build based on b7e81df

dev/condest.html

@@ -9,7 +9,7 @@
 30.0
 
 julia> opnorm1(invlyapop(A))
-3.7666666666666706</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/f732b0f0585cd23b912dd8987d71f6539d1b708b/src/condest.jl#L1-L26">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.opnorm1est" href="#MatrixEquations.opnorm1est"><code>MatrixEquations.opnorm1est</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">γ = opnorm1est(op)</code></pre><p>Compute <code>γ</code>, a lower bound of the <code>1</code>-norm of the square linear operator <code>op</code>, using reverse communication based computations to evaluate <code>op * x</code> and <code>op&#39; * x</code>. It is expected that in most cases <span>$\gamma &gt; \|op\|_1/10$</span>, which is usually acceptable for estimating the condition numbers of linear operators.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
+3.7666666666666706</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/b7e81dffa1af8f914182da29239560eee7934bd3/src/condest.jl#L1-L26">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.opnorm1est" href="#MatrixEquations.opnorm1est"><code>MatrixEquations.opnorm1est</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">γ = opnorm1est(op)</code></pre><p>Compute <code>γ</code>, a lower bound of the <code>1</code>-norm of the square linear operator <code>op</code>, using reverse communication based computations to evaluate <code>op * x</code> and <code>op&#39; * x</code>. It is expected that in most cases <span>$\gamma &gt; \|op\|_1/10$</span>, which is usually acceptable for estimating the condition numbers of linear operators.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
 3×3 Array{Float64,2}:
  -6.0  -2.0   1.0
   5.0   1.0  -1.0
@@ -19,7 +19,7 @@
 18.0
 
 julia&gt; opnorm1est(invlyapop(A))
-3.76666666666667</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/f732b0f0585cd23b912dd8987d71f6539d1b708b/src/condest.jl#L50-L72">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.oprcondest" href="#MatrixEquations.oprcondest"><code>MatrixEquations.oprcondest</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">rcond = oprcondest(op, opinv; exact = false)</code></pre><p>Compute <code>rcond</code>, an estimation of the <code>1</code>-norm reciprocal condition number of a linear operator <code>op</code>, where <code>opinv</code> is the inverse operator <code>inv(op)</code>. The estimate is computed as <span>$\text{rcond} = 1 / (\|op\|_1\|opinv\|_1)$</span>, using estimates of the <code>1</code>-norm, if <code>exact = false</code>, or computed exact values of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p><p><em>Note:</em> No check is performed to verify that <code>opinv = inv(op)</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
+3.76666666666667</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/b7e81dffa1af8f914182da29239560eee7934bd3/src/condest.jl#L50-L72">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.oprcondest" href="#MatrixEquations.oprcondest"><code>MatrixEquations.oprcondest</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">rcond = oprcondest(op, opinv; exact = false)</code></pre><p>Compute <code>rcond</code>, an estimation of the <code>1</code>-norm reciprocal condition number of a linear operator <code>op</code>, where <code>opinv</code> is the inverse operator <code>inv(op)</code>. The estimate is computed as <span>$\text{rcond} = 1 / (\|op\|_1\|opinv\|_1)$</span>, using estimates of the <code>1</code>-norm, if <code>exact = false</code>, or computed exact values of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p><p><em>Note:</em> No check is performed to verify that <code>opinv = inv(op)</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
 3×3 Array{Float64,2}:
  -6.0  -2.0   1.0
   5.0   1.0  -1.0
@@ -35,7 +35,7 @@
 0.008849557522123885
 
