SRC/slamswlq.f(3)

SRC/slamswlq.f(3)

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NAME 🎀

SRC/slamswlq.f — literally the name, that’s the whole section. Slay.

SYNOPSIS ✨

Functions/Subroutines

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

Function/Subroutine Documentation 💖

subroutine slamswlq (character side, character trans, integer m, integer n,
     integer k, integer mb, integer nb, real, dimension( lda, * ) a, integer
     lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldc, * )
     c, integer ldc, real, dimension( * ) work, integer lwork, integer info)
     SLAMSWLQ

Purpose 🔥

SLAMSWLQ 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

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

Parameters 🫶

• SIDE — SIDE is CHARACTER*1; = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right. 💅
• TRANS — TRANS is CHARACTER*1; = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.
• M — M is INTEGER; the number of rows of the matrix C. M >=0.
• N — N is INTEGER; the number of columns of the matrix C. N >= 0.
• K — K is INTEGER; the number of elementary reflectors whose product defines the matrix Q. M >= K >= 0;
• MB — MB is INTEGER; the row block size to be used in the blocked LQ. M >= MB >= 1
• NB — NB is INTEGER; the column block size to be used in the blocked LQ. NB > M.
• A — A is REAL array, dimension (LDA,M) if SIDE = 'L', (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 SLASWLQ in the first k rows of its array argument A.
• LDA — LDA is INTEGER; the leading dimension of the array A. LDA >= max(1,K).
• T — T is REAL array, dimension ( M * Number of blocks(CEIL(N-K/NB-K)), The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.
• LDT — LDT is INTEGER; the leading dimension of the array T. LDT >= MB.
• C — C is REAL array, dimension (LDC,N). On entry, the M-by-N matrix C. On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
• LDC — LDC is INTEGER; the leading dimension of the array C. LDC >= max(1,M).
• WORK — (workspace) REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
• LWORK — LWORK is INTEGER; the dimension of the array WORK. If MIN(M,N,K) = 0, LWORK >= 1. 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 minimal 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.
• INFO — INFO is INTEGER; = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author 🌸

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Further Details 🔥

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 Further Details in 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 the description of Sequential TSQR in Section 2.2 of [1].

[1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
    J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
    SIAM J. Sci. Comput, vol. 34, no. 1, 2012

📍 Definition at line 198 of file slamswlq.f.

Author 🤖

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