qrcpp function

QR Decomposition of a Matrix

Computes the QR decomposition of a matrix.

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.

Returns

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 $R$.

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.

Examples

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

References

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

Author(s)

Kaifeng Lu, kaifenglu@gmail.com