Eigen
3.4.99 (git rev 199c5f2b47eb1f8e5a2d20e60f07e97cd95a6ba6)

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 multithreading. 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 \( A^T A \) for the first four solvers based on Cholesky and LU, as denoted by the * symbol (topright 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 

LLT  0.05  0.42  5.83  374.55  6.79 ^{*}  30.15 ^{*}  236.34 ^{*}  3847.17 ^{*} 
LDLT  0.07 (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) ^{*} 
PartialPivLU  0.08 (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) ^{*} 
FullPivLU  0.1 (x1.9)  4.48 (x10.6)  281.33 (x48.2)    6.83 (x1) ^{*}  32.67 (x1.1) ^{*}  498.25 (x2.1) ^{*}   
HouseholderQR  0.19 (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) 
ColPivHouseholderQR  0.23 (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) 
CompleteOrthogonalDecomposition  0.23 (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) 
FullPivHouseholderQR  0.23 (x4.3)  4.64 (x11)  289.1 (x49.6)    69.38 (x10.2)  446.73 (x14.8)  4852.12 (x20.5)   
JacobiSVD  1.01 (x18.6)  71.43 (x168.4)      113.81 (x16.7)  1179.66 (x39.1)     
BDCSVD  1.07 (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 leastsquare solving for overconstrained problems, and the reported timing include the cost to form the symmetric covariance matrix \( A^T A \).
Observations:
The above table has been generated by the bench/dense_solvers.cpp file, feelfree to hack it to generate a table matching your hardware, compiler, and favorite problem sizes.