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@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.1","generation_timestamp":"2024-02-22T05:38:19","documenter_version":"1.2.1"}}
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dev/basics/SampledIntegralProblem/index.html

@@ -24,13 +24,13 @@
  1.0</code></pre><p>Now, we can integrate this data set as follows:</p><pre><code class="language-julia hljs">problem = SampledIntegralProblem(y, x)
 method = TrapezoidalRule()
 solve(problem, method)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">retcode: Success
-u: 0.33379501385041543</code></pre><p>The exact answer is of course <span>$</span> 1/3 <span>$</span>.</p><h2 id="Details"><a class="docs-heading-anchor" href="#Details">Details</a><a id="Details-1"></a><a class="docs-heading-anchor-permalink" href="#Details" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="SciMLBase.SampledIntegralProblem" href="#SciMLBase.SampledIntegralProblem"><code>SciMLBase.SampledIntegralProblem</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Defines a integral problem over pre-sampled data. Documentation Page: https://docs.sciml.ai/Integrals/stable/</p><p><strong>Mathematical Specification of a data Integral Problem</strong></p><p>Sampled integral problems are defined as:</p><p class="math-container">\[\sum_i w_i y_i\]</p><p>where <code>y_i</code> are sampled values of the integrand, and <code>w_i</code> are weights assigned by a quadrature rule, which depend on sampling points <code>x</code>.</p><p><strong>Problem Type</strong></p><p><strong>Constructors</strong></p><pre><code class="nohighlight hljs">SampledIntegralProblem(y::AbstractArray, x::AbstractVector; dim=ndims(y), kwargs...)</code></pre><ul><li>y: The sampled integrand, must be a subtype of <code>AbstractArray</code>. It is assumed that the values of <code>y</code> along dimension <code>dim</code> correspond to the integrand evaluated at sampling points <code>x</code></li><li>x: Sampling points, must be a subtype of <code>AbstractVector</code>.</li><li>dim: Dimension along which to integrate. Defaults to the last dimension of <code>y</code>.</li><li>kwargs: Keyword arguments copied to the solvers.</li></ul><p><strong>Fields</strong></p><p>The fields match the names of the constructor arguments.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/SciMLBase.jl/blob/v2.26.1/src/problems/basic_problems.jl#L515">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="CommonSolve.solve-Tuple{SampledIntegralProblem, SciMLBase.AbstractIntegralAlgorithm}" href="#CommonSolve.solve-Tuple{SampledIntegralProblem, SciMLBase.AbstractIntegralAlgorithm}"><code>CommonSolve.solve</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">solve(prob::SampledIntegralProblem, alg::SciMLBase.AbstractIntegralAlgorithm; kwargs...)</code></pre><p><strong>Keyword Arguments</strong></p><p>There are no keyword arguments used to solve <code>SampledIntegralProblem</code>s</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/45937631f6d2c5a9824045c74d2f64043388cf04/src/common.jl#L155-L163">source</a></section></article><h3 id="Non-equidistant-grids"><a class="docs-heading-anchor" href="#Non-equidistant-grids">Non-equidistant grids</a><a id="Non-equidistant-grids-1"></a><a class="docs-heading-anchor-permalink" href="#Non-equidistant-grids" title="Permalink"></a></h3><p>If the sampling points <code>x</code> are provided as an <code>AbstractRange</code> (constructed with the <code>range</code> function for example), faster methods are used that take advantage of the fact that the points are equidistantly spaced. Otherwise, general methods are used for non-uniform grids.</p><p>Example:</p><pre><code class="language-julia hljs">f = x -&gt; x^7
+u: 0.33379501385041543</code></pre><p>The exact answer is of course <span>$</span> 1/3 <span>$</span>.</p><h2 id="Details"><a class="docs-heading-anchor" href="#Details">Details</a><a id="Details-1"></a><a class="docs-heading-anchor-permalink" href="#Details" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="SciMLBase.SampledIntegralProblem" href="#SciMLBase.SampledIntegralProblem"><code>SciMLBase.SampledIntegralProblem</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Defines a integral problem over pre-sampled data. Documentation Page: https://docs.sciml.ai/Integrals/stable/</p><p><strong>Mathematical Specification of a data Integral Problem</strong></p><p>Sampled integral problems are defined as:</p><p class="math-container">\[\sum_i w_i y_i\]</p><p>where <code>y_i</code> are sampled values of the integrand, and <code>w_i</code> are weights assigned by a quadrature rule, which depend on sampling points <code>x</code>.</p><p><strong>Problem Type</strong></p><p><strong>Constructors</strong></p><pre><code class="nohighlight hljs">SampledIntegralProblem(y::AbstractArray, x::AbstractVector; dim=ndims(y), kwargs...)</code></pre><ul><li>y: The sampled integrand, must be a subtype of <code>AbstractArray</code>. It is assumed that the values of <code>y</code> along dimension <code>dim</code> correspond to the integrand evaluated at sampling points <code>x</code></li><li>x: Sampling points, must be a subtype of <code>AbstractVector</code>.</li><li>dim: Dimension along which to integrate. Defaults to the last dimension of <code>y</code>.</li><li>kwargs: Keyword arguments copied to the solvers.</li></ul><p><strong>Fields</strong></p><p>The fields match the names of the constructor arguments.