# ?ormbr

Multiplies an arbitrary real matrix by the real orthogonal matrix Q or PT determined by ?gebrd.

## Syntax

FORTRAN 77:

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

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

Fortran 95:

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

## Description

Given an arbitrary real matrix C, this routine forms one of the matrix products `Q*C`, `QT*C`, `C*Q`, `C*Q,T`, `P*C`, `PT*C`, `C*P`, `C*PT`, where Q and P are orthogonal matrices computed by a call to sgebrd/dgebrd. The routine overwrites the product on C.

## Input Parameters

In the descriptions below, r denotes the order of Q or PT:

If `side = 'L'`, `r = m`; if `side = 'R'`, `r = n`.

vect

  CHARACTER*1. Must be 'Q' or 'P'.

  If `vect = 'Q'`, then Q or QT is applied to C.

  If `vect = 'P'`, then P or PT is applied to C.

side

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

  If `side = 'L'`, multipliers are applied to C from the left.

  If `side = 'R'`, they are applied to C from the right.

trans

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

  If `trans = 'N'`, then Q or P is applied to C.

  If `trans = 'T'`, then QT or PT is applied to C.

m

  INTEGER. The number of rows in C.

n

  INTEGER. The number of columns in C.

k

  INTEGER. One of the dimensions of A in ?gebrd:

  If `vect = 'Q'`, the number of columns in A;

  If `vect = 'P'`, the number of rows in A.

  Constraints: m ≥ 0, n ≥ 0, k ≥ 0.

a, c, work

  REAL for sormbr

  DOUBLE PRECISION for dormbr.

  Arrays:

  a(lda,*) is the array a as returned by ?gebrd.

  Its second dimension must be at least max(1, min(r,k)) for `vect = 'Q'`, or max(1, r)) for `vect = 'P'`.

  c(ldc,*) holds the matrix C.

  Its second dimension 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, r) if `vect = 'Q'`;

  lda ≥ max(1, min(r,k)) if `vect = 'P'`.

ldc

  INTEGER. The first dimension of c; ldc ≥ max(1, m).

tau

  REAL for sormbr

  DOUBLE PRECISION for dormbr.

  Array, DIMENSION at least max (1, min(r, k)).

  For `vect = 'Q'`, the array tauq as returned by ?gebrd. For `vect = 'P'`, the array taup as returned by ?gebrd.

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`, `C*Q,T`, `P*C`, `PT*C`, `C*P`, or `C*PT`, as specified by vect, 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 ormbr interface are the following:

a

  Holds the matrix A of size (r,min(nq,k)) where

  `r = nq`, if `vect = 'Q'`,

  `r = min(nq,k)`, if `vect = 'P'`,

  `nq = m`, if `side = 'L'`,

  `nq = n`, if `side = 'R'`,

  `k = m`, if `vect = 'P'`,

  `k = n`, if `vect = 'Q'`.

tau

  Holds the vector of length min(nq,k).

c

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

vect

  Must be 'Q' or 'P'. The default value is 'Q'.

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` for `side = 'L'`, or

`lwork = m*blocksize` for `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 computed product differs from the exact product by a matrix E such that `||E||2 = O(ε)*||C||2`.

The total number of floating-point operations is approximately

`2*n*k(2*m - k)` if `side = 'L'` and `m ≥ k`;

`2*m*k(2*n - k)` if `side = 'R'` and `n ≥ k`;

`2*m2*n` if `side = 'L'` and `m < k`;

`2*n2*m` if `side = 'R'` and `n < k`.

The complex counterpart of this routine is ?unmbr.

Parent topic: Singular Value Decomposition