics.hessian","text":"hessian(O, vars; simplify)\n\n\nA helper function for computing the Hessian of the expression O with respect to an array of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.sparsehessian","page":"Derivatives and Differentials","title":"Symbolics.sparsehessian","text":"sparsehessian(op, vars; simplify, full)\n\n\nA helper function for computing the sparse Hessian of an expression with respect to an array of variable expressions.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Symbolics.sparsehessian_vals","page":"Derivatives and Differentials","title":"Symbolics.sparsehessian_vals","text":"sparsehessian_vals(op, vars, I, J; simplify)\n\n\nA helper function for computing the values of the sparse Hessian of an expression with respect to an array of variable expressions given the sparsity structure.\n\n\n\n\n\n","category":"function"},{"location":"manual/derivatives/#Adding-Analytical-Derivatives","page":"Derivatives and Differentials","title":"Adding Analytical Derivatives","text":"","category":"section"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"There are many derivatives pre-defined by DiffRules.jl. For example,","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"using Symbolics\n@variables x y z\nf(x,y,z) = x^2 + sin(x+y) - z","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"f automatically has the derivatives defined via the tracing mechanism. It will do this by directly building the internals of your function and differentiating that.","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"However, often you may want to define your own derivatives so that way automatic Jacobian etc. calculations can utilize this information. This can allow for more succinct versions of the derivatives to be calculated to scale to larger systems. You can define derivatives for your function via the dispatch:","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"# `N` arguments are accepted by the relevant method of `my_function`\nSymbolics.derivative(::typeof(my_function), args::NTuple{N,Any}, ::Val{i})","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"where i means that it's the derivative with respect to the ith argument. args is the array of arguments, so, for example, if your function is f(x,t), then args = [x,t]. You should return an Term for the derivative of your function.","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"For example, sin(t)'s derivative (by t) is given by the following:","category":"page"},{"location":"manual/derivatives/","page":"Derivatives and Differentials","title":"Derivatives and Differentials","text":"Symbolics.derivative(::typeof(sin), args::NTuple{1,Any}, ::Val{1}) = cos(args[1])","category":"page"},{"location":"#Symbolics.jl","page":"Home","title":"Symbolics.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Symbolics.jl is a fast and modern Computer Algebra System (CAS) for a fast and modern programming language (Julia). The goal is to have a high-performance and parallelized symbolic algebra system that is directly extendable in the same language as that of the users.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install Symbolics.jl, use the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(\"Symbolics\")","category":"page"},{"location":"#Citation","page":"Home","title":"Citation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you use Symbolics.jl, please cite this paper","category":"page"},{"location":"","page":"Home","title":"Home","text":"@article{gowda2021high,\n title={High-performance symbolic-numerics via multiple dispatch},\n author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher},\n journal={arXiv preprint arXiv:2105.03949},\n year={2021}\n}","category":"page"},{"location":"#Feature-Summary","page":"Home","title":"Feature Summary","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Because Symbolics.jl is built into the Julia language and works with its dispatches, generic functions in Base Julia will work with symbolic expressions! Make matrices of symbolic expressions and multiply them: it will just work. Take the LU-factorization. Etc. Thus, see the Julia Documentation for a large list of functionality available in Symbolics.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"A general list of the features is:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Symbolic arithmetic with type information and multiple dispatch\nSymbolic polynomials and trigonometric functions\nPattern matching, simplification and substitution\nDifferentiation\nSymbolic linear algebra (factorizations, inversion, determinants, eigencomputations, etc.)\nDiscrete math (representations of summations, products, binomial coefficients, etc.)