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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Integration on an annulus · FastTransforms.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="FastTransforms.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">FastTransforms.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../dev/">Development</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>Integration on an annulus</a></li><li><a class="tocitem" href="../automaticdifferentiation/">Automatic differentiation through spherical harmonic transforms</a></li><li><a class="tocitem" href="../chebyshev/">Chebyshev transform</a></li><li><a class="tocitem" href="../disk/">Holomorphic integration on the unit disk</a></li><li><a class="tocitem" href="../nonlocaldiffusion/">Nonlocal diffusion on <span>$\mathbb{S}^2$</span></a></li><li><a class="tocitem" href="../padua/">Padua transform</a></li><li><a class="tocitem" href="../sphere/">Spherical harmonic addition theorem</a></li><li><a class="tocitem" href="../spinweighted/">Spin-weighted spherical harmonics</a></li><li><a class="tocitem" href="../subspaceangles/">Subspace angles</a></li><li><a class="tocitem" href="../triangle/">Calculus on the reference triangle</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Integration on an annulus</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Integration on an annulus</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaApproximation/FastTransforms.jl/blob/master/examples/annulus.jl" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Integration-on-an-annulus-1"><a class="docs-heading-anchor" href="#Integration-on-an-annulus-1">Integration on an annulus</a><a class="docs-heading-anchor-permalink" href="#Integration-on-an-annulus-1" title="Permalink"></a></h1><p>In this example, we explore integration of the function:</p><div>\[ f(x,y) = \frac{x^3}{x^2+y^2-\frac{1}{4}},\]</div><p>over the annulus defined by <span>$\{(r,\theta) : \rho &lt; r &lt; 1, 0 &lt; \theta &lt; 2\pi\}$</span> with parameter <span>$\rho = \frac{2}{3}$</span>. We will calculate the integral:</p><div>\[ \int_0^{2\pi}\int_{\frac{2}{3}}^1 f(r\cos\theta,r\sin\theta)^2r{\rm\,d}r{\rm\,d}\theta,\]</div><p>by analyzing the function in an annulus polynomial series. We analyze the function on an <span>$N\times M$</span> tensor product grid defined by:</p><div>\[\begin{aligned}
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Integration on an annulus · FastTransforms.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="FastTransforms.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">FastTransforms.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../dev/">Development</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>Integration on an annulus</a></li><li><a class="tocitem" href="../automaticdifferentiation/">Automatic differentiation through spherical harmonic transforms</a></li><li><a class="tocitem" href="../chebyshev/">Chebyshev transform</a></li><li><a class="tocitem" href="../disk/">Holomorphic integration on the unit disk</a></li><li><a class="tocitem" href="../halfrange/">Half-range Chebyshev polynomials</a></li><li><a class="tocitem" href="../nonlocaldiffusion/">Nonlocal diffusion on <span>$\mathbb{S}^2$</span></a></li><li><a class="tocitem" href="../padua/">Padua transform</a></li><li><a class="tocitem" href="../sphere/">Spherical harmonic addition theorem</a></li><li><a class="tocitem" href="../spinweighted/">Spin-weighted spherical harmonics</a></li><li><a class="tocitem" href="../subspaceangles/">Subspace angles</a></li><li><a class="tocitem" href="../triangle/">Calculus on the reference triangle</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Integration on an annulus</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Integration on an annulus</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaApproximation/FastTransforms.jl/blob/master/examples/annulus.jl" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Integration-on-an-annulus-1"><a class="docs-heading-anchor" href="#Integration-on-an-annulus-1">Integration on an annulus</a><a class="docs-heading-anchor-permalink" href="#Integration-on-an-annulus-1" title="Permalink"></a></h1><p>In this example, we explore integration of the function:</p><div>\[ f(x,y) = \frac{x^3}{x^2+y^2-\frac{1}{4}},\]</div><p>over the annulus defined by <span>$\{(r,\theta) : \rho &lt; r &lt; 1, 0 &lt; \theta &lt; 2\pi\}$</span> with parameter <span>$\rho = \frac{2}{3}$</span>. We will calculate the integral:</p><div>\[ \int_0^{2\pi}\int_{\frac{2}{3}}^1 f(r\cos\theta,r\sin\theta)^2r{\rm\,d}r{\rm\,d}\theta,\]</div><p>by analyzing the function in an annulus polynomial series. We analyze the function on an <span>$N\times M$</span> tensor product grid defined by:</p><div>\[\begin{aligned}
