Eigen  3.3.4
Benchmark of dense decompositions

This page presents a speed comparison of the dense matrix decompositions offered by Eigen for a wide range of square matrices and overconstrained problems.

For a more general overview on the features and numerical robustness of linear solvers and decompositions, check this table .

This benchmark has been run on a laptop equipped with an Intel core i7 @ 2,6 GHz, and compiled with clang with AVX and FMA instruction sets enabled but without multi-threading. It uses single precision float numbers. For double, you can get a good estimate by multiplying the timings by a factor 2.

The square matrices are symmetric, and for the overconstrained matrices, the reported timmings include the cost to compute the symmetric covariance matrix for the first four solvers based on Cholesky and LU, as denoted by the * symbol (top-right corner part of the table). Timings are in milliseconds, and factors are relative to the LLT decomposition which is the fastest but also the least general and robust.

solver/size 8x8 100x100 1000x1000 4000x4000 10000x8 10000x100 10000x1000 10000x4000
LLT0.050.425.83374.556.79 *30.15 *236.34 *3847.17 *
LDLT0.07 (x1.3)0.65 (x1.5)26.86 (x4.6)2361.18 (x6.3)6.81 (x1) *31.91 (x1.1) *252.61 (x1.1) *5807.66 (x1.5) *
PartialPivLU0.08 (x1.5)0.69 (x1.6)15.63 (x2.7)709.32 (x1.9)6.81 (x1) *31.32 (x1) *241.68 (x1) *4270.48 (x1.1) *
FullPivLU0.1 (x1.9)4.48 (x10.6)281.33 (x48.2)-6.83 (x1) *32.67 (x1.1) *498.25 (x2.1) *-
HouseholderQR0.19 (x3.5)2.18 (x5.2)23.42 (x4)1337.52 (x3.6)34.26 (x5)129.01 (x4.3)377.37 (x1.6)4839.1 (x1.3)
ColPivHouseholderQR0.23 (x4.3)2.23 (x5.3)103.34 (x17.7)9987.16 (x26.7)36.05 (x5.3)163.18 (x5.4)2354.08 (x10)37860.5 (x9.8)
CompleteOrthogonalDecomposition0.23 (x4.3)2.22 (x5.2)99.44 (x17.1)10555.3 (x28.2)35.75 (x5.3)169.39 (x5.6)2150.56 (x9.1)36981.8 (x9.6)
FullPivHouseholderQR0.23 (x4.3)4.64 (x11)289.1 (x49.6)-69.38 (x10.2)446.73 (x14.8)4852.12 (x20.5)-
JacobiSVD1.01 (x18.6)71.43 (x168.4)--113.81 (x16.7)1179.66 (x39.1)--
BDCSVD1.07 (x19.7)21.83 (x51.5)331.77 (x56.9)18587.9 (x49.6)110.53 (x16.3)397.67 (x13.2)2975 (x12.6)48593.2 (x12.6)

*: 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 .

Observations:

The above table has been generated by the bench/dense_solvers.cpp file, feel-free to hack it to generate a table matching your hardware, compiler, and favorite problem sizes.