Cholesky Decompsition

While symmetric, positive definite matrices are rather special, they occur quite frequently in some applications, so their special factorization, called Cholesky decomposition, is good to know about. When you can use it, Cholesky decomposition is about a factor of two faster than alternative methods for solving linear equations.Instead of seeking arbitrary lower and upper triangular factors L and U, Cholesky decomposition constructs a lower triangular matrix L whose transpose L^T can itself serve as the upper triangular part. In other words we replace equation:

This factorization is sometimes referred to as “taking the square root” of the matrix A. The

components of LT are of course related to those of L by:

Writing out this equation in components one readily obtains the analogs of equations:

and

If you apply the last two equations in the order i = 1, 2, . . . ,N, you will see

that the L’s that occur on the right-hand side are already determined by the time they are

needed.

Source:

http//:www.mpi-hd.mpg.de/astrophysik/HEA/internal/nuermical_Recipes/f2-9.pdf