Solve a matrix equation using the generalized Cholesky decompostion
This function solves the equation Ax=b for x, given b and the generalized Cholesky decompostion of A. If only the first argument is given, then a G-inverse of A is returned.
a generalized cholesky decompostion of a matrix, as
returned by the
a numeric vector or matrix, that forms the right-hand side of the equation.
solve the problem for the full (orignal) matrix, or for the cholesky matrix.
other arguments are ignored
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.
This routine solves either the original problem Ay=b
full argument) or the subproblem sqrt(D)L'y=b.
b is missing it returns the inverse of
A or L, respectively.
b is not present, the inverse of
a is returned, otherwise the solution to
1 2 3 4 5 6 7 8 9 10
# 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) ginv <- solve(ch1, full=FALSE) # generalized inverse of ch1 tinv <- solve(ch1, full=TRUE) # generalized inverse of temp all.equal(temp %*% tinv %*% temp, temp)
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