build based on a09f2d3

dev/compare/index.html

@@ -567,4 +567,4 @@
   2.187066 seconds (8.27 k allocations: 76.855 MiB, 0.36% compilation time)
 
 julia> @time itp(xq, yq; method = Hiyoshi(2));
- 13.762652 seconds (9.26 k allocations: 76.920 MiB, 0.06% compilation time)</code></pre><h1 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h1><p>Overall, the smooth interpolants have the best performance, with <code>Farin(1)</code> and <code>Hiyoshi(2)</code> typically beating most interpolants. <code>Hiyoshi(2)</code> is much slower than the other interpolants, though, and <code>Farin(1)</code> may be a preferable interpolant if <span>$C^1$</span> continuity at the data sites is sufficient. For generating derivatives, the <code>Direct()</code> seems to beat the results with the <code>Iterative()</code> method in most situations.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../swiss/">« Switzerland Elevation Data</a><a class="docs-footer-nextpage" href="../interpolation_math/">Interpolation Details »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Thursday 7 September 2023 08:13">Thursday 7 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+ 13.762652 seconds (9.26 k allocations: 76.920 MiB, 0.06% compilation time)</code></pre><h1 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h1><p>Overall, the smooth interpolants have the best performance, with <code>Farin(1)</code> and <code>Hiyoshi(2)</code> typically beating most interpolants. <code>Hiyoshi(2)</code> is much slower than the other interpolants, though, and <code>Farin(1)</code> may be a preferable interpolant if <span>$C^1$</span> continuity at the data sites is sufficient. For generating derivatives, the <code>Direct()</code> seems to beat the results with the <code>Iterative()</code> method in most situations.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../swiss/">« Switzerland Elevation Data</a><a class="docs-footer-nextpage" href="../interpolation_math/">Interpolation Details »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Sunday 10 September 2023 01:51">Sunday 10 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/differentiation/index.html

@@ -248,4 +248,4 @@
 7.479964687679311
 
 julia> εH_nohull = rrmserr(f′′.(_x, _y), to_mat.(Hg), ∂, _x, _y)
-38.884740966379056</code></pre><p>The errors are smaller, though not by much.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../interpolation/">« Interpolation</a><a class="docs-footer-nextpage" href="../swiss/">Switzerland Elevation Data »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Thursday 7 September 2023 08:13">Thursday 7 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+38.884740966379056</code></pre><p>The errors are smaller, though not by much.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../interpolation/">« Interpolation</a><a class="docs-footer-nextpage" href="../swiss/">Switzerland Elevation Data »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Sunday 10 September 2023 01:51">Sunday 10 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/differentiation_math/index.html

