DGERQF(l) — LAPACK routine (version 2.0)

NAME

DGERQF - compute an RQ factorization of a real M-by-N matrix A

SYNOPSIS

SUBROUTINE DGERQF(
M, N, A, LDA, TAU, WORK, LWORK, INFO )

INTEGER INFO, LDA, LWORK, M, N

DOUBLE PRECISION A( LDA, ∗ ), TAU( ∗ ), WORK( LWORK )

PURPOSE

DGERQF computes an RQ factorization of a real M-by-N matrix A: A = R ∗ Q.

ARGUMENTS

M (input) INTEGER
The number of rows of the matrix A. M >= 0.

N (input) INTEGER
The number of columns of the matrix A. N >= 0.

A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if m <= n, the upper triangle of the subarray A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R; if m >= n, the elements on and above the (m-n)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M).

TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further Details).

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,M). For optimum performance LWORK >= M∗NB, where NB is the optimal blocksize.

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau ∗ v ∗ v’

where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).