\nLogical and Boolean expressions\nSymbolic equation solving and conversion to arbitrary precision\nSupport for non-standard algebras (non-commutative symbols and customizable rulesets)\nSpecial functions (list provided by SpecialFunctions.jl)\nAutomatic conversion of Julia code to symbolic code\nGeneration of (high performance and parallel) functions from symbolic expressions\nFast automated sparsity detection and generation of sparse Jacobians and Hessians","category":"page"},{"location":"","page":"Home","title":"Home","text":"and much more.","category":"page"},{"location":"#Extension-Packages","page":"Home","title":"Extension Packages","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Below is a list of known extension packages. If you would like your package to be listed here, feel free to open a pull request!","category":"page"},{"location":"","page":"Home","title":"Home","text":"ModelingToolkit.jl: Symbolic representations of common numerical systems\nOrdinary differential equations\nStochastic differential equations\nPartial differential equations\nNonlinear systems\nOptimization problems\nOptimal Control\nCausal and acausal modeling (Simulink/Modelica)\nAutomated model transformation, simplification, and composition\nCatalyst.jl: Symbolic representations of chemical reactions\nSymbolically build and represent large systems of chemical reactions\nGenerate code for ODEs, SDEs, continuous-time Markov Chains, and more\nSimulate the models using the SciML ecosystem with O(1) Gillespie methods\nDataDrivenDiffEq.jl: Automatic identification of equations from data\nAutomated construction of ODEs and DAEs from data\nRepresentations of Koopman operators and Dynamic Mode Decomposition (DMD)\nSymbolicRegression.jl: Distributed High-Performance symbolic regression\nParallelized generic algorithms for finding equations from data\nPareto frontier-based scoring\nReversePropagation.jl: Source-to-source reverse mode automatic differentiation\nAutomated tracing of code and construction of backpropagation equations\nComposes with symbolic transformation and simplification functionality","category":"page"},{"location":"#Reproducibility","page":"Home","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"using TOML\nusing Markdown\nversion = TOML.parse(read(\"../../Project.toml\", String))[\"version\"]\nname = TOML.parse(read(\"../../Project.toml\", String))[\"name\"]\nlink_manifest = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Manifest.toml\"\nlink_project = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Project.toml\"\nMarkdown.parse(\"\"\"You can also download the\n[manifest]($link_manifest)\nfile and the\n[project]($link_project)\nfile.\n\"\"\")","category":"page"},{"location":"examples/perturbation/#perturb_alg","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"","category":"section"},{"location":"examples/perturbation/#Background","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Background","text":"","category":"section"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Symbolics.jl is a fast and modern Computer Algebra System (CAS) written in the Julia Programming Language. It is an integral part of the SciML ecosystem of differential equation solvers and scientific machine learning packages. While Symbolics.jl is primarily designed for modern scientific computing (e.g., auto-differentiation, machine learning), it is a powerful CAS that can also be useful for classic scientific computing. One such application is using the perturbation theory to solve algebraic and differential equations.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Perturbation methods are a collection of techniques to solve intractable problems that generally don't have a closed solution but depend on a tunable parameter and have closed or easy solutions for some values of the parameter. The main idea is to assume a solution as a power series in the tunable parameter (say ϵ), such that ϵ = 0 corresponds to an easy solution.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We will discuss the general steps of the perturbation methods to solve algebraic (this tutorial) and differential equations","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"The hallmark of the perturbation method is the generation of long and convoluted intermediate equations, which are subjected to algorithmic and mechanical manipulations. Therefore, these problems are well suited for CAS. In fact, CAS software packages have been used to help with the perturbation calculations since the early 1970s.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"In this tutorial, our goal is to show how to use a mix of symbolic manipulations (Symbolics.jl) and numerical methods to solve simple perturbation problems.","category":"page"},{"location":"examples/perturbation/#Solving-the-Quintic","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Solving the Quintic","text":"","category":"section"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We start with the “hello world!” analog of the perturbation problems, solving the quintic (fifth-order) equations. We want to find a real valued x such that x^5 + x = 1. According to the Abel's theorem, a general quintic equation does not have a closed form solution. Of course, we can easily solve this equation numerically; for example, by using the Newton's method. We use the following implementation of the Newton's method:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"using Symbolics, SymbolicUtils\n\nfunction solve_newton(f, x, x₀; abstol=1e-8, maxiter=50)\n xₙ = Float64(x₀)\n fₙ₊₁ = x - f / Symbolics.derivative(f, x)\n\n for i = 1:maxiter\n xₙ₊₁ = substitute(fₙ₊₁, Dict(x => xₙ))\n if abs(xₙ₊₁ - xₙ) < abstol\n return xₙ₊₁\n else\n xₙ = xₙ₊₁\n end\n end\n return xₙ₊₁\nend","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"In this code, Symbolics.derivative(eq, x) does exactly what it names implies: it calculates the symbolic derivative of eq (a Symbolics.jl expression) with respect to x (a Symbolics.jl variable). We use Symbolics.substitute(eq, D) to evaluate the update formula by substituting variables or sub-expressions (defined in a dictionary D) in eq. It should be noted that substitute is the workhorse of our code and will be used multiple times in the rest of these tutorials. solve_newton is written with simplicity and clarity in mind, and not performance.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Let's go back to our quintic. We can define a Symbolics variable as @variables x and then solve the equation solve_newton(x^5 + x - 1, x, 1.0) (here, x₀ = 1.0 is our first guess). The answer is 0.7549. Now, let's see how we can solve the same problem using the perturbation methods.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We introduce a tuning parameter epsilon into our equation: x^5 + epsilon x = 1. If epsilon = 1, we get our original problem. For epsilon = 0, the problem transforms to an easy one: x^5 = 1 which has an exact real solution x = 1 (and four complex solutions which we ignore here). We expand x as a power series on epsilon:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" x(epsilon) = a_0 + a_1 epsilon + a_2 epsilon^2 + O(epsilon^3)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"a_0","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"is the solution of the easy equation, therefore a_0 = 1. Substituting into the original problem,","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" (a_0 + a_1 epsilon + a_2 epsilon^2)^5 + epsilon (a_0 + a_1 epsilon + a_2 epsilon^2) - 1 = 0","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Expanding the equations, we get","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" epsilon (1 + 5 a_1) + epsilon^2 (a_1 + 5 a_2 + 10 a_1^2) + 𝑂(epsilon^3) = 0","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"This equation should hold for each power of epsilon. Therefore,","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" 1 + 5 a_1 = 0","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"and","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" a_1 + 5 a_2 + 10 a_1^2 = 0","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"This system of equations does not initially seem to be linear because of the presence of terms like 10 a_1^2, but upon closer inspection is found to be linear (this is a feature of the perturbation methods). In addition, the system is in a triangular form, meaning the first equation depends only on a_1, the second one on a_1 and a_2, such that we can replace the result of a_1 from the first one into the second equation and remove the non-linear term. We solve the first equation to get a_1 = -frac15. Substituting in the second one and solve for a_2:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" a_2 = frac(-frac15 + 10(-(frac15)²)5 = -frac125","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Finally,","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" x(epsilon) = 1 - fracepsilon5 - fracepsilon^225 + O(epsilon^3)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Solving the original problem, x(1) = 076, compared to 0.7548 calculated numerically. We can improve the accuracy by including more terms in the expansion of x. However, the calculations, while straightforward, become messy and intractable to do manually very quickly. This is why a CAS is very helpful to solve perturbation problems.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Now, let's see how we can do these calculations in Julia. Let n be the order of the expansion. We start by defining the symbolic variables:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"n = 2\n@variables ϵ a[1:n]","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Then, we define","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"x = 1 + a[1]*ϵ + a[2]*ϵ^2","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"The next step is to substitute x in the problem equation","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" eq = x^5 + ϵ*x - 1","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"The expanded form of eq is","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"expand(eq)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We need a way to get the coefficients of different powers of ϵ. Function collect_powers(eq, x, ns) returns the powers of variable x in expression eq. Argument ns is the range of the powers.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"function collect_powers(eq, x, ns; max_power=100)\n eq = substitute(expand(eq), Dict(x^j => 0 for j=last(ns)+1:max_power))\n\n eqs = []\n for i in ns\n powers = Dict(x^j => (i==j ? 1 : 0) for j=1:last(ns))\n push!(eqs, substitute(eq, powers))\n end\n eqs\nend","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"To return the coefficients of ϵ and ϵ^2 in eq, we can write","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"eqs = collect_powers(eq, ϵ, 1:2)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"A few words on how collect_powers works, It uses substitute to find the coefficient of a given power of x by passing a Dict with all powers of x set to 0, except the target power which is set to 1. For example, the following expression returns the coefficient of ϵ^2 in eq,","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"substitute(expand(eq), Dict(\n ϵ => 0,\n ϵ^2 => 1,\n ϵ^3 => 0,\n ϵ^4 => 0,\n ϵ^5 => 0,\n ϵ^6 => 0,\n ϵ^7 => 0,\n ϵ^8 => 0)\n)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Back to our problem. Having the coefficients of the powers of ϵ, we can set each equation in eqs to 0 (remember, we rearrange the problem such that eq is 0) and solve the system of linear equations to find the numerical values of the coefficients. Symbolics.jl has a function Symbolics.solve_for that can solve systems of linear equations. However, the presence of higher-order terms in eqs prevents Symbolics.solve_for(eqs .