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</code></pre><p>Then we wait for the friendly folks at <a href="https://github.com/JuliaPackaging">JuliaPackaging</a> to merge the pull request to Yggdrasil, triggering a new release of the <a href="https://github.com/JuliaBinaryWrappers/FastTransforms_jll.jl">FastTransforms_jll.jl</a> meta package that stores all precompiled binaries. With this release, we update the FastTransforms.jl <a href="https://github.com/JuliaApproximation/FastTransforms.jl/blob/master/Project.toml">Project.toml</a> to point to the latest release and register the new version.</p><p>Since development of Yggdrasil is quite rapid, a fork may easily become stale. Git permits the developer to forcibly make a master branch on a fork even with upstream master:</p><pre><code class="language-none">git fetch upstream
git checkout master
git reset --hard upstream/master
git push origin master --force</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../generated/annulus/">Integration on an annulus »</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="Tuesday 7 May 2024 07:10">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
git push origin master --force</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../generated/annulus/">Integration on an annulus »</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="Tuesday 7 May 2024 07:13">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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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
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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="Tuesday 7 May 2024 07:10">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</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="Tuesday 7 May 2024 07:13">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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λ: 0.5010383094167955 and the 2-norm: 1.0000000000187363
λ: 0.5010383094167954 and the 2-norm: 1.0000000000000002
λ: 0.5010383094167955 and the 2-norm: 0.9999999999999999
</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="../annulus/">« Integration on an annulus</a><a class="docs-footer-nextpage" href="../chebyshev/">Chebyshev transform »</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="Tuesday 7 May 2024 07:10">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
</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="../annulus/">« Integration on an annulus</a><a class="docs-footer-nextpage" href="../chebyshev/">Chebyshev transform »</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="Tuesday 7 May 2024 07:13">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
2 changes: 1 addition & 1 deletion dev/generated/chebyshev/index.html
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f = exp.(p_2)
f̌ = chebyshevutransform(f, Val(2))
f̃ = x -&gt; [sin((k+1)*acos(x))/sin(acos(x)) for k=0:n-3]&#39; * f̌
f̃(0.1) ≈ exp(0.1)</code></pre><pre><code class="language-none">true</code></pre><p>Second kind polynomials <span>$\to$</span> second kind points</p><pre><code class="language-julia">ichebyshevutransform(f̌, Val(2)) ≈ exp.(p_2)</code></pre><pre><code class="language-none">true</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="../automaticdifferentiation/">« Automatic differentiation through spherical harmonic transforms</a><a class="docs-footer-nextpage" href="../disk/">Holomorphic integration on the unit disk »</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="Tuesday 7 May 2024 07:10">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
f̃(0.1) ≈ exp(0.1)</code></pre><pre><code class="language-none">true</code></pre><p>Second kind polynomials <span>$\to$</span> second kind points</p><pre><code class="language-julia">ichebyshevutransform(f̌, Val(2)) ≈ exp.(p_2)</code></pre><pre><code class="language-none">true</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="../automaticdifferentiation/">« Automatic differentiation through spherical harmonic transforms</a><a class="docs-footer-nextpage" href="../disk/">Holomorphic integration on the unit disk »</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="Tuesday 7 May 2024 07:13">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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0.000104898 -1.01818e-18 -4.97719e-5 -7.29653e-19 -1.14056e-5 5.87956e-19 -1.12136e-5 4.60098e-19 -8.949e-6 4.53171e-19 -5.82665e-6 -2.44462e-19 -3.11536e-6 -6.1681e-19 -1.37708e-6 5.61031e-19 -5.04943e-7 -8.46536e-20 -1.52739e-7 -3.66863e-19 -3.7074e-8 -2.38249e-19 -6.54014e-9 1.24858e-19 -4.49338e-10 7.25589e-19 2.34081e-10 -4.87718e-19 1.39557e-10 -5.94023e-19
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1.54054e-5 1.23828e-18 1.00144e-6 1.84463e-19 1.23964e-6 -8.43152e-20 1.21824e-6 5.74481e-20 9.71898e-7 -1.03935e-19 6.32656e-7 -8.20175e-20 3.38215e-7 1.5389e-19 1.49487e-7 -1.12048e-19 5.48103e-8 -2.06987e-19 1.65791e-8 6.79198e-20 4.02426e-9 1.22785e-19 7.09969e-10 -6.46596e-20 4.88112e-11 -5.51333e-20 -2.53943e-11 1.23995e-19 -1.51441e-11 7.96269e-20
-1.5778e-18 0.000145153 1.32424e-19 0.000143445 -7.6147e-20 6.55677e-5 2.22587e-19 -1.45469e-5 3.01577e-19 -5.3233e-5 3.22253e-20 -5.18305e-5 -8.36085e-22 -3.33247e-5 -1.86065e-19 -1.60719e-5 -4.43286e-19 -6.0251e-6 8.63083e-20 -1.74358e-6 5.65738e-21 -3.62096e-7 4.33784e-20 -3.58901e-8 -1.49965e-19 9.57958e-9 1.3012e-20 6.5884e-9 3.28648e-19 1.80042e-9</code></pre><p>The Dunkl-Xu coefficients are useful for integration. The integral of <span>$f(x,y)$</span> over the disk should be <span>$\pi/2$</span> by harmonicity. The coefficient of <span>$P_{0,0}$</span> multiplied by <code>√π</code> is:</p><pre><code class="language-julia">U[1, 1]*sqrt(π)</code></pre><pre><code class="language-none">1.5707955409153043</code></pre><p>Using an orthonormal basis, the integral of <span>$[f(x,y)]^2$</span> over the disk is approximately the square of the 2-norm of the coefficients:</p><pre><code class="language-julia">norm(U)^2, π/(2*sqrt(2))*log1p(sqrt(2))</code></pre><pre><code class="language-none">(0.978937045726087, 0.9789599179781414)</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="../chebyshev/">« Chebyshev transform</a><a class="docs-footer-nextpage" href="../halfrange/">Half-range Chebyshev polynomials »</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="Tuesday 7 May 2024 07:10">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
-1.5778e-18 0.000145153 1.32424e-19 0.000143445 -7.6147e-20 6.55677e-5 2.22587e-19 -1.45469e-5 3.01577e-19 -5.3233e-5 3.22253e-20 -5.18305e-5 -8.36085e-22 -3.33247e-5 -1.86065e-19 -1.60719e-5 -4.43286e-19 -6.0251e-6 8.63083e-20 -1.74358e-6 5.65738e-21 -3.62096e-7 4.33784e-20 -3.58901e-8 -1.49965e-19 9.57958e-9 1.3012e-20 6.5884e-9 3.28648e-19 1.80042e-9</code></pre><p>The Dunkl-Xu coefficients are useful for integration. The integral of <span>$f(x,y)$</span> over the disk should be <span>$\pi/2$</span> by harmonicity. The coefficient of <span>$P_{0,0}$</span> multiplied by <code>√π</code> is:</p><pre><code class="language-julia">U[1, 1]*sqrt(π)</code></pre><pre><code class="language-none">1.5707955409153043</code></pre><p>Using an orthonormal basis, the integral of <span>$[f(x,y)]^2$</span> over the disk is approximately the square of the 2-norm of the coefficients:</p><pre><code class="language-julia">norm(U)^2, π/(2*sqrt(2))*log1p(sqrt(2))</code></pre><pre><code class="language-none">(0.978937045726087, 0.9789599179781414)</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="../chebyshev/">« Chebyshev transform</a><a class="docs-footer-nextpage" href="../halfrange/">Half-range Chebyshev polynomials »</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="Tuesday 7 May 2024 07:13">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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