gchol: Generalized Cholesky decompostion

Description

Perform the generalized Cholesky decompostion of a real symmetric matrix. The function and R documentation are included from the archived kinship package that can be found at http://cran.r-project.org/web/packages/kinship/index.html.

Usage

gchol(x, tolerance=1e-10)

Arguments

Value

an object of class gchol containing the generalized Cholesky decompostion. It has the appearance of a lower triangular matrix.

Details

A symmetric matrix A can be decomposed as LDL', where L is a lower triangular matrix with 1's on the diagonal, L' is the transpose of L, and D is diagonal. The inverse of L is also lower-triangular, with 1's on the diagonal. If all elements of D are positive, then A must be symmetric positive definite (SPD), and the solution can be reduced the usual Cholesky decomposition U'U where U is upper triangular and U = sqrt(D) L'.

The main advantage of the generalized form is that it admits of matrices that are not of full rank: D will contain zeros marking the redundant columns, and the rank of A is the number of non-zero columns. If all elements of D are zero or positive, then A is a non-negative definite (NND) matrix. The generalized form also has the (quite minor) numerical advantage of not requiring square roots during its calculation. To extract the components of the decompostion, use the diag and as.matrix functions.

The solve has a method for gchol decompostions, and there are gchol methods for block diagonal symmetric (bdsmatrix) matrices as well.