lrstat (version 0.2.13)

qrcpp: QR Decomposition of a Matrix

Description

Computes the QR decomposition of a matrix.

Usage

qrcpp(X, tol = 1e-12)

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 \(R\).

Arguments

X

A numeric matrix whose QR decomposition is to be computed.

tol

The tolerance for detecting linear dependencies in the columns of X.

Author

Kaifeng Lu,

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 \(j\)th Householder vector are stored in \(A((j+1):m, j)\). The permutation \(P\) is encoded in an integer vector pivot.

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)