/home/abui.../zlaswlq.f(3)

💅 ZLASWLQ 💅

Manual name: Library Functions Manