From 6113b38b53ae0322a3ed4c6fb93407abc49695f5 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 7 Mar 2024 13:32:06 +0000 Subject: [PATCH] build based on e437e29 --- stable | 2 +- v0.2 | 2 +- v0.2.2/assets/documenter.js | 331 ++++++++++++++++++++++ v0.2.2/assets/search.js | 267 +++++++++++++++++ v0.2.2/assets/themes/documenter-dark.css | 7 + v0.2.2/assets/themes/documenter-light.css | 9 + v0.2.2/assets/themeswap.js | 66 +++++ v0.2.2/assets/warner.js | 49 ++++ v0.2.2/index.html | 32 +++ v0.2.2/search/index.html | 2 + v0.2.2/search_index.js | 3 + v0.2.2/siteinfo.js | 1 + versions.js | 2 +- 13 files changed, 770 insertions(+), 3 deletions(-) create mode 100644 v0.2.2/assets/documenter.js create mode 100644 v0.2.2/assets/search.js create mode 100644 v0.2.2/assets/themes/documenter-dark.css create mode 100644 v0.2.2/assets/themes/documenter-light.css create mode 100644 v0.2.2/assets/themeswap.js create mode 100644 v0.2.2/assets/warner.js create mode 100644 v0.2.2/index.html create mode 100644 v0.2.2/search/index.html create mode 100644 v0.2.2/search_index.js create mode 100644 v0.2.2/siteinfo.js diff --git a/stable b/stable index eac0a14..eb9e63b 120000 --- a/stable +++ b/stable @@ -1 +1 @@ -v0.2.1 \ No newline at end of file +v0.2.2 \ No newline at end of file diff --git a/v0.2 b/v0.2 index eac0a14..eb9e63b 120000 --- a/v0.2 +++ b/v0.2 @@ -1 +1 @@ -v0.2.1 \ No newline at end of file +v0.2.2 \ No newline at end of file diff --git a/v0.2.2/assets/documenter.js b/v0.2.2/assets/documenter.js new file mode 100644 index 0000000..6adfbbb --- /dev/null +++ b/v0.2.2/assets/documenter.js @@ -0,0 +1,331 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fas", "fa-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-times"); + 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-times"); + }, 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() { + $('#documenter .docs-navbar').headroom({ + "tolerance": {"up": 10, "down": 10}, + }); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// 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 + $(document).keyup(function(e) { + if (e.keyCode == 27) settings.removeClass('is-active'); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// 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 = $("#documenter .docs-menu .is-active").get(0); + if(typeof active !== 'undefined') { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +function set_theme(theme) { + var active = null; + var disabled = []; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + var themename = ss.ownerNode.getAttribute("data-theme-name"); + if(themename === null) continue; // ignore non-theme stylesheets + // Find the active theme + if(themename === theme) active = ss; + else disabled.push(ss); + } + if(active !== null) { + active.disabled = false; + if(active.ownerNode.getAttribute("data-theme-primary") === null) { + document.getElementsByTagName('html')[0].className = "theme--" + theme; + } else { + document.getElementsByTagName('html')[0].className = ""; + } + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + + // Store the theme in localStorage + if(typeof(window.localStorage) !== "undefined") { + window.localStorage.setItem("documenter-theme", theme); + } else { + console.error("Browser does not support window.localStorage"); + } +} + +// Theme picker setup +$(document).ready(function() { + // onchange callback + $('#documenter-themepicker').change(function themepick_callback(ev){ + var themename = $('#documenter-themepicker option:selected').attr('value'); + set_theme(themename); + }); + + // 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); + }) + } else { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = $("html").hasClass(`theme--${e.value}`); + }) + } + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function() { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function(x) { + target_href = version_selector_select.children("option:selected").get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { + var option = $(""); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== 'undefined') { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); + DOC_VERSIONS.forEach(function(each) { + var version_url = documenterBaseURL + "/../" + each; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $(""); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}) + +}) diff --git a/v0.2.2/assets/search.js b/v0.2.2/assets/search.js new file mode 100644 index 0000000..c133f74 --- /dev/null +++ b/v0.2.2/assets/search.js @@ -0,0 +1,267 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', + 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + } +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { + +$(document).ready(function() { + // parseUri 1.2.2 + // (c) Steven Levithan + // MIT License + function parseUri (str) { + var o = parseUri.options, + m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), + uri = {}, + i = 14; + + while (i--) uri[o.key[i]] = m[i] || ""; + + uri[o.q.name] = {}; + uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { + if ($1) uri[o.q.name][$1] = $2; + }); + + return uri; + }; + parseUri.options = { + strictMode: false, + key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], + q: { + name: "queryKey", + parser: /(?:^|&)([^&=]*)=?([^&]*)/g + }, + parser: { + strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, + loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ + } + }; + + $("#search-form").submit(function(e) { + e.preventDefault() + }) + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'does', + 'either', + 'ever', + 'every', + 'from', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'into', + 'it', + 'its', + 'just', + 'least', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'who', + 'whom', + 'why', + 'will', + 'would', + 'yet', + 'you', + 'your' + ]) + + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" + // would not find anything if searching for "add!", only for the entire qualification + lunr.tokenizer.separator = /[\s\-\.]+/ + + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') + }) + } + + lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') + lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') + + var index = lunr(function () { + this.ref('location') + this.field('title',{boost: 100}) + this.field('text') + documenterSearchIndex['docs'].forEach(function(e) { + this.add(e) + }, this) + }) + var store = {} + + documenterSearchIndex['docs'].forEach(function(e) { + store[e.location] = {title: e.title, category: e.category, page: e.page} + }) + + $(function(){ + searchresults = $('#documenter-search-results'); + searchinfo = $('#documenter-search-info'); + searchbox = $('#documenter-search-query'); + searchform = $('.docs-search'); + sidebar = $('.docs-sidebar'); + function update_search(querystring) { + tokens = lunr.tokenizer(querystring) + results = index.query(function (q) { + tokens.forEach(function (t) { + q.term(t.toString(), { + fields: ["title"], + boost: 100, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["title"], + boost: 10, + usePipeline: true, + editDistance: 2, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["text"], + boost: 1, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + }) + }) + searchinfo.text("Number of results: " + results.length) + searchresults.empty() + results.forEach(function(result) { + data = store[result.ref] + link = $(''+data.title+'') + link.attr('href', documenterBaseURL+'/'+result.ref) + if (data.category != "page"){ + cat = $('('+data.category+', '+data.page+')') + } else { + cat = $('('+data.category+')') + } + li = $('
  • ').append(link).append(" ").append(cat) + searchresults.append(li) + }) + } + + function update_search_box() { + querystring = searchbox.val() + update_search(querystring) + } + + searchbox.keyup(_.debounce(update_search_box, 250)) + searchbox.change(update_search_box) + + // Disable enter-key form submission for the searchbox on the search page + // and just re-run search rather than refresh the whole page. + searchform.keypress( + function(event){ + if (event.which == '13') { + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + update_search_box(); + event.preventDefault(); + } + } + ); + + search_query_uri = parseUri(window.location).queryKey["q"] + if(search_query_uri !== undefined) { + search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) + searchbox.val(search_query) + } + update_search_box(); + }) +}) + +}) diff --git a/v0.2.2/assets/themes/documenter-dark.css b/v0.2.2/assets/themes/documenter-dark.css new file mode 100644 index 0000000..c94a294 --- /dev/null +++ b/v0.2.2/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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.button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content 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    Edit on GitHub

    MultivariateBases

    MultivariateBases.jl is a standardized API for multivariate polynomial bases based on the MultivariatePolynomials API.

    MultivariateBases.AbstractPolynomialBasis — Type
    abstract type AbstractPolynomialBasis end

    Polynomial basis of a subspace of the polynomials [Section~3.1.5, BPT12].

    [BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics, 2012.

    source
    MultivariateBases.maxdegree_basis — Function
    maxdegree_basis(B::Type{<:AbstractPolynomialBasis}, variables, maxdegree::Int)

    Return the basis of type B generating all polynomials of degree up to maxdegree with variables variables.

    source
    MultivariateBases.basis_covering_monomials — Function
    basis_covering_monomials(B::Type{<:AbstractPolynomialBasis}, monos::AbstractVector{<:AbstractMonomial})

    Return the minimal basis of type B that can generate all polynomials of the monomial basis generated by monos.

    Examples

    For example, to generate all the polynomials with nonzero coefficients for the monomials x^4 and x^2, we need three polynomials as otherwise, we generate polynomials with nonzero constant term.

    julia> using DynamicPolynomials
+
+julia> @polyvar x
+(x,)
+
+julia> basis_covering_monomials(ChebyshevBasis, [x^2, x^4])
+ChebyshevBasisFirstKind([1.0, -1.0 + 2.0x², 1.0 - 8.0x² + 8.0x⁴])
    source
    MultivariateBases.FixedPolynomialBasis — Type
    struct FixedPolynomialBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis
+    polynomials::PV
+end

    Polynomial basis with the polynomials of the vector polynomials. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.

    source
    MultivariateBases.OrthonormalCoefficientsBasis — Type
    struct OrthonormalCoefficientsBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis
+    polynomials::PV
+end

    Polynomial basis with the polynomials of the vector polynomials that are orthonormal with respect to the inner produce derived from the inner product of their coefficients. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.

    source

    Monomial basis

    MultivariateBases.MonomialBasis — Type
    struct MonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis
+    monomials::MV
+end

    Monomial basis with the monomials of the vector monomials. For instance, MonomialBasis([1, x, y, x^2, x*y, y^2]) is the monomial basis for the subspace of quadratic polynomials in the variables x, y.

    This basis is orthogonal under a scalar product defined with the complex Gaussian measure as density. Once normalized so as to be orthonormal with this scalar product, one get ths ScaledMonomialBasis.

    source
    MultivariateBases.ScaledMonomialBasis — Type
    struct ScaledMonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis
+    monomials::MV
+end

    Scaled monomial basis (see [Section 3.1.5, BPT12]) with the monomials of the vector monomials. Given a monomial $x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ of degree $d = \sum_{i=1}^n \alpha_i$, the corresponding polynomial of the basis is

    \[{d \choose \alpha}^{\frac{1}{2}} x^{\alpha} \quad \text{ where } \quad +{d \choose \alpha} = \frac{d!}{\alpha_1! \alpha_2! \cdots \alpha_n!}.\]

    For instance, create a polynomial with the basis $[xy^2, xy]$ creates the polynomial $\sqrt{3} a xy^2 + \sqrt{2} b xy$ where a and b are new JuMP decision variables. Constraining the polynomial $axy^2 + bxy$ to be zero with the scaled monomial basis constrains a/√3 and b/√2 to be zero.

    This basis is orthonormal under the scalar product:

    \[\langle f, g \rangle = \int_{\mathcal{C}^n} f(z) \overline{g(z)} d\nu_n\]

    where $\nu_n$ is the Gaussian measure on $\mathcal{C}^n$ with the density $\pi^{-n} \exp(-\lVert z \rVert^2)$. See [Section 4; B07] for more details.

    [BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics (2012).

    [B07] Barvinok, Alexander. Integration and optimization of multivariate polynomials by restriction onto a random subspace. Foundations of Computational Mathematics 7.2 (2007): 229-244.

    source

    Orthogonal basis

    MultivariateBases.AbstractMultipleOrthogonalBasis — Type
    abstract type AbstractMultipleOrthogonalBasis{P} <: AbstractPolynomialVectorBasis{P, Vector{P}} end

    Polynomial basis such that $\langle p_i(x), p_j(x) \rangle = 0$ if $i \neq j$ where

    \[\langle p(x), q(x) \rangle = \int p(x)q(x) w(x) dx\]

    where the weight is a product of weight functions $w(x) = w_1(x_1)w_2(x_2) \cdots w_n(x_n)$ in each variable. The polynomial of the basis are product of univariate polynomials: $p(x) = p_1(x_1)p_2(x_2) \cdots p_n(x_n)$. where the univariate polynomials of variable x_i form an univariate orthogonal basis for the weight function w_i(x_i). Therefore, they satisfy the recurrence relation

    \[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

    where reccurence_first_coef gives a_k, reccurence_second_coef gives b_k, reccurence_third_coef gives c_k and reccurence_deno_coef gives d_k.

    source
    MultivariateBases.univariate_orthogonal_basis — Function
    univariate_orthogonal_basis(B::Type{<:AbstractMultipleOrthogonalBasis},
+                            variable::MP.AbstractVariable, degree::Integer)

    Return the vector of univariate polynomials of the basis B up to degree with variable variable.

    source
    MultivariateBases.reccurence_first_coef — Function
    reccurence_first_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

    Return a_{degree} in recurrence equation

    \[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

    source
    MultivariateBases.reccurence_second_coef — Function
    reccurence_second_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

    Return b_{degree} in recurrence equation

    \[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

    source
    MultivariateBases.reccurence_third_coef — Function
    reccurence_third_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

