?opmtr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?opmtr

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

Syntax

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)

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

Include Files

mkl.fi, lapack.f90

Description

The routine multiplies a real matrix C by Q or Q^T, where Q is the orthogonal matrix Q formed by sptrd when reducing a packed real symmetric matrix A to tridiagonal form: A = Q*T*Q^T. 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, Q^T*C, C*Q, or C*Q^T (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 Q^T is applied to C from the left.

If side = 'R', Q or Q^T 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 Q^T.

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 leading dimension of c; ldc≥ max(1, n) .

Output Parameters

c

Overwritten by the product Q*C, Q^T*C, C*Q, or C*Q^T (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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine opmtr interface are the following:

ap

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 with the number of elements 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: LAPACK Computational Routines

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