qrcpp: QR Decomposition of a Matrix

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

QR Decomposition of a Matrix

Description

Computes the QR decomposition of a matrix.

Usage

qrcpp(x, tol = 1e-12)

Arguments

x	A numeric matrix whose QR decomposition is to be computed.
tol	The tolerance for detecting linear dependencies in the columns of x.

Details

This function performs Householder QR with column pivoting: Given an m-by-n matrix A with m \geq n, the following algorithm computes r = \textrm{rank}(A) and the factorization Q^T A P equal to

|	R_{11}	R_{12}	|	r
|	0	0	|	m-r
	r	n-r

with Q = H_1 \cdots H_r and P = P_1 \cdots P_r. The upper triangular part of A is overwritten by the upper triangular part of R and components (j+1):m of the jth Householder vector are stored in A((j+1):m, j). The permutation P is encoded in an integer vector pivot.

Value

A list with the following components:

• qr: A matrix with the same dimensions as x. The upper triangle contains the R of the decomposition and the lower triangle contains Householder vectors (stored in compact form).

• rank: The rank of x as computed by the decomposition.

• pivot: The column permutation for the pivoting strategy used during the decomposition.

• Q: The complete m-by-m orthogonal matrix Q.

• R: The complete m-by-n upper triangular matrix R.

Author(s)

Kaifeng Lu, [e-mail]

References

Gene N. Golub and Charles F. Van Loan. Matrix Computations, second edition. Baltimore, Maryland: The John Hopkins University Press, 1989, p.235.

Examples

hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, `+`) }
h9 <- hilbert(9)
qrcpp(h9)

lrstat documentation built on Oct. 18, 2024, 9:06 a.m.