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@@ -5,4 +5,4 @@
 VL = eigvecs(S,left=true)</code></pre><p>The results are currently unreliable if the Frobenius norm of <code>A</code> is very small or very large, so scale if necessary (see Balancing, below).</p><p>Eigenvectors are not currently available for the &quot;real Schur&quot; forms. But don&#39;t despair; one can convert a standard quasi-triangular real <code>Schur</code> into a complex <code>Schur</code> with the <code>triangularize</code> function provided here.</p><h2 id="Balancing"><a class="docs-heading-anchor" href="#Balancing">Balancing</a><a id="Balancing-1"></a><a class="docs-heading-anchor-permalink" href="#Balancing" title="Permalink"></a></h2><p>The accuracy of eigenvalues and eigenvectors may be improved for some matrices by use of a similarity transform which reduces the matrix norm.  This is done by default in the <code>eigen!</code> method, and may also be handled explicitly via the <code>balance!</code> function provided here:</p><pre><code class="language-julia hljs">Ab, B = balance!(copy(A))
 S = schur(Ab)
 v = eigvecs(S)
-lmul!(B, v) # to get the eigenvectors of A</code></pre><p>More details are in the function docstring. Although the balancing function also does permutations to isolate trivial subspaces, the Schur routines do not yet exploit this opportunity for reduced workload.</p><h2 id="Generalized-problems"><a class="docs-heading-anchor" href="#Generalized-problems">Generalized problems</a><a id="Generalized-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Generalized-problems" title="Permalink"></a></h2><p>The generalized Schur decomposition is provided [via <code>schur(A,B)</code>, or <code>ggschur!(A,B)</code>], for real and complex element types. No specific implementations are currently provided for Hermitian/Symmetric problems.</p><p>Eigenvectors are also available for complex element types, via <code>eigvecs(S::GeneralizedSchur)</code>.</p><h2 id="Reordering"><a class="docs-heading-anchor" href="#Reordering">Reordering</a><a id="Reordering-1"></a><a class="docs-heading-anchor-permalink" href="#Reordering" title="Permalink"></a></h2><p>An invariant (viz. deflating) subspace may be extracted via the <code>ordschur</code> function. This package provides this for ordinary Schur factorizations (real and complex) and for complex generalized Schur factorizations.</p><h2 id="Acknowledgements"><a class="docs-heading-anchor" href="#Acknowledgements">Acknowledgements</a><a id="Acknowledgements-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgements" title="Permalink"></a></h2><p>This package includes translations from <a href="http://www.netlib.org/lapack/index.html">LAPACK</a> code, and incorporates or elaborates several methods from Andreas Noack&#39;s <a href="http://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl">GenericLinearAlgebra.jl</a> package.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="library/">Library »</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="Saturday 1 November 2025 03:00">Saturday 1 November 2025</span>. Using Julia version 1.12.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+lmul!(B, v) # to get the eigenvectors of A</code></pre><p>More details are in the function docstring. Although the balancing function also does permutations to isolate trivial subspaces, the Schur routines do not yet exploit this opportunity for reduced workload.</p><h2 id="Generalized-problems"><a class="docs-heading-anchor" href="#Generalized-problems">Generalized problems</a><a id="Generalized-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Generalized-problems" title="Permalink"></a></h2><p>The generalized Schur decomposition is provided [via <code>schur(A,B)</code>, or <code>ggschur!(A,B)</code>], for real and complex element types. No specific implementations are currently provided for Hermitian/Symmetric problems.</p><p>Eigenvectors are also available for complex element types, via <code>eigvecs(S::GeneralizedSchur)</code>.</p><h2 id="Reordering"><a class="docs-heading-anchor" href="#Reordering">Reordering</a><a id="Reordering-1"></a><a class="docs-heading-anchor-permalink" href="#Reordering" title="Permalink"></a></h2><p>An invariant (viz. deflating) subspace may be extracted via the <code>ordschur</code> function. This package provides this for ordinary Schur factorizations (real and complex) and for complex generalized Schur factorizations.</p><h2 id="Acknowledgements"><a class="docs-heading-anchor" href="#Acknowledgements">Acknowledgements</a><a id="Acknowledgements-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgements" title="Permalink"></a></h2><p>This package includes translations from <a href="http://www.netlib.org/lapack/index.html">LAPACK</a> code, and incorporates or elaborates several methods from Andreas Noack&#39;s <a href="http://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl">GenericLinearAlgebra.jl</a> package.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="library/">Library »</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="Monday 3 November 2025 15:17">Monday 3 November 2025</span>. Using Julia version 1.12.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>