 julia&gt; 1/opnorm1(lyapop(A))/opnorm1(invlyapop(A))
-0.008849557522123885</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/f732b0f0585cd23b912dd8987d71f6539d1b708b/src/condest.jl#L161-L192">source</a></section><section><div><pre><code class="nohighlight hljs">rcond = oprcondest(op; exact = false)</code></pre><p>Compute <code>rcond</code>, an estimation of the <code>1</code>-norm reciprocal condition number of a linear operator <code>op</code>, where <code>op</code> is one of the defined Lyapunov or Sylvester operators. The estimate is computed as <span>$\text{rcond} = 1 / (\|op\|_1\|inv(op)\|_1)$</span>, using estimates of the <code>1</code>-norm, if <code>exact = false</code>, or computed exact values of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/f732b0f0585cd23b912dd8987d71f6539d1b708b/src/condest.jl#L196-L205">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.opsepest" href="#MatrixEquations.opsepest"><code>MatrixEquations.opsepest</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">sep = opsepest(opinv; exact = false)</code></pre><p>Compute <code>sep</code>, an estimation of the <code>1</code>-norm separation of a linear operator <code>op</code>, where <code>opinv</code> is the inverse operator <code>inv(op)</code>. The estimate is computed as <span>$\text{sep}  = 1 / \|opinv\|_1$</span> , using an estimate of the <code>1</code>-norm, if <code>exact = false</code>, or the computed exact value of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p><p>The separation of the operator <code>op</code> is defined as</p><p class="math-container">\[\text{sep} = \displaystyle\min_{X\neq 0} \frac{\|op*X\|}{\|X\|}.\]</p><p>An estimate of the reciprocal condition number of <code>op</code> can be computed as <span>$\text{sep}/\|op\|_1$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
+0.008849557522123885</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/b7e81dffa1af8f914182da29239560eee7934bd3/src/condest.jl#L161-L192">source</a></section><section><div><pre><code class="nohighlight hljs">rcond = oprcondest(op; exact = false)</code></pre><p>Compute <code>rcond</code>, an estimation of the <code>1</code>-norm reciprocal condition number of a linear operator <code>op</code>, where <code>op</code> is one of the defined Lyapunov or Sylvester operators. The estimate is computed as <span>$\text{rcond} = 1 / (\|op\|_1\|inv(op)\|_1)$</span>, using estimates of the <code>1</code>-norm, if <code>exact = false</code>, or computed exact values of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/b7e81dffa1af8f914182da29239560eee7934bd3/src/condest.jl#L196-L205">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MatrixEquations.opsepest" href="#MatrixEquations.opsepest"><code>MatrixEquations.opsepest</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">sep = opsepest(opinv; exact = false)</code></pre><p>Compute <code>sep</code>, an estimation of the <code>1</code>-norm separation of a linear operator <code>op</code>, where <code>opinv</code> is the inverse operator <code>inv(op)</code>. The estimate is computed as <span>$\text{sep}  = 1 / \|opinv\|_1$</span> , using an estimate of the <code>1</code>-norm, if <code>exact = false</code>, or the computed exact value of the <code>1</code>-norm, if <code>exact = true</code>. The <code>exact = true</code> option is not recommended for large order operators.</p><p>The separation of the operator <code>op</code> is defined as</p><p class="math-container">\[\text{sep} = \displaystyle\min_{X\neq 0} \frac{\|op*X\|}{\|X\|}.\]</p><p>An estimate of the reciprocal condition number of <code>op</code> can be computed as <span>$\text{sep}/\|op\|_1$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = [-6. -2. 1.; 5. 1. -1; -4. -2. -1.]
 3×3 Array{Float64,2}:
  -6.0  -2.0   1.0
   5.0   1.0  -1.0
@@ -51,4 +51,4 @@
 0.26548672566371656
 
 julia&gt; 1/opnorm1(invlyapop(A))
-0.26548672566371656</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/f732b0f0585cd23b912dd8987d71f6539d1b708b/src/condest.jl#L111-L147">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="sylvkr.html">« Matrix Equation Solvers using Kronecker-product Expansions</a><a class="docs-footer-nextpage" href="meoperators.html">Linear Operators Related to Matrix Equation Solvers »</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 0.27.25 on <span class="colophon-date" title="Tuesday 12 September 2023 13:26">Tuesday 12 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+0.26548672566371656</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/andreasvarga/MatrixEquations.jl/blob/b7e81dffa1af8f914182da29239560eee7934bd3/src/condest.jl#L111-L147">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="sylvkr.html">« Matrix Equation Solvers using Kronecker-product Expansions</a><a class="docs-footer-nextpage" href="meoperators.html">Linear Operators Related to Matrix Equation Solvers »</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 0.27.25 on <span class="colophon-date" title="Tuesday 12 September 2023 13:29">Tuesday 12 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>