# asa006

asa006, a MATLAB code which computes the Cholesky factor of a symmetric positive definite (SPD) matrix.

This is a version of Applied Statistics Algorithm 6.

If A is a symmetric positive definite matrix, then there is an upper triangular matrix U with the property that

A = U' * U
The matrix U is known as the Cholesky factor of A, and can be used to easily solve linear systems involving A or compute the inverse of A.

The algorithm implemented here uses a compressed storage for both the matrix A and the factor U. This saves some storage, but can make computations a little awkward.

### Languages:

asa006 is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

asa007, a MATLAB code which computes the inverse of a symmetric positive definite (SPD) matrix, and uses a version of ASA006 for for Cholesky factorization.

asa047, a MATLAB code which implements the Nelder-Mead minimization algorithm, and uses a version of ASA006 for Cholesky factorization.

toeplitz_cholesky, a MATLAB code which computes the Cholesky factorization of a nonnegative definite symmetric Toeplitz matrix.

### Reference:

1. PR Freeman,
Remark AS R44: A Remark on AS 6 and AS7: Triangular decomposition of a symmetric matrix and Inversion of a positive semi-definite symmetric matrix,
Applied Statistics,
Volume 31, Number 3, 1982, pages 336-339.
2. Michael Healy,
Algorithm AS 6: Triangular decomposition of a symmetric matrix,
Applied Statistics,
Volume 17, Number 2, 1968, pages 195-197.

### Source Code:

• cholesky.m computes the Cholesky factorization of a symmetric positive definite (SPD) matrix.
• subchl.m computes the Cholesky factorization of a submatrix of a symmetric positive definite (SPD) matrix.

Last revised on 20 June 2021.