# ?orgrq

Generates the real matrix Q of the RQ factorization formed by ?gerqf.

## Syntax

FORTRAN 77:

call sorgrq(m, n, k, a, lda, tau, work, lwork, info)

call dorgrq(m, n, k, a, lda, tau, work, lwork, info)

Fortran 95:

call orgrq(a, tau [,info])

## Description

The routine generates an m-by-n real matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors `H(i)` of order n: `Q = H(1)* H`(2)`*...*`H(k)``as returned by the routines sgerqf/dgerqf. Use this routine after a call to sgerqf/dgerqf.

## Input Parameters

m

  INTEGER. The number of rows of the matrix Q (`m≥ 0`).

n

  INTEGER. The number of columns of the matrix Q (`n≥ m`).

k

  INTEGER. The number of elementary reflectors whose product defines the matrix Q (`m≥ k≥ 0`).

a, tau, work

  REAL for sorgrq

  DOUBLE PRECISION for dorgrq

  Arrays: a(lda,*), tau(*).

  On entry, the (m - k + i)-th row of a must contain the vector which defines the elementary reflector `H(i)`, for i = 1,2,...,k, as returned by sgerqf/dgerqf in the last k rows of its array argument a;

  tau(i) must contain the scalar factor of the elementary reflector `H(i)`, as returned by sgerqf/dgerqf;

  The second dimension of a must be at least max(1, n).

  The dimension of tau must be at least max(1, k).

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; at least max(1, m).

lwork

  INTEGER. The size of the work array; at least max(1, m).

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  Overwritten by the m-by-n matrix Q.

`work(1)`

  If `info = 0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine orgrq interface are the following:

a

  Holds the matrix A of size (m,n).

tau

  Holds the vector of length (k).

## Application Notes

For better performance, try using `lwork =m*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The complex counterpart of this routine is ?ungrq.

Parent topic: Orthogonal Factorizations