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Improve dense linear solver docs with practical guidance

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libeigen/eigen!2395

Co-authored-by: Rasmus Munk Larsen <rmlarsen@gmail.com>
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Rasmus Munk Larsen
2026-04-05 21:40:42 -07:00
parent 8eabfb5342
commit bde3a68bae
5 changed files with 128 additions and 73 deletions
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doc/DenseDecompositionBenchmark.dox
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@@ -30,10 +30,11 @@ Timings are in \b milliseconds, and factors are relative to the LLT decompositio
<a name="note_ls">\b *: </a> This decomposition do not support direct least-square solving for over-constrained problems, and the reported timing include the cost to form the symmetric covariance matrix \f$ A^T A \f$.
\b Observations:
+ LLT is always the fastest solvers.
+ LLT is always the fastest solver.
+ For largely over-constrained problems, the cost of Cholesky/LU decompositions is dominated by the computation of the symmetric covariance matrix.
+ For large problem sizes, only the decomposition implementing a cache-friendly blocking strategy scale well. Those include LLT, PartialPivLU, HouseholderQR, and BDCSVD. This explain why for a 4k x 4k matrix, HouseholderQR is faster than LDLT. In the future, LDLT and ColPivHouseholderQR will also implement blocking strategies.
+ For large problem sizes, only the decompositions implementing a cache-friendly blocking strategy scale well. Those include LLT, PartialPivLU, HouseholderQR, and BDCSVD. This explains why for a 4k x 4k matrix, HouseholderQR is faster than LDLT.
+ CompleteOrthogonalDecomposition is based on ColPivHouseholderQR and they thus achieve the same level of performance.
+ FullPivLU and FullPivHouseholderQR are dramatically slower for large matrices due to the lack of blocking, and are not shown for the 4k x 4k case.
The above table was originally generated by a benchmark tool. Feel free to write your own benchmark to generate a table matching your hardware, compiler, and favorite problem sizes.