    Return c_{degree} in recurrence equation

    \[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

    source
    MultivariateBases.reccurence_deno_coef — Function
    reccurence_deno_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

    Return d_{degree} in recurrence equation

    \[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

    source
    MultivariateBases.ProbabilistsHermiteBasis — Type
    struct ProbabilistsHermiteBasis{P} <: AbstractHermiteBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2/2)$ over the interval $[-\infty, \infty]$.

    source
    MultivariateBases.PhysicistsHermiteBasis — Type
    struct PhysicistsHermiteBasis{P} <: AbstractHermiteBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2)$ over the interval $[-\infty, \infty]$.

    source
    MultivariateBases.LaguerreBasis — Type
    struct LaguerreBasis{P} <: AbstractMultipleOrthogonalBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x)$ over the interval $[0, \infty]$.

    source
    MultivariateBases.AbstractGegenbauerBasis — Type
    struct AbstractGegenbauerBasis{P} <: AbstractMultipleOrthogonalBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = (1 - x^2)^{\alpha - 1/2}$ over the interval $[-1, 1]$.

    source
    MultivariateBases.LegendreBasis — Type
    struct LegendreBasis{P} <: AbstractGegenbauerBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = 1$ over the interval $[-1, 1]$.

    source
    MultivariateBases.ChebyshevBasisFirstKind — Type
    struct ChebyshevBasisFirstKind{P} <: AbstractChebyshevBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = \frac{1}{\sqrt{1 - x^2}}$ over the interval $[-1, 1]$.

    source
    MultivariateBases.ChebyshevBasisSecondKind — Type
    struct ChebyshevBasisSecondKind{P} <: AbstractChebyshevBasis{P}
+    polynomials::Vector{P}
+end

    Orthogonal polynomial with respect to the univariate weight function $w(x) = \sqrt{1 - x^2}$ over the interval $[-1, 1]$.

    source
    diff --git a/v0.2.2/search/index.html b/v0.2.2/search/index.html new file mode 100644 index 0000000..6ed62e7 --- /dev/null +++ b/v0.2.2/search/index.html @@ -0,0 +1,2 @@ + +Search · MultivariateBases

    Loading search...