r_n &amp; = \sqrt{\cos^2\left[(n+\tfrac{1}{2})\pi/2N\right] + \rho^2 \sin^2\left[(n+\tfrac{1}{2})\pi/2N\right]},\quad{\rm for}\quad 0\le n &lt; N,\quad{\rm and}\\
\theta_m &amp; = 2\pi m/M,\quad{\rm for}\quad 0\le m &lt; M;
\end{aligned}\]</div><p>we convert the function samples to Chebyshev×Fourier coefficients using <code>plan_annulus_analysis</code>; and finally, we transform the Chebyshev×Fourier coefficients to annulus polynomial coefficients using <code>plan_ann2cxf</code>.</p><p>For the storage pattern of the arrays, please consult the <a href="https://MikaelSlevinsky.github.io/FastTransforms">documentation</a>.</p><pre><code class="language-julia">using FastTransforms, LinearAlgebra, Plots
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-1.96208e-17 7.71573e-19 0.00458845 -2.03822e-18 1.86344e-17 3.19547e-18 0.00369818 -1.86152e-18 -3.29302e-18 8.11326e-19 -4.97022e-21 -1.77139e-18 1.47439e-18 -6.8012e-18 2.63512e-18 -4.24712e-18 5.7017e-18 -7.2048e-18 7.38785e-18 -4.72462e-18 1.59607e-17 -5.27319e-18 1.44348e-17 -1.67467e-18 -4.59916e-19 8.00009e-19 5.99498e-18 -3.34661e-18 -2.03062e-18
1.3498e-17 1.71852e-19 -0.00150441 7.97602e-19 -9.38596e-18 -3.21855e-18 -0.00121128 -4.47164e-19 -1.76175e-17 -4.83245e-18 6.52956e-18 -5.47568e-18 5.27089e-18 -2.9175e-18 5.23985e-18 -5.70705e-18 -3.147e-18 -3.85137e-19 -3.05417e-18 -5.31797e-19 -4.09175e-18 3.59849e-18 4.48894e-18 3.4972e-18 1.74387e-18 1.53405e-18 8.8938e-18 4.33464e-18 5.85819e-18
5.24011e-17 -4.20047e-18 0.000517281 -2.62968e-18 1.29799e-17 -9.06153e-18 0.000424358 -4.57265e-18 2.49476e-18 -6.80939e-18 -6.07451e-18 -2.399e-18 1.03771e-18 6.6648e-19 -1.96393e-17 1.69496e-20 -1.42128e-18 5.37575e-18 -1.16727e-17 3.57204e-19 -7.01129e-18 2.48838e-18 -1.17285e-17 4.39475e-19 5.83824e-18 -1.63971e-18 -3.91773e-19 1.39577e-18 1.02075e-17
-8.95802e-18 6.39707e-19 -0.000162126 3.55753e-18 7.44168e-18 4.28763e-18 -5.40419e-5 5.16629e-18 8.97839e-18 5.4486e-18 3.6724e-18 4.64571e-18 -1.67417e-18 6.04678e-19 5.73104e-18 -2.13942e-18 -1.44968e-17 -3.57457e-18 1.24167e-17 -5.67995e-18 -1.09651e-17 -4.98049e-18 6.22788e-18 -2.89642e-18 -1.52622e-17 -2.30036e-18 1.82749e-18 -1.26575e-18 -9.91876e-18</code></pre><p>The annulus coefficients are useful for integration. The integral of <span>$[f(x,y)]^2$</span> over the annulus is approximately the square of the 2-norm of the coefficients:</p><pre><code class="language-julia">norm(U)^2, 5π/8*(1675/4536+9*log(3)/32-3*log(7)/32)</code></pre><pre><code class="language-none">(0.9735516844404257, 0.973547572736036)</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dev/">« Development</a><a class="docs-footer-nextpage" href="../automaticdifferentiation/">Automatic differentiation through spherical harmonic transforms »</a></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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Friday 15 March 2024 20:14">Friday 15 March 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
-8.95802e-18 6.39707e-19 -0.000162126 3.55753e-18 7.44168e-18 4.28763e-18 -5.40419e-5 5.16629e-18 8.97839e-18 5.4486e-18 3.6724e-18 4.64571e-18 -1.67417e-18 6.04678e-19 5.73104e-18 -2.13942e-18 -1.44968e-17 -3.57457e-18 1.24167e-17 -5.67995e-18 -1.09651e-17 -4.98049e-18 6.22788e-18 -2.89642e-18 -1.52622e-17 -2.30036e-18 1.82749e-18 -1.26575e-18 -9.91876e-18</code></pre><p>The annulus coefficients are useful for integration. The integral of <span>$[f(x,y)]^2$</span> over the annulus is approximately the square of the 2-norm of the coefficients:</p><pre><code class="language-julia">norm(U)^2, 5π/8*(1675/4536+9*log(3)/32-3*log(7)/32)</code></pre><pre><code class="language-none">(0.9735516844404257, 0.973547572736036)</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../dev/">« Development</a><a class="docs-footer-nextpage" href="../automaticdifferentiation/">Automatic differentiation through spherical harmonic transforms »</a></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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Thursday 21 March 2024 16:09">Thursday 21 March 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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