sherman.morrison

Sherman-Morrison formula

Description

The Sherman-Morrison formula gives a convenient expression for the inverse of the rank 1 update \((\mathbf{A} + \mathbf{bd}^T)\) where \(\mathbf{A}\) is a \(n\times n\) matrix and \(\mathbf{b}\) , \(\mathbf{d}\) are \(n\) -dimensional vectors. Thus

\[(\mathbf{A} + \mathbf{bd}^T)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{bd}^T\mathbf{A}^{-1}}{1 + \mathbf{d}^T\mathbf{A}^{-1}\mathbf{b}}.\]

Usage

sherman.morrison(a, b, d = b, inverted = FALSE)

Arguments

Argument	Description
a	a numeric matrix.
b	a numeric vector.
d	a numeric vector.
inverted	logical. If TRUE , a is supposed to contain its inverse .

Details

Method of sherman.morrison calls BLAS level 2 subroutines DGEMV and DGER for computational efficiency.

Value

a square matrix of the same order as a .

Examples

n <- 10
ones <- rep(1, n)
a <- 0.5 * diag(n)
z <- sherman.morrison(a, ones, 0.5 * ones)
z