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/SciMLBase.jl/blob/v2.26.1/src/problems/basic_problems.jl#L515">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="CommonSolve.solve-Tuple{SampledIntegralProblem, SciMLBase.AbstractIntegralAlgorithm}" href="#CommonSolve.solve-Tuple{SampledIntegralProblem, SciMLBase.AbstractIntegralAlgorithm}"><code>CommonSolve.solve</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">solve(prob::SampledIntegralProblem, alg::SciMLBase.AbstractIntegralAlgorithm; kwargs...)</code></pre><p><strong>Keyword Arguments</strong></p><p>There are no keyword arguments used to solve <code>SampledIntegralProblem</code>s</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/8313934c7dd8ab37da5ceceb63107f5dd93ca436/src/common.jl#L155-L163">source</a></section></article><h3 id="Non-equidistant-grids"><a class="docs-heading-anchor" href="#Non-equidistant-grids">Non-equidistant grids</a><a id="Non-equidistant-grids-1"></a><a class="docs-heading-anchor-permalink" href="#Non-equidistant-grids" title="Permalink"></a></h3><p>If the sampling points <code>x</code> are provided as an <code>AbstractRange</code> (constructed with the <code>range</code> function for example), faster methods are used that take advantage of the fact that the points are equidistantly spaced. Otherwise, general methods are used for non-uniform grids.</p><p>Example:</p><pre><code class="language-julia hljs">f = x -&gt; x^7
 x = [0.0; sort(rand(1000)); 1.0]
 y = f.(x)
 problem = SampledIntegralProblem(y, x)
 method = TrapezoidalRule()
 solve(problem, method)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">retcode: Success
-u: 0.1250040468299159</code></pre><h3 id="Evaluating-multiple-integrals-at-once"><a class="docs-heading-anchor" href="#Evaluating-multiple-integrals-at-once">Evaluating multiple integrals at once</a><a id="Evaluating-multiple-integrals-at-once-1"></a><a class="docs-heading-anchor-permalink" href="#Evaluating-multiple-integrals-at-once" title="Permalink"></a></h3><p>If the provided data set <code>y</code> is a multidimensional array, the integrals are evaluated across only one of its axes. For performance reasons, the last axis of the array <code>y</code> is chosen by default, but this can be modified with the <code>dim</code> keyword argument to the problem definition.</p><pre><code class="language-julia hljs">f1 = x -&gt; x^2
+u: 0.12500297775123098</code></pre><h3 id="Evaluating-multiple-integrals-at-once"><a class="docs-heading-anchor" href="#Evaluating-multiple-integrals-at-once">Evaluating multiple integrals at once</a><a id="Evaluating-multiple-integrals-at-once-1"></a><a class="docs-heading-anchor-permalink" href="#Evaluating-multiple-integrals-at-once" title="Permalink"></a></h3><p>If the provided data set <code>y</code> is a multidimensional array, the integrals are evaluated across only one of its axes. For performance reasons, the last axis of the array <code>y</code> is chosen by default, but this can be modified with the <code>dim</code> keyword argument to the problem definition.</p><pre><code class="language-julia hljs">f1 = x -&gt; x^2
 f2 = x -&gt; x^3
 f3 = x -&gt; x^4
 x = range(0, 1, length = 20)
@@ -47,10 +47,10 @@
 y = f.(x)
 problem = SampledIntegralProblem(y, x)
 method = TrapezoidalRule()
-solve(problem, method)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/45937631f6d2c5a9824045c74d2f64043388cf04/src/algorithms_sampled.jl#L3-L19">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Integrals.SimpsonsRule" href="#Integrals.SimpsonsRule"><code>Integrals.SimpsonsRule</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SimpsonsRule</code></pre><p>Struct for evaluating an integral via the Simpson&#39;s composite 1/3-3/8 rule over <code>AbstractRange</code>s (evenly spaced points) and Simpson&#39;s composite 1/3 rule for non-equidistant grids.</p><p>Example with equidistant data:</p><pre><code class="nohighlight hljs">using Integrals
+solve(problem, method)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/8313934c7dd8ab37da5ceceb63107f5dd93ca436/src/algorithms_sampled.jl#L3-L19">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Integrals.SimpsonsRule" href="#Integrals.SimpsonsRule"><code>Integrals.SimpsonsRule</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SimpsonsRule</code></pre><p>Struct for evaluating an integral via the Simpson&#39;s composite 1/3-3/8 rule over <code>AbstractRange</code>s (evenly spaced points) and Simpson&#39;s composite 1/3 rule for non-equidistant grids.</p><p>Example with equidistant data:</p><pre><code class="nohighlight hljs">using Integrals
 f = x -&gt; x^2
 x = range(0, 1, length=20)
 y = f.(x)
 problem = SampledIntegralProblem(y, x)
 method = SimpsonsRule()
-solve(problem, method)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/45937631f6d2c5a9824045c74d2f64043388cf04/src/algorithms_sampled.jl#L22-L40">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../IntegralFunction/">« Integral Functions</a><a class="docs-footer-nextpage" href="../solve/">Common Solver Options (Solve Keyword Arguments) »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Thursday 22 February 2024 05:38">Thursday 22 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+solve(problem, method)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/Integrals.jl/blob/8313934c7dd8ab37da5ceceb63107f5dd93ca436/src/algorithms_sampled.jl#L22-L40">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../IntegralFunction/">« Integral Functions</a><a class="docs-footer-nextpage" href="../solve/">Common Solver Options (Solve Keyword Arguments) »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Thursday 22 February 2024 05:38">Thursday 22 February 2024</span>. Using Julia version 1.10.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>