solve.bdsmatrix: Solve a matrix equation using the generalized Cholesky decompostion

Description

This function solves the equation Ax=b for x, when A is a block diagonal sparse matrix (an object of class bdsmatrix).

Usage

# S3 method for bdsmatrix
solve(a, b, full=TRUE, tolerance=1e-10, ...)

Value

if argument b is not present, the inverse of a is returned, otherwise the solution to matrix equation. The equation is solved using a generalized Cholesky decomposition.

Arguments

a: a block diagonal sparse matrix object
b: a numeric vector or matrix, that forms the right-hand side of the equation.
full: if true, return the full inverse matrix; if false return only that portion corresponding to the blocks. This argument is ignored if b is present. If the bdsmatrix a has a non-sparse portion, i.e., if the rmat component is present, then the inverse of a will not be block-diagonal sparse. In this case setting full=F returns only a portion of the inverse. The elements that are returned are those of the full inverse, but the off-diagonal elements that are not returned would not have been zero.
tolerance: the tolerance for detecting singularity in the a matrix
...: other arguments are ignored

Details

The matrix a consists of a block diagonal sparse portion with an optional dense border. The inverse of a, which is to be computed if y is not provided, will have the same block diagonal structure as a only if there is no dense border, otherwise the resulting matrix will not be sparse.

However, these matrices may often be very large, and a non sparse version of one of them will require gigabytes of even terabytes of space. For one of the common computations (degrees of freedom in a penalized model) only those elements of the inverse that correspond to the non-zero part of a are required; the full=F option returns only that portion of the (block diagonal portion of) the inverse matrix.

Examples

Run this code

tmat <- bdsmatrix(c(3,2,2,4), 
              c(22,1,2,21,3,20,19,4,18,17,5,16,15,6,7, 8,14,9,10,13,11,12),
              matrix(c(1,0,1,1,0,0,1,1,0,1,0,10,0,
                       0,1,1,0,1,1,0,1,1,0,1,0,10), ncol=2))
dim(tmat)
solve(tmat, cbind(1:13, rep(1,13)))