/** \eigenManualPage DenseDecompositionBenchmark 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 \link TopicLinearAlgebraDecompositions table \endlink.
This benchmark has been run on a laptop equipped with an Intel core i7 \@ 2,6 GHz, and compiled with clang with \b AVX and \b FMA instruction sets enabled but without multi-threading.
It uses \b single \b precision \b 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 \f$ A^T A \f$ for the first four solvers based on Cholesky and LU, as denoted by the \b * symbol (top-right corner part of the table).
Timings are in \b milliseconds, and factors are relative to the LLT decomposition which is the fastest but also the least general and robust.
<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.
+ 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.
+ CompleteOrthogonalDecomposition is based on ColPivHouseholderQR and they thus achieve the same level of performance.
The above table has been generated by the <a href="https://gitlab.com/libeigen/eigen/raw/master/bench/dense_solvers.cpp">bench/dense_solvers.cpp</a> file, feel-free to hack it to generate a table matching your hardware, compiler, and favorite problem sizes.
