## Contents
1 Definition |

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

**Note #1**

Actual performance gain may be even higher than two because pivoting required by LU decomposition results in some performance penalty.

Main applications of Cholesky decomposition are linear systems and SPD/HPD matrix inversion.

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:

- It makes effective use of CPU cache without explicitly specifying its size. With each iteration matrices become smaller until they fits completely in cache. Such algorithms are called 'cache oblivious'.
- Recursive algorithm makes extensive use of ALGLIB BLAS - mostly of the Level 3 BLAS, the most optimized part of ALGLIB package. The matrices being multiplied are usually square (which is more suited for optimization than a low rank updates generated by LAPACK routines).

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 by spdmatrixcholesky and hpdmatrixcholesky 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*.

C++ | `trfac` subpackage | |

C# | `trfac` subpackage |

*This article is intended for personal use only.*

C# source.

C++ source.

C++ source. MPFR/GMP is used.

GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.

FreePascal version.

Delphi version.

VB.NET version.

VBA version.

Python version (CPython and IronPython are supported).

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