# ?opmtr

Multiplies a real matrix by the real orthogonal matrix Q determined by ?sptrd.

## Syntax

FORTRAN 77:

call sopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info)

call dopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info)

Fortran 95:

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

## Description

The routine multiplies a real matrix C by Q or QT, where Q is the orthogonal matrix Q formed by ?sptrd when reducing a packed real symmetric matrix A to tridiagonal form: `A = Q*T*QT`. Use this routine after a call to ?sptrd.

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

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

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

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.

uplo

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

  Use the same uplo as supplied to ?sptrd.

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`).

ap, tau, c, work

  REAL for sopmtr

  DOUBLE PRECISION for dopmtr.

  ap and tau are the arrays returned by ?sptrd.

  The dimension of ap must be at least max(1, r(r+1)/2).

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

  c(ldc,*) contains the matrix C.

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

  work(*) is a workspace array.

  The dimension of work must be at least

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

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

ldc

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

## Output Parameters

c

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

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 opmtr interface are the following:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (r*(r+1)/2), where

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

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

tau

  Holds the vector of length (r-1).

c

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

side

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

uplo

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

trans

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

## Application Notes

The computed product differs from the exact product by a matrix E such that `||E||2 = O(ε) ||C||2`, where ε is the machine precision.

The total number of floating-point operations is approximately `2*m2*n` if `side = 'L'`, or `2*n2*m` if `side = 'R'`.

The complex counterpart of this routine is ?upmtr.

Parent topic: Symmetric Eigenvalue Problems