?ormbr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?ormbr

Multiplies an arbitrary real matrix by the real orthogonal matrix Q or P^T determined by ?gebrd.

Syntax

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)

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

Include Files

mkl.fi, lapack.f90

Description

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

Input Parameters

In the descriptions below, r denotes the order of Q or P^T:

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

vect

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

If vect = 'Q', then Q or Q^T is applied to C.

If vect = 'P', then P or P_T 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 Q^T or P^T 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 leading dimension of a. Constraints:

lda≥ max(1, r) if vect = 'Q';

lda≥ max(1, min(r,k)) if vect = 'P'.

ldc

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

tau

REAL for sormbr

DOUBLE PRECISION for dormbr.

Array, size 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, Q^T*C, C*Q, C*Q,^T, P*C, P^T*C, C*P, or C*P^T, 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.

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 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: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ormbr