# ?ormqr

Multiplies a real matrix by the orthogonal matrix Q of the QR factorization formed by ?geqrf or ?geqpf.

## Syntax

FORTRAN 77:

call sormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)

call dormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)

Fortran 95:

call ormqr(a, tau, c [,side] [,trans] [,info])

## Description

The routine multiplies a real matrix C by Q or Q T, where Q is the orthogonal matrix Q of the QR factorization formed by the routines sgeqrf/dgeqrf or sgeqpf/dgeqpf.

Depending on the parameters side and trans, the routine can form one of the matrix products `Q*C`, `QT*C`, `C*Q`, or `C*QT` (overwriting the result on C).

## Input Parameters

side

  CHARACTER*1. Must be either 'L' or 'R'.

  If `side ='L'`, Q or QT is applied to C from the left.

  If `side ='R'`, Q or QT is applied to C from the right.

trans

  CHARACTER*1. Must be either 'N' or 'T'.

  If `trans ='N'`, the routine multiplies C by Q.

  If `trans ='T'`, the routine multiplies C by QT.

m

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

n

  INTEGER. The number of columns in C (`n ≥ 0`).

k

  INTEGER. The number of elementary reflectors whose product defines the matrix Q. Constraints:

  `0 ≤ k ≤ m` if `side ='L'`;

  `0 ≤ k ≤ n` if `side ='R'`.

a, tau, c, work

  REAL for sgeqrf

  DOUBLE PRECISION for dgeqrf.

  Arrays:

  a(lda,*) and tau(*) are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf. The second dimension of a must be at least max(1, k). The dimension of tau must be at least max(1, k).

  c(ldc,*) contains the matrix C.

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

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

  INTEGER. The first dimension of a. Constraints:

  `lda ≥ max(1, m)` if `side = 'L'`;

  `lda ≥ max(1, n)` if `side = 'R'`.

ldc

  INTEGER. The first dimension of c. Constraint:

  ldc ≥ max(1, m).

lwork

  INTEGER. The size of the work array. Constraints:

  `lwork ≥ max(1, n)` if `side = 'L'`;

  `lwork ≥ max(1, m)` if `side = 'R'`.

  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

c

  Overwritten by the product `Q*C`, `QT*C`, `C*Q`, or `C*QT` (as specified by side and trans).

`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 ormqr interface are the following:

a

  Holds the matrix A of size (r,k).

  `r = m` if `side = 'L'`.

  `r = n` if `side = 'R'`.

tau

  Holds the vector of length (k).

c

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

side

  Must be 'L' or 'R'. The default value is 'L'.

trans

  Must be 'N' or 'T'. The default value is 'N'.

## Application Notes

For better performance, try using `lwork = n*blocksize` (if `side = 'L'`) or `lwork = m*blocksize` (if `side = 'R'`) 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 ?unmqr.

Parent topic: Orthogonal Factorizations