chol2inv-methods: Inverse from Cholesky Factor

In Matrix: Sparse and Dense Matrix Classes and Methods

Inverse from Cholesky Factor

Description

Given formally upper and lower triangular matrices U and L, compute (U' U)^{-1} and (L L')^{-1}, respectively.

This function can be seen as way to compute the inverse of a symmetric positive definite matrix given its Cholesky factor. Equivalently, it can be seen as a way to compute (X' X)^{-1} given the R part of the QR factorization of X, if R is constrained to have positive diagonal entries.

Usage

chol2inv(x, ...)
## S4 method for signature 'dtrMatrix'
chol2inv(x, ...)
## S4 method for signature 'dtCMatrix'
chol2inv(x, ...)
## S4 method for signature 'generalMatrix'
chol2inv(x, uplo = "U", ...)

Arguments

x	a square matrix or Matrix, typically the result of a call to chol. If x is square but not (formally) triangular, then only the upper or lower triangle is considered, depending on optional argument uplo if x is a Matrix.
uplo	a string, either "U" or "L", indicating which triangle of x contains the Cholesky factor. The default is "U", to be consistent with chol2inv from base.
...	further arguments passed to or from methods.

Value

A matrix, symmetricMatrix, or diagonalMatrix representing the inverse of the positive definite matrix whose Cholesky factor is x. The result is a traditional matrix if x is a traditional matrix, dense if x is dense, and sparse if x is sparse.

See Also

The default method from base, chol2inv, called for traditional matrices x.

Generic function chol, for computing the upper triangular Cholesky factor L' of a symmetric positive semidefinite matrix.

Generic function solve, for solving linear systems and (as a corollary) for computing inverses more generally.

Examples

(A <- Matrix(cbind(c(1, 1, 1), c(1, 2, 4), c(1, 4, 16))))
(R <- chol(A))
(L <- t(R))
(R2i <- chol2inv(R))
(L2i <- chol2inv(R))
stopifnot(exprs = {
    all.equal(R2i, tcrossprod(solve(R)))
    all.equal(L2i,  crossprod(solve(L)))
    all.equal(as(R2i %*% A, "matrix"), diag(3L)) # the identity 
    all.equal(as(L2i %*% A, "matrix"), diag(3L)) # ditto
})

Matrix documentation built on Oct. 19, 2024, 1:08 a.m.