~ 0, a) from workings properly. Instead, we can exploit the fact that our system is in a triangular form and start by solving eqs[1] for a₁ and then substitute this in eqs[2] and solve for a₂, and so on. This cascading process is done by function solve_coef(eqs, ps):","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"function solve_coef(eqs, ps)\n vals = Dict()\n\n for i = 1:length(ps)\n eq = substitute(eqs[i], vals)\n vals[ps[i]] = Symbolics.solve_for(eq ~ 0, ps[i])\n end\n vals\nend","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Here, eqs is an array of expressions (assumed to be equal to 0) and ps is an array of variables. The result is a dictionary of variable => value pairs. We apply solve_coef to eqs to get the numerical values of the parameters:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"vals = solve_coef(eqs, a)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Finally, we substitute back the values of a in the definition of x as a function of 𝜀. Note that 𝜀 is a number (usually Float64), whereas ϵ is a symbolic variable.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"X = 𝜀 -> 1 + vals[a[1]]*𝜀 + vals[a[2]]*𝜀^2","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Therefore, the solution to our original problem becomes X(1), which is equal to 0.76. We can use larger values of n to improve the accuracy of estimations.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"n x\n1 0.8\n2 0.76\n3 0.752\n4 0.752\n5 0.7533\n6 0.7543\n7 0.7548\n8 0.7550","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Remember, the numerical value is 0.7549. The two functions collect_powers and solve_coef(eqs, a) are used in all the examples in this and the next tutorial.","category":"page"},{"location":"examples/perturbation/#Solving-the-Kepler's-Equation","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Solving the Kepler's Equation","text":"","category":"section"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Historically, the perturbation methods were first invented to solve orbital calculations of the Moon and the planets. In homage to this history, our second example has a celestial theme. Our goal is solving the Kepler's equation:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":" E - esin(E) = M","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"where e is the eccentricity of the elliptical orbit, M is the mean anomaly, and E (unknown) is the eccentric anomaly (the angle between the position of a planet in an elliptical orbit and the point of periapsis). This equation is central to solving two-body Keplerian orbits.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Similar to the first example, it is easy to solve this problem using the Newton's method. For example, let e = 001671 (the eccentricity of the Earth) and M = pi2. We have solve_newton(x - e*sin(x) - M, x, M) equals to 1.5875 (compared to π/2 = 1.5708). Now, we try to solve the same problem using the perturbation techniques (see function test_kepler).","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"For e = 0, we get E = M. Therefore, we can use e as our perturbation parameter. For consistency with other problems, we also rename e to epsilon and E to x.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"From here on, we use the helper function def_taylor to define Taylor's series by calling it as x = def_taylor(ϵ, a, 1), where the arguments are, respectively, the perturbation variable, which is an array of coefficients (starting from the coefficient of epsilon^1), and an optional constant term.","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"def_taylor(x, ps) = sum([a*x^i for (i,a) in enumerate(ps)])\ndef_taylor(x, ps, p₀) = p₀ + def_taylor(x, ps)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We start by defining the variables (assuming n = 3):","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"n = 3\n@variables ϵ M a[1:n]\nx = def_taylor(ϵ, a, M)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We further simplify by substituting sin with its power series using the expand_sin helper function:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"expand_sin(x, n) = sum([(isodd(k) ? -1 : 1)*(-x)^(2k-1)/factorial(2k-1) for k=1:n])","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"To test,","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"expand_sin(0.1, 10) ≈ sin(0.1)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"The problem equation is","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"eq = x - ϵ * expand_sin(x, n) - M","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"We follow the same process as the first example. We collect the coefficients of the powers of ϵ","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"eqs = collect_powers(eq, ϵ, 1:n)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"and then solve for a:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"vals = solve_coef(eqs, a)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"Finally, we substitute vals back in x:","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"x′ = substitute(x, vals)\nX = (𝜀, 𝑀) -> substitute(x′, Dict(ϵ => 𝜀, M => 𝑀))\nX(0.01671, π/2)","category":"page"},{"location":"examples/perturbation/","page":"Mixed Symbolic-Numeric Perturbation Theory","title":"Mixed Symbolic-Numeric Perturbation Theory","text":"The result is 1.5876, compared to the numerical value of 1.5875. 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GitHub