      diff --git a/v0.2.2/search_index.js b/v0.2.2/search_index.js new file mode 100644 index 0000000..3aa72e0 --- /dev/null +++ b/v0.2.2/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"","page":"Introduction","title":"Introduction","text":"CurrentModule = MultivariateBases\nDocTestSetup = quote\n using MultivariateBases\nend","category":"page"},{"location":"#MultivariateBases","page":"Introduction","title":"MultivariateBases","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"MultivariateBases.jl is a standardized API for multivariate polynomial bases based on the MultivariatePolynomials API.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"AbstractPolynomialBasis\nmaxdegree_basis\nbasis_covering_monomials\nFixedPolynomialBasis\nOrthonormalCoefficientsBasis","category":"page"},{"location":"#MultivariateBases.AbstractPolynomialBasis","page":"Introduction","title":"MultivariateBases.AbstractPolynomialBasis","text":"abstract type AbstractPolynomialBasis end\n\nPolynomial basis of a subspace of the polynomials [Section~3.1.5, BPT12].\n\n[BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics, 2012.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.maxdegree_basis","page":"Introduction","title":"MultivariateBases.maxdegree_basis","text":"maxdegree_basis(B::Type{<:AbstractPolynomialBasis}, variables, maxdegree::Int)\n\nReturn the basis of type B generating all polynomials of degree up to maxdegree with variables variables.\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.basis_covering_monomials","page":"Introduction","title":"MultivariateBases.basis_covering_monomials","text":"basis_covering_monomials(B::Type{<:AbstractPolynomialBasis}, monos::AbstractVector{<:AbstractMonomial})\n\nReturn the minimal basis of type B that can generate all polynomials of the monomial basis generated by monos.\n\nExamples\n\nFor example, to generate all the polynomials with nonzero coefficients for the monomials x^4 and x^2, we need three polynomials as otherwise, we generate polynomials with nonzero constant term.\n\njulia> using DynamicPolynomials\n\njulia> @polyvar x\n(x,)\n\njulia> basis_covering_monomials(ChebyshevBasis, [x^2, x^4])\nChebyshevBasisFirstKind([1.0, -1.0 + 2.0x², 1.0 - 8.0x² + 8.0x⁴])\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.FixedPolynomialBasis","page":"Introduction","title":"MultivariateBases.FixedPolynomialBasis","text":"struct FixedPolynomialBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis\n polynomials::PV\nend\n\nPolynomial basis with the polynomials of the vector polynomials. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.OrthonormalCoefficientsBasis","page":"Introduction","title":"MultivariateBases.OrthonormalCoefficientsBasis","text":"struct OrthonormalCoefficientsBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis\n polynomials::PV\nend\n\nPolynomial basis with the polynomials of the vector polynomials that are orthonormal with respect to the inner produce derived from the inner product of their coefficients. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.\n\n\n\n\n\n","category":"type"},{"location":"#Monomial-basis","page":"Introduction","title":"Monomial basis","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"MonomialBasis\nScaledMonomialBasis","category":"page"},{"location":"#MultivariateBases.MonomialBasis","page":"Introduction","title":"MultivariateBases.MonomialBasis","text":"struct MonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis\n monomials::MV\nend\n\nMonomial basis with the monomials of the vector monomials. For instance, MonomialBasis([1, x, y, x^2, x*y, y^2]) is the monomial basis for the subspace of quadratic polynomials in the variables x, y.\n\nThis basis is orthogonal under a scalar product defined with the complex Gaussian measure as density. Once normalized so as to be orthonormal with this scalar product, one get ths ScaledMonomialBasis.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.ScaledMonomialBasis","page":"Introduction","title":"MultivariateBases.ScaledMonomialBasis","text":"struct ScaledMonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis\n monomials::MV\nend\n\nScaled monomial basis (see [Section 3.1.5, BPT12]) with the monomials of the vector monomials. Given a monomial x^alpha = x_1^alpha_1 cdots x_n^alpha_n of degree d = sum_i=1^n alpha_i, the corresponding polynomial of the basis is\n\nd choose alpha^frac12 x^alpha quad text where quad\nd choose alpha = fracdalpha_1 alpha_2 cdots alpha_n\n\nFor instance, create a polynomial with the basis xy^2 xy creates the polynomial sqrt3 a xy^2 + sqrt2 b xy where a and b are new JuMP decision variables. Constraining the polynomial axy^2 + bxy to be zero with the scaled monomial basis constrains a/√3 and b/√2 to be zero.\n\nThis basis is orthonormal under the scalar product:\n\nlangle f g rangle = int_mathcalC^n f(z) overlineg(z) dnu_n\n\nwhere nu_n is the Gaussian measure on mathcalC^n with the density pi^-n exp(-lVert z rVert^2). See [Section 4; B07] for more details.\n\n[BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics (2012).\n\n[B07] Barvinok, Alexander. Integration and optimization of multivariate polynomials by restriction onto a random subspace. Foundations of Computational Mathematics 7.2 (2007): 229-244.