?laswlq

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?laswlq

Computes blocked Short-Wide LQ matrix factorization.

call slaswlq(m, n, mb, nb, a, lda, t, ldt, work, lwork, info)

call dlaswlq(m, n, mb, nb, a, lda, t, ldt, work, lwork, info)

call claswlq(m, n, mb, nb, a, lda, t, ldt, work, lwork, info)

call zlaswlq(m, n, mb, nb, a, lda, t, ldt, work, lwork, info)

Description

?laswlq computes a blocked Short-Wide LQ (SWLQ) factorization of an m-by-n matrix A, where n≥m: A = L*Q.

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 bygelqt, 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 tplqt. For more details of the overall algorithm, see [DEMMEL12].

Input Parameters

m

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

n

INTEGER. The number of columns of the matrix A. n≥m≥ 0.

mb

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

nb

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

a

REAL for slaswlq

DOUBLE PRECISION for dlaswlq

COMPLEX for claswlq

COMPLEX*16 for zlaswlq

Array of size (lda, n). On entry, the m-by-n matrix A.

lda

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

ldt

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

lwork

INTEGER. The dimension of the array work. lwork≥mb * m.

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

a

On exit, the elements on and below the diagonal of the array contain the n-by-n lower triangular matrix L; the elements above the diagonal represent Q by the rows of blocked V.

t

REAL for slaswlq

DOUBLE PRECISION for dlaswlq

COMPLEX for claswlq

COMPLEX*16 for zlaswlq

Array of size (ldt, n * Number_of_row_blocks), where Number_of_row_blocks = ceiling((n - m)/(nb-m)). The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

work

REAL for slaswlq

DOUBLE PRECISION for dlaswlq

COMPLEX for claswlq

COMPLEX*16 for zlaswlq

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

?laswlq