Generalized Cholesky Decomposition

DESCRIPTION:

Perform the generalized Cholesky decomposition of a real symmetric matrix.

USAGE:

gchol(x, tolerance=1e-10)

REQUIRED ARGUMENTS:

x
the symmetric matrix to be factored.

OPTIONAL ARGUMENTS:

tolerance
the numeric tolerance for detection of singular columns in x.

VALUE:

an object of class gchol containing the generalized Cholesky decomposition. 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 decomposition, use the diag and as.matrix functions.

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

SEE ALSO:

, .

EXAMPLES:

# 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 8x8 symmetric
temp <- smat[c(1:5, 5:8), c(1:5, 5:8)]
ch1 <- gchol(temp)

print(as.matrix(ch1))   # Print out L
print(diag(ch1))        # Print out D
aeq <- function(x,y) all.equal(as.vector(x), as.vector(y))
aeq(diag(ch1)[6], 0)    # Check that it has a zero in the proper place

ginv <- solve(ch1)    # See if I get a generalized inverse
aeq(temp %*% ginv %*% temp, temp)
aeq(ginv %*% temp %*% ginv, ginv)