\n\n\n\n\n\n","category":"type"},{"location":"#Orthogonal-basis","page":"Introduction","title":"Orthogonal basis","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"AbstractMultipleOrthogonalBasis\nunivariate_orthogonal_basis\nreccurence_first_coef\nreccurence_second_coef\nreccurence_third_coef\nreccurence_deno_coef\nProbabilistsHermiteBasis\nPhysicistsHermiteBasis\nLaguerreBasis\nAbstractGegenbauerBasis\nLegendreBasis\nChebyshevBasisFirstKind\nChebyshevBasisSecondKind","category":"page"},{"location":"#MultivariateBases.AbstractMultipleOrthogonalBasis","page":"Introduction","title":"MultivariateBases.AbstractMultipleOrthogonalBasis","text":"abstract type AbstractMultipleOrthogonalBasis{P} <: AbstractPolynomialVectorBasis{P, Vector{P}} end\n\nPolynomial basis such that langle p_i(x) p_j(x) rangle = 0 if i neq j where\n\nlangle p(x) q(x) rangle = int p(x)q(x) w(x) dx\n\nwhere the weight is a product of weight functions w(x) = w_1(x_1)w_2(x_2) cdots w_n(x_n) in each variable. The polynomial of the basis are product of univariate polynomials: p(x) = p_1(x_1)p_2(x_2) cdots p_n(x_n). where the univariate polynomials of variable x_i form an univariate orthogonal basis for the weight function w_i(x_i). Therefore, they satisfy the recurrence relation\n\nd_k p_k(x_i) = (a_k x_i + b_k) p_k-1(x_i) + c_k p_k-2(x_i)\n\nwhere reccurence_first_coef gives a_k, reccurence_second_coef gives b_k, reccurence_third_coef gives c_k and reccurence_deno_coef gives d_k.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.univariate_orthogonal_basis","page":"Introduction","title":"MultivariateBases.univariate_orthogonal_basis","text":"univariate_orthogonal_basis(B::Type{<:AbstractMultipleOrthogonalBasis},\n variable::MP.AbstractVariable, degree::Integer)\n\nReturn the vector of univariate polynomials of the basis B up to degree with variable variable.\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.reccurence_first_coef","page":"Introduction","title":"MultivariateBases.reccurence_first_coef","text":"reccurence_first_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)\n\nReturn a_{degree} in recurrence equation\n\nd_k p_k(x_i) = (a_k x_i + b_k) p_k-1(x_i) + c_k p_k-2(x_i)\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.reccurence_second_coef","page":"Introduction","title":"MultivariateBases.reccurence_second_coef","text":"reccurence_second_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)\n\nReturn b_{degree} in recurrence equation\n\nd_k p_k(x_i) = (a_k x_i + b_k) p_k-1(x_i) + c_k p_k-2(x_i)\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.reccurence_third_coef","page":"Introduction","title":"MultivariateBases.reccurence_third_coef","text":"reccurence_third_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)\n\nReturn c_{degree} in recurrence equation\n\nd_k p_k(x_i) = (a_k x_i + b_k) p_k-1(x_i) + c_k p_k-2(x_i)\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.reccurence_deno_coef","page":"Introduction","title":"MultivariateBases.reccurence_deno_coef","text":"reccurence_deno_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)\n\nReturn d_{degree} in recurrence equation\n\nd_k p_k(x_i) = (a_k x_i + b_k) p_k-1(x_i) + c_k p_k-2(x_i)\n\n\n\n\n\n","category":"function"},{"location":"#MultivariateBases.ProbabilistsHermiteBasis","page":"Introduction","title":"MultivariateBases.ProbabilistsHermiteBasis","text":"struct ProbabilistsHermiteBasis{P} <: AbstractHermiteBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = exp(-x^22) over the interval -infty infty.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.PhysicistsHermiteBasis","page":"Introduction","title":"MultivariateBases.PhysicistsHermiteBasis","text":"struct PhysicistsHermiteBasis{P} <: AbstractHermiteBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = exp(-x^2) over the interval -infty infty.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.LaguerreBasis","page":"Introduction","title":"MultivariateBases.LaguerreBasis","text":"struct LaguerreBasis{P} <: AbstractMultipleOrthogonalBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = exp(-x) over the interval 0 infty.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.AbstractGegenbauerBasis","page":"Introduction","title":"MultivariateBases.AbstractGegenbauerBasis","text":"struct AbstractGegenbauerBasis{P} <: AbstractMultipleOrthogonalBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = (1 - x^2)^alpha - 12 over the interval -1 1.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.LegendreBasis","page":"Introduction","title":"MultivariateBases.LegendreBasis","text":"struct LegendreBasis{P} <: AbstractGegenbauerBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = 1 over the interval -1 1.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.ChebyshevBasisFirstKind","page":"Introduction","title":"MultivariateBases.ChebyshevBasisFirstKind","text":"struct ChebyshevBasisFirstKind{P} <: AbstractChebyshevBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = frac1sqrt1 - x^2 over the interval -1 1.\n\n\n\n\n\n","category":"type"},{"location":"#MultivariateBases.ChebyshevBasisSecondKind","page":"Introduction","title":"MultivariateBases.ChebyshevBasisSecondKind","text":"struct ChebyshevBasisSecondKind{P} <: AbstractChebyshevBasis{P}\n polynomials::Vector{P}\nend\n\nOrthogonal polynomial with respect to the univariate weight function w(x) = sqrt1 - x^2 over the interval -1 1.\n\n\n\n\n\n","category":"type"}] +} diff --git a/v0.2.2/siteinfo.js b/v0.2.2/siteinfo.js new file mode 100644 index 0000000..c12bcff --- /dev/null +++ b/v0.2.2/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.2.2"; diff --git a/versions.js b/versions.js index 04b5794..f630691 100644 --- a/versions.js +++ b/versions.js @@ -4,5 +4,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v0.2.1"; +var DOCUMENTER_NEWEST = "v0.2.2"; var DOCUMENTER_STABLE = "stable";