Automated Sparse Parallelism of Julia Functions via Tracing

Because the Symbolics.jl expressions obey Julia semantics, one can directly transform existing Julia functions into Symbolics.jl symbolic representations of the function by simply inputting the symbolic values into the function and using what is returned. For example, let's take the following numerical PDE discretization:

using Symbolics, LinearAlgebra, SparseArrays, Plots
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+
+const Mx = Array(Tridiagonal([1.0 for i in 1:N-1], [-2.0 for i in 1:N], [1.0 for i in 1:N-1]))
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+    cat(dA, dB, dC, dims=3)
+end
f (generic function with 1 method)

We can build the Symbolics version of this model by tracing the model function:

# Define the initial condition as normal arrays
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+du = simplify.(f(collect(u), nothing, 0.0))
+vec(du)[1:10]

\[ \begin{equation} +\left[ +\begin{array}{c} + - u_{1}ˏ_{1}ˏ_1 + 100 \left( - 4 u_{1}ˏ_{1}ˏ_1 + 2 \left( u_{1}ˏ_{2}ˏ_1 + u_{2}ˏ_{1}ˏ_1 \right) \right) + u_{1}ˏ_{1}ˏ_3 - u_{1}ˏ_{1}ˏ_1 u_{1}ˏ_{1}ˏ_2 \\ +100 \left( u_{1}ˏ_{1}ˏ_1 - 4 u_{2}ˏ_{1}ˏ_1 + 2 u_{2}ˏ_{2}ˏ_1 + u_{3}ˏ_{1}ˏ_1 \right) - u_{2}ˏ_{1}ˏ_1 + u_{2}ˏ_{1}ˏ_3 - u_{2}ˏ_{1}ˏ_1 u_{2}ˏ_{1}ˏ_2 \\ +100 \left( u_{2}ˏ_{1}ˏ_1 - 4 u_{3}ˏ_{1}ˏ_1 + 2 u_{3}ˏ_{2}ˏ_1 + u_{4}ˏ_{1}ˏ_1 \right) - u_{3}ˏ_{1}ˏ_1 + u_{3}ˏ_{1}ˏ_3 - u_{3}ˏ_{1}ˏ_1 u_{3}ˏ_{1}ˏ_2 \\ +100 \left( u_{3}ˏ_{1}ˏ_1 - 4 u_{4}ˏ_{1}ˏ_1 + 2 u_{4}ˏ_{2}ˏ_1 + u_{5}ˏ_{1}ˏ_1 \right) - u_{4}ˏ_{1}ˏ_1 + u_{4}ˏ_{1}ˏ_3 - u_{4}ˏ_{1}ˏ_1 u_{4}ˏ_{1}ˏ_2 \\ +100 \left( u_{4}ˏ_{1}ˏ_1 - 4 u_{5}ˏ_{1}ˏ_1 + 2 u_{5}ˏ_{2}ˏ_1 + u_{6}ˏ_{1}ˏ_1 \right) - u_{5}ˏ_{1}ˏ_1 + u_{5}ˏ_{1}ˏ_3 - u_{5}ˏ_{1}ˏ_1 u_{5}ˏ_{1}ˏ_2 \\ +100 \left( u_{5}ˏ_{1}ˏ_1 - 4 u_{6}ˏ_{1}ˏ_1 + 2 u_{6}ˏ_{2}ˏ_1 + u_{7}ˏ_{1}ˏ_1 \right) - u_{6}ˏ_{1}ˏ_1 + u_{6}ˏ_{1}ˏ_3 - u_{6}ˏ_{1}ˏ_1 u_{6}ˏ_{1}ˏ_2 \\ +100 \left( u_{6}ˏ_{1}ˏ_1 - 4 u_{7}ˏ_{1}ˏ_1 + 2 u_{7}ˏ_{2}ˏ_1 + u_{8}ˏ_{1}ˏ_1 \right) - u_{7}ˏ_{1}ˏ_1 + u_{7}ˏ_{1}ˏ_3 - u_{7}ˏ_{1}ˏ_1 u_{7}ˏ_{1}ˏ_2 \\ +100 \left( u_{7}ˏ_{1}ˏ_1 - 4 u_{8}ˏ_{1}ˏ_1 + 2 u_{8}ˏ_{2}ˏ_1 + u_{9}ˏ_{1}ˏ_1 \right) - u_{8}ˏ_{1}ˏ_1 + u_{8}ˏ_{1}ˏ_3 - u_{8}ˏ_{1}ˏ_1 u_{8}ˏ_{1}ˏ_2 \\ +100 \left( u_1_{0}ˏ_{1}ˏ_1 + u_{8}ˏ_{1}ˏ_1 - 4 u_{9}ˏ_{1}ˏ_1 + 2 u_{9}ˏ_{2}ˏ_1 \right) - u_{9}ˏ_{1}ˏ_1 + u_{9}ˏ_{1}ˏ_3 - u_{9}ˏ_{1}ˏ_1 u_{9}ˏ_{1}ˏ_2 \\ + - u_1_{0}ˏ_{1}ˏ_1 + 100 \left( - 4 u_1_{0}ˏ_{1}ˏ_1 + 2 u_1_{0}ˏ_{2}ˏ_1 + u_1_{1}ˏ_{1}ˏ_1 + u_{9}ˏ_{1}ˏ_1 \right) + u_1_{0}ˏ_{1}ˏ_3 - u_1_{0}ˏ_{1}ˏ_1 u_1_{0}ˏ_{1}ˏ_2 \\ +\end{array} +\right] +\end{equation} + \]

The output, here the in-place modified du, is a symbolic representation of each output of the function. We can then utilize this in the Symbolics functionality. For example, let's build a parallel version of f first:

fastf = eval(Symbolics.build_function(du,u,
+            parallel=Symbolics.MultithreadedForm())[2])
#13 (generic function with 1 method)

Now let's compute the sparse Jacobian function and compile a fast multithreaded version:

jac = Symbolics.sparsejacobian(vec(du), vec(u))
+row,col,val = findnz(jac)
+scatter(row,col,legend=false,ms=1,c=:black)
Example block output
fjac = eval(Symbolics.build_function(jac,u,
+            parallel=Symbolics.MultithreadedForm())[2])
#15 (generic function with 1 method)

