Cholesky decomposition, along with LU decomposition, is one of the most popular triangular factorizations. Cholesky decomposition of a symmetric positive definite (SPD) matrix (or Hermitian positive definite, HPD) is quite similar to LU decomposition: A is represented as A = LL H (or, that essentially the same, as A = U HU). However there are differences too. First, there is no pivoting (but factorization is stable). Second, instead of two matrices (L and U) we have only one matrix multiplied by itself. Finally, Chelesky decomposition require two times less operation than LU decomposition of SPD/HPD matrix of the same size.
Actual performance gain may be even higher than two because pivoting required by LU decomposition results in some performance penalty.
ALGLIB package implements recursive version of Cholesky decomposition similar to the ATLAS one. Decomposition of NxN matrix is reduced to a sequence of approximately 0.5Nx0.5N decompositions. Recursion is stopped when matrix size becomes small enough to fit in the smallest L1 cache of the modern CPUs. At this point, non-recursive version is called. This approach has the following advantages:
These features allows us to achieve the maximum performance possible with our compiler/language. However, different compilers and different programming languages have different limitations. As for linear algebra algorithms, C++ version of ALGLIB is the most efficient implementation. Implementations in other languages have significantly lower performance.
Cholesky decomposition is calculated byand subroutines. Because A is symmetric/Hermitian, only upper or lower triangle may be specified. Correspondingly, as result we will get either lower triangular L or upper triangular U.
This article is intended for personal use only.
C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
Python version (CPython and IronPython are supported).