?ormtr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?ormtr

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

Syntax

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

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

call ormtr(a, 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 sytrd when reducing a real symmetric matrix A to tridiagonal form: A = Q*T*Q^T. Use this routine after a call to ?sytrd.

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 ?sytrd.

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).

a, c, tau, work

REAL for sormtr

DOUBLE PRECISION for dormtr

a(lda,*) and tau are the arrays returned by ?sytrd.

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

The size 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, its dimension max(1, lwork).

lda

INTEGER. The leading dimension of a; lda≥ max(1, r).

ldc

INTEGER. The leading dimension of c; 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, Q^T*C, C*Q, or C*Q^T (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.

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 ormtr interface are the following:

a

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

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' 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*m²*n, if side = 'L', or 2*n²*m, if side = 'R'.

The complex counterpart of this routine is unmtr.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ormtr