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Doc: Add the missing decompositions to the dense catalogue

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libeigen/eigen!2836

Closes #2072

Co-authored-by: Rasmus Munk Larsen <rmlarsen@gmail.com>
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Rasmus Munk Larsen
2026-08-16 18:53:52 -07:00
co-authored by Rasmus Munk Larsen
parent bbac95e9ec
commit e453d30949
1 changed files with 75 additions and 0 deletions
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doc/TopicLinearAlgebraDecompositions.dox
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@@ -76,6 +76,18 @@ To get an overview of the true relative speed of the different decompositions, c
</tr>
<tr>
<td>RandColPivHouseholderQR</td>
<td>-</td>
<td>Fast</td>
<td>Good</td>
<td>Yes</td>
<td>Orthogonalization, least squares for overdetermined systems</td>
<td>Yes (and does least squares)</td>
<td>Good</td>
<td>Blocking, randomized pivot selection</td>
</tr>
<tr class="alt">
<td>FullPivHouseholderQR</td>
<td>-</td>
<td>Slow (no blocking)</td>
@@ -123,6 +135,18 @@ To get an overview of the true relative speed of the different decompositions, c
<td>-</td>
</tr>
<tr>
<td>BunchKaufman</td>
<td>Self-adjoint (possibly indefinite)</td>
<td>Very fast</td>
<td>Good</td>
<td>-</td>
<td>Inertia</td>
<td>Yes</td>
<td>Good</td>
<td>Blocking</td>
</tr>
<tr><th class="inter" colspan="9">\n Singular values and eigenvalues decompositions</th></tr>
<tr>
@@ -161,6 +185,18 @@ To get an overview of the true relative speed of the different decompositions, c
<td><em>Closed forms for 2x2 and 3x3</em></td>
</tr>
<tr>
<td>TridiagonalEigenSolver</td>
<td>Symmetric tridiagonal</td>
<td>Fast</td>
<td>Good</td>
<td>-</td>
<td>Eigenvalues/vectors, subsets of the spectrum</td>
<td>-</td>
<td>Good</td>
<td>Vectorization, Explicit MT</td>
</tr>
<tr>
<td>ComplexEigenSolver</td>
<td>Square</td>
@@ -197,6 +233,18 @@ To get an overview of the true relative speed of the different decompositions, c
<td>-</td>
</tr>
<tr class="alt">
<td>GeneralizedEigenSolver</td>
<td>Square and real</td>
<td>Slow-very slow<sup><a href="#note2">2</a></sup></td>
<td>Depends on condition number</td>
<td>-</td>
<td>Generalized eigenvalues/vectors</td>
<td>-</td>
<td>Average</td>
<td>-</td>
</tr>
<tr><th class="inter" colspan="9">\n Helper decompositions</th></tr>
<tr>
@@ -223,6 +271,30 @@ To get an overview of the true relative speed of the different decompositions, c
<td>-</td>
</tr>
<tr>
<td>RealQZ</td>
<td>Square and real (pair of matrices)</td>
<td>Average-slow<sup><a href="#note2">2</a></sup></td>
<td>Depends on condition number</td>
<td>-</td>
<td>Generalized eigenvalues of a pencil</td>
<td>-</td>
<td>Average</td>
<td>-</td>
</tr>
<tr class="alt">
<td>ComplexQZ</td>
<td>Square (pair of matrices)</td>
<td>Slow-very slow<sup><a href="#note2">2</a></sup></td>
<td>Depends on condition number</td>
<td>-</td>
<td>Generalized eigenvalues of a pencil</td>
<td>-</td>
<td>Average</td>
<td>-</td>
</tr>
<tr class="alt">
<td>Tridiagonalization</td>
<td>Self-adjoint</td>
@@ -263,6 +335,9 @@ The following recommendations apply to the most common use cases:
\li <b>Symmetric positive definite systems:</b> Use \b LLT. It is the fastest solver and has excellent
numerical properties for this class of problems. For semidefinite or nearly singular symmetric systems,
use \b LDLT.
\li <b>Symmetric indefinite systems:</b> Use \b BunchKaufman. It extends the Cholesky family to
indefinite self-adjoint matrices with a backward-stable pivoting strategy at LDLT-like speed,
and also reveals the inertia of the matrix.
\li <b>General invertible systems:</b> Use \b PartialPivLU. It uses cache-friendly blocking and implicit
multi-threading, making it the fastest general-purpose solver. Partial pivoting is sufficient for
virtually all practical problems.