gchol: Generalized Cholesky decompostion

Description

Perform the generalized Cholesky decompostion of a real symmetric matrix.

Usage

gchol(x, tolerance=1e-10)

Value

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

Arguments

x: the symmetric matrix to be factored
tolerance: the numeric tolerance for detection of singular columns in x.

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.

Examples

Run this code

# Create a matrix that is symmetric, but not positive definite
#   The matrix temp has column 6 redundant with cols 1-5
smat <- matrix(1:64, ncol=8)
smat <- smat + t(smat) + diag(rep(20,8))  #smat is 8 by 8 symmetric
temp <-  smat[c(1:5, 5:8), c(1:5, 5:8)]
ch1  <- gchol(temp)

print(as.matrix(ch1), digits=4)   # print out L
print(diag(ch1))        # Note the zero at position 6

ginv <- solve(ch1)    # generalized inverse
diag(ginv)            # also has column 6 marked as singular