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# sgerqf

```
NAME
SGERQF - compute an RQ factorization of a real M-by-N matrix
A

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

INTEGER        INFO, LDA, LWORK, M, N

REAL           A( LDA, * ), TAU( * ), WORK( LWORK )

PURPOSE
SGERQF 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) REAL 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 subdi-
agonal 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) REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see
Further Details).

WORK    (workspace) REAL 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).
```