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

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.

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.

Examples

Run this code

(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
})

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