?lamswlq

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?lamswlq

Multiplies a general real matrix by a real orthogonal matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization.

call slamswlq(side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)

call dlamswlq(side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)

call clamswlq(side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)

call zlamswlq(side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)

Description

?lamswlq overwrites the general real m-by-n matrix C with

	`side = 'L'`	`side = 'R'`
`trans = 'N'`	`Q`*`C`	`C`*`Q`
`trans = 'T'`	`Q`^T*`C`	`C`*`Q`^T
`trans = 'C'`	`Q`^H*`C`	`C`*`Q`^H

where Q is a real orthogonal matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization (?laswlq).

Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices: Q = Q(1) * Q(2) * . . . * Q(k), where each Q(i) zeros out upper diagonal entries of a block of nb rows of A:

Q(1) zeros out the upper diagonal entries of rows 1:nb of A,

Q(2) zeros out the bottom mb-n rows of rows [1:m, nb + 1:2*nb - m] of A,

Q(3) zeros out the bottom mb-n rows of rows [1:m, 2*nb-m + 1:3*nb - 2*m] of A . . . .

Q(1) is computed by gelqt, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:mb of A, and by upper triangular block reflectors, stored in array t(1:ldt, 1:n). For more information, see gelqt.

Q(i) for i > 1 is computed by tplqt, which represents Q(i) by Householder vectors stored in columns [(i - 1)*(nb - m) + m + 1:i*(nb - m) + m] of A, and by upper triangular block reflectors, stored in array t(1:ldt, (i - 1)*m + 1:i*m). The last Q(k) may use fewer rows. For more information see Further Details in tplqt. For more details of the overall algorithm, see [DEMMEL12].

Input Parameters

side

CHARACTER*1.

If side = 'L': apply op(Q) from the left;

if side = 'R': apply op(Q) from the right.

trans

CHARACTER*1.

If trans = 'N': No transpose, op(Q) = Q;

if trans = 'T': Transpose, op(Q) = Q^T;

if trans= 'C': Transpose, op(Q) = Q^H.

m

INTEGER. The number of rows of the matrix C. m≥ 0.

n

INTEGER. The number of columns of the matrix C. n≥m.

k

INTEGER. The number of elementary reflectors whose product defines the matrix Q. m≥k≥ 0;

mb

INTEGER. The row block size to be used in the blocked QR. m≥mb≥ 1

nb

INTEGER. The block size to be used in the blocked QR. nb > m.

a

REAL for slamswlq

DOUBLE PRECISION for dlamswlq

COMPLEX for clamswlq

COMPLEX*16 for zlamswlq

Array of size (lda, m) if side = 'L' or (lda, n) if side = 'R'. The i-th row must contain the vector which defines the blocked elementary reflector H(i), for i = 1, 2, ..., k, as returned by ?laswlq in the first k rows of its array argument a.

lda

INTEGER. The leading dimension of the array a.lda≥ max(1, k).

t

REAL for slamswlq

DOUBLE PRECISION for dlamswlq

COMPLEX for clamswlq

COMPLEX*16 for zlamswlq

Array of size (m * Number of blocks(ceiling(n - k/nb - k))), The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks as described previously.

ldt

INTEGER. The leading dimension of the array t. ldt≥mb.

c

REAL for slamswlq

DOUBLE PRECISION for dlamswlq

COMPLEX for clamswlq

COMPLEX*16 for zlamswlq

Array of size (ldc,n). On entry, the m-by-n matrix C.

ldc

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

lwork

INTEGER. The size of the array work. If side = 'L', lwork≥ max(1, nb) * mb; if side = 'R', lwork≥ max(1, m) * mb. 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.

Output Parameters

c

On exit, c is overwritten by op(Q)*C or C*op(Q).

work

REAL for slamswlq

DOUBLE PRECISION for dlamswlq

COMPLEX for clamswlq

COMPLEX*16 for zlamswlq

Workspace array of size (max(1, lwork)).

info

INTEGER.

info = 0: successful exit.

info < 0: if info = -i, the i-th argument had an illegal value.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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