?latsqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?latsqr

Computes a blocked Tall-Skinny QR matrix factorization.

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

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

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

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

Description

?latsqr computes a blocked Tall-Skinny QR (TSQR) factorization of an m-by-n matrix A, where m≥n: A = Q*R.

TSQR performs QR 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 subdiagonal entries of a block of mb rows of A:

Q(1) zeros out the subdiagonal entries of rows 1:MB of A,

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

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

Q(1) is computed by geqrt, 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 geqrt.

Q(i) for i > 1 is computed by tpqrt, which represents Q(i) by Householder vectors stored in rows [(i - 1)*(mb - n) + n + 1:i*(mb - n) + n] of a, and by upper triangular block reflectors, stored in array t(1:ldt, (i - 1)*n + 1:i*n). The last Q(k) may use fewer rows. For more information, see tpqrt. 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. m≥n≥ 0.

mb

INTEGER. The row block size to be used in the blocked QR. mb > n.

nb

INTEGER. The column block size to be used in the blocked QR. n≥nb≥ 1.

a

REAL for slatsqr

DOUBLE PRECISION for dlatsqr

COMPLEX for clatsqr

COMPLEX*16 for zlatsqr

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≥nb.

lwork

INTEGER. The dimension of the array work. lwork≥nb*n. 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 above the diagonal of the array contain the n-by-n upper triangular matrix R and the elements below the diagonal represent Q by the columns of blocked V.

t

REAL for slatsqr

DOUBLE PRECISION for dlatsqr

COMPLEX for clatsqr

COMPLEX*16 for zlatsqr

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

work

REAL for slatsqr

DOUBLE PRECISION for dlatsqr

COMPLEX for clatsqr

COMPLEX*16 for zlatsqr

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

?latsqr