@@ -21,4 +21,4 @@
 \overline{\boldsymbol g}_1 &= \gamma_i^\prime g_{i1}, \\
 \overline{\boldsymbol g}_2 &= \gamma_i^\prime g_{i2}, \\
 \boldsymbol{\bm\theta} &= \begin{bmatrix} \frac{\partial f(\boldsymbol x_0)}{\partial x} & \frac{\partial f(\boldsymbol x_0)}{\partial y} & \frac{\partial^2 f(\boldsymbol x_0)}{\partial x^2} & \frac{\partial f(\boldsymbol x_0)}{\partial y^2} & \frac{\partial f(\boldsymbol x_0)}{\partial x\partial y} \end{bmatrix}^T.
-\end{align*}\]</p><p>To solve this linear system, let</p><p class="math-container">\[\boldsymbol D = \begin{bmatrix} \overline{\boldsymbol A} \\ \overline{\boldsymbol B} \\ \overline{\boldsymbol C} \end{bmatrix}, \quad \boldsymbol c = \begin{bmatrix} \overline{\boldsymbol w} \\ \overline{\boldsymbol g}_1 \\ \overline{\boldsymbol g}_2 \end{bmatrix},\]</p><p>so that <span>$\boldsymbol D^T\boldsymbol D\boldsymbol\theta = \boldsymbol D^T\boldsymbol c$</span>. These are just the normal equations for <span>$\boldsymbol D\boldsymbol \theta = \boldsymbol c$</span>, thus we can estimate the gradients and Hessians by simply solving <span>$\boldsymbol D\boldsymbol \theta = \boldsymbol c$</span>.</p><h1 id="Generation-Away-from-the-Data-Sites"><a class="docs-heading-anchor" href="#Generation-Away-from-the-Data-Sites">Generation Away from the Data Sites</a><a id="Generation-Away-from-the-Data-Sites-1"></a><a class="docs-heading-anchor-permalink" href="#Generation-Away-from-the-Data-Sites" title="Permalink"></a></h1><p>It is possible to extend these ideas so that we can approximate the derivative at any point <span>$\boldsymbol x_0 \in \mathcal C(\boldsymbol X)$</span>. Using the associated interpolant, simply approximate <span>$z_0$</span> with the value of the interpolant at <span>$\boldsymbol x_0$</span>, and then replace <span>$W_i$</span> by <span>$\lambda_i/\|\boldsymbol x_i-\boldsymbol x_0\|$</span>, where <span>$\lambda_i$</span> is the Sibson coordinate at <span>$\boldsymbol x_i$</span> relative to <span>$\boldsymbol x_0$</span>. If using a direct approach to approximate gradients and Hessians, Sibson coordinates cannot be used (because you can&#39;t extend the weights out to <span>$N_0^2$</span>) and so <span>$W_i$</span> remains as is in that case. Note that the <span>$N_0$</span> neighbourhoods are now the sets of natural neighbours.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../interpolation_math/">« Interpolation Details</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Thursday 7 September 2023 08:13">Thursday 7 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{align*}\]</p><p>To solve this linear system, let</p><p class="math-container">\[\boldsymbol D = \begin{bmatrix} \overline{\boldsymbol A} \\ \overline{\boldsymbol B} \\ \overline{\boldsymbol C} \end{bmatrix}, \quad \boldsymbol c = \begin{bmatrix} \overline{\boldsymbol w} \\ \overline{\boldsymbol g}_1 \\ \overline{\boldsymbol g}_2 \end{bmatrix},\]</p><p>so that <span>$\boldsymbol D^T\boldsymbol D\boldsymbol\theta = \boldsymbol D^T\boldsymbol c$</span>. These are just the normal equations for <span>$\boldsymbol D\boldsymbol \theta = \boldsymbol c$</span>, thus we can estimate the gradients and Hessians by simply solving <span>$\boldsymbol D\boldsymbol \theta = \boldsymbol c$</span>.</p><h1 id="Generation-Away-from-the-Data-Sites"><a class="docs-heading-anchor" href="#Generation-Away-from-the-Data-Sites">Generation Away from the Data Sites</a><a id="Generation-Away-from-the-Data-Sites-1"></a><a class="docs-heading-anchor-permalink" href="#Generation-Away-from-the-Data-Sites" title="Permalink"></a></h1><p>It is possible to extend these ideas so that we can approximate the derivative at any point <span>$\boldsymbol x_0 \in \mathcal C(\boldsymbol X)$</span>. Using the associated interpolant, simply approximate <span>$z_0$</span> with the value of the interpolant at <span>$\boldsymbol x_0$</span>, and then replace <span>$W_i$</span> by <span>$\lambda_i/\|\boldsymbol x_i-\boldsymbol x_0\|$</span>, where <span>$\lambda_i$</span> is the Sibson coordinate at <span>$\boldsymbol x_i$</span> relative to <span>$\boldsymbol x_0$</span>. If using a direct approach to approximate gradients and Hessians, Sibson coordinates cannot be used (because you can&#39;t extend the weights out to <span>$N_0^2$</span>) and so <span>$W_i$</span> remains as is in that case. Note that the <span>$N_0$</span> neighbourhoods are now the sets of natural neighbours.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../interpolation_math/">« Interpolation Details</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.24 on <span class="colophon-date" title="Sunday 10 September 2023 01:51">Sunday 10 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>