build based on 7d21e5e

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.6.7","generation_timestamp":"2024-01-31T21:47:13","documenter_version":"1.2.1"}}
+{"documenter":{"julia_version":"1.6.7","generation_timestamp":"2024-02-02T08:30:54","documenter_version":"1.2.1"}}

dev/examples/index.html

@@ -8,7 +8,7 @@
 for k in 1:N-3
     x[k+3] = 10/3*x[k+2] - 3*x[k+1] + 2/3*x[k]
 end
-plot(4:N, abs.(x.-sqrt(2))[4:N], label=&quot;numerical error&quot;, yaxis=:log, background_color=:transparent, foreground_color=:gray)</code></pre><img src="151bbf28.svg" alt="Example block output"/><p>The backward recurrence, on the other hand, will be dominated by the eigenvalue <span>$\frac{1}{3}$</span> instead. While there may be other algorithms that are stable in this case, they are not trivial and vary across different problems.</p><h2 id="Pseudo-stablisation-methods"><a class="docs-heading-anchor" href="#Pseudo-stablisation-methods">Pseudo-stablisation methods</a><a id="Pseudo-stablisation-methods-1"></a><a class="docs-heading-anchor-permalink" href="#Pseudo-stablisation-methods" title="Permalink"></a></h2><p>The idea of pseudo-stablisation is to use a high precision to achieve a given error tolerance despite the instability. The precision choice is smart such that no extra care is taken by the end user, resulting in a &quot;stable&quot; behavior. Needless to say, pseudo-stablisation only works on problems where the initial values and the recurrence coefficients can be computed to arbitrary precision.</p><p>The current iteration of the strategy focuses on linear problems, where the end error scales linearly with the initial error which is considered as machine epsilon. To get the scaling factor, a test recurrence with random initial values is run. The random initial values ensure that the initial error is significant and the leading eigenvalue immediately takes over. The estimated scaling factor is then the ratio between the end norm of the test run and the initial norm.</p><p>The test recurrence may sound like a performance trade-off, but since it is run under a low precision, the cost is negligible.</p><h3 id="Linear-recursive-Sequence-2"><a class="docs-heading-anchor" href="#Linear-recursive-Sequence-2">Linear-recursive Sequence</a><a class="docs-heading-anchor-permalink" href="#Linear-recursive-Sequence-2" title="Permalink"></a></h3><p>First, we shall recognise that the linear recursive sequence is basically a 1D stencil recurrence with the stencil</p><pre><code class="language-julia hljs">stencil = (CartesianIndex(-3), CartesianIndex(-2), CartesianIndex(-1));</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(CartesianIndex(-3,), CartesianIndex(-2,), CartesianIndex(-1,))</code></pre><p>with the coefficients</p><pre><code class="language-julia hljs">coef0(m) = 2//3
+plot(4:N, abs.(x.-sqrt(2))[4:N], label=&quot;numerical error&quot;, yaxis=:log, background_color=:transparent, foreground_color=:gray)</code></pre><img src="1a71ce91.svg" alt="Example block output"/><p>The backward recurrence, on the other hand, will be dominated by the eigenvalue <span>$\frac{1}{3}$</span> instead. While there may be other algorithms that are stable in this case, they are not trivial and vary across different problems.</p><h2 id="Pseudo-stablisation-methods"><a class="docs-heading-anchor" href="#Pseudo-stablisation-methods">Pseudo-stablisation methods</a><a id="Pseudo-stablisation-methods-1"></a><a class="docs-heading-anchor-permalink" href="#Pseudo-stablisation-methods" title="Permalink"></a></h2><p>The idea of pseudo-stablisation is to use a high precision to achieve a given error tolerance despite the instability. The precision choice is smart such that no extra care is taken by the end user, resulting in a &quot;stable&quot; behavior. Needless to say, pseudo-stablisation only works on problems where the initial values and the recurrence coefficients can be computed to arbitrary precision.</p><p>The current iteration of the strategy focuses on linear problems, where the end error scales linearly with the initial error which is considered as machine epsilon. To get the scaling factor, a test recurrence with random initial values is run. The random initial values ensure that the initial error is significant and the leading eigenvalue immediately takes over. The estimated scaling factor is then the ratio between the end norm of the test run and the initial norm.</p><p>The test recurrence may sound like a performance trade-off, but since it is run under a low precision, the cost is negligible.</p><h3 id="Linear-recursive-Sequence-2"><a class="docs-heading-anchor" href="#Linear-recursive-Sequence-2">Linear-recursive Sequence</a><a class="docs-heading-anchor-permalink" href="#Linear-recursive-Sequence-2" title="Permalink"></a></h3><p>First, we shall recognise that the linear recursive sequence is basically a 1D stencil recurrence with the stencil</p><pre><code class="language-julia hljs">stencil = (CartesianIndex(-3), CartesianIndex(-2), CartesianIndex(-1));</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(CartesianIndex(-3,), CartesianIndex(-2,), CartesianIndex(-1,))</code></pre><p>with the coefficients</p><pre><code class="language-julia hljs">coef0(m) = 2//3
 coef1(m) = -3
 coef2(m) = 10//3
 coefs = (coef0, coef1, coef2)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(Main.coef0, Main.coef1, Main.coef2)</code></pre><p>They will work together to define the recurrence</p><pre><code class="language-julia hljs"># calculate x[m]
@@ -36,6 +36,6 @@
  1.4142135623730951
  1.4142135623730951
  1.4142135623730951
- 1.414213562373095
- 1.414213562373095
- 1.414213562373095</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../docstrings/">« Docstrings</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="Wednesday 31 January 2024 21:47">Wednesday 31 January 2024</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+ 1.4142135623730951
+ 1.4142135623730951
+ 1.4142135623730951</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../docstrings/">« Docstrings</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="Friday 2 February 2024 08:30">Friday 2 February 2024</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/index.html

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