?lamtsqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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?lamtsqr

Multiplies a general matrix by the product of blocked elementary reflectors computed by tall skinny QR factorization (?latsqr)

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

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

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

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

Description

?lamtsqr overwrites the general real or complexm-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 tall skinny QR factorization (?latsqr). Tall-Skinny QR (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

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. m≥n≥ 0.

k

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

mb

INTEGER. The block size to be used in the blocked QR. mb > n. (Must be the same as in ?latsqr)

nb

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

a

REAL for slamtsqr

DOUBLE PRECISION for dlamtsqr

COMPLEX for clamtsqr

COMPLEX*16 for zlamtsqr

Array of size (lda, k). The i-th column must contain the vector which defines the blocked elementary reflector H(i), for i = 1,2,...,k, as returned by ?latsqr in the first k columns of its array argument a.

lda

INTEGER. The leading dimension of the array a.

If side = 'L', lda≥ max(1, m);

if side = 'R', lda≥ max(1, n).

t

REAL for slamtsqr

DOUBLE PRECISION for dlamtsqr

COMPLEX for clamtsqr

COMPLEX*16 for zlamtsqr

Array of size (n * Number of blocks(ceiling(m-k/mb-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≥nb.

c

REAL for slamtsqr

DOUBLE PRECISION for dlamtsqr

COMPLEX for clamtsqr

COMPLEX*16 for zlamtsqr

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 dimension of the array work. If side = 'L', lwork≥ max(1, n)*nb; if side = 'R', lwork≥ max(1, mb)*nb. 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 slamtsqr

DOUBLE PRECISION for dlamtsqr

COMPLEX for clamtsqr

COMPLEX*16 for zlamtsqr

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

?lamtsqr

Description

Input Parameters

Output Parameters