It takes awhile for this to generate, but the results will be worth it! Now let's set up the parabolic PDE to be solved by DifferentialEquations.jl. We will set up the vanilla version and the sparse multithreaded version:

using OrdinaryDiffEq
+u0 = zeros(N, N, 3)
+MyA = zeros(N, N);
+AMx = zeros(N, N);
+DA = zeros(N, N);
+prob = ODEProblem(f, u0, (0.0, 10.0))
+fastprob = ODEProblem(ODEFunction((du, u, p, t) -> fastf(du, u),
+                                   jac = (du, u, p, t) -> fjac(du, u),
+                                   jac_prototype = similar(jac, Float64)),
+                                   u0, (0.0, 10.0))
ODEProblem with uType Array{Float64, 3} and tType Float64. In-place: true
+timespan: (0.0, 10.0)
+u0: 32×32×3 Array{Float64, 3}:
+[:, :, 1] =
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ ⋮                        ⋮              ⋱  ⋮                        ⋮    
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+
+[:, :, 2] =
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ ⋮                        ⋮              ⋱  ⋮                        ⋮    
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+
+[:, :, 3] =
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ ⋮                        ⋮              ⋱  ⋮                        ⋮    
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0

Let's see the timing difference:

using BenchmarkTools
+#@btime solve(prob, TRBDF2()); # 33.073 s (895404 allocations: 23.87 GiB)
+#warning the following solve takes a long time to compile, but afterwards is very fast.
+#@btime solve(fastprob, TRBDF2()); # 209.670 ms (8208 allocations: 109.25 MiB)

Boom, an automatic 157x acceleration that grows as the size of the problem increases!

diff --git a/v5.27.1/tutorials/converting_to_C/index.html b/v5.27.1/tutorials/converting_to_C/index.html new file mode 100644 index 000000000..6acd3bbcf --- /dev/null +++ b/v5.27.1/tutorials/converting_to_C/index.html @@ -0,0 +1,28 @@ + +Automatic Conversion of Julia Code to C Functions · Symbolics.jl
GitHub

Automatic Conversion of Julia Code to C Functions

Since Symbolics.jl can trace Julia code into Symbolics IR that can be built and compiled via build_function to C, this gives us a nifty way to automatically generate C functions from Julia code! To see this in action, let's start with the Lotka-Volterra equations:

using Symbolics
+function lotka_volterra!(du, u, p, t)
+  x, y = u
+  α, β, δ, γ = p
+  du[1] = dx = α*x - β*x*y
+  du[2] = dy = -δ*y + γ*x*y
+end
lotka_volterra! (generic function with 1 method)

Now we trace this into Symbolics:

@variables t du[1:2] u[1:2] p[1:4]
+du = collect(du)
+lotka_volterra!(du, u, p, t)
+du

\[ \begin{equation} +\left[ +\begin{array}{c} +p_1 u_1 - p_2 u_1 u_2 \\ + - p_3 u_2 + p_4 u_1 u_2 \\ +\end{array} +\right] +\end{equation} + \]

and then we build the function:

build_function(du, u, p, t, target=Symbolics.CTarget())
"#include <math.h>\nvoid diffeqf(double* du, const double* RHS1, const double* RHS2, const double RHS3) {\n  du[0] = RHS2[0] * RHS1[0] + -1 * RHS2[1] * RHS1[0] * RHS1[1];\n  du[1] = -1 * RHS2[2] * RHS1[1] + RHS2[3] * RHS1[0] * RHS1[1];\n}\n"

If we want to compile this, we do expression=Val{false}:

f = build_function(du, u, p, t, target=Symbolics.CTarget(), expression=Val{false})
RuntimeGeneratedFunction(#=in Symbolics=#, #=using Symbolics=#, :((du, u, p, t)->begin
+          #= /home/runner/work/Symbolics.jl/Symbolics.jl/src/build_function.jl:809 =#
+          ccall(("diffeqf", "/tmp/jl_dcVunonY1m"), Cvoid, (Ptr{Float64}, Ptr{Float64}, Ptr{Float64}, Float64), du, u, p, t)
+      end))

now we check it computes the same thing:

du = rand(2); du2 = rand(2)
+u = rand(2)
+p = rand(4)
+t = rand()
+f(du, u, p, t)
+lotka_volterra!(du2, u, p, t)
+du == du2 # true!
true
diff --git a/versions.js b/versions.js index 1c1c6e9f9..f415923c6 100644 --- a/versions.js +++ b/versions.js @@ -40,5 +40,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v5.27.0"; +var DOCUMENTER_NEWEST = "v5.27.1"; var DOCUMENTER_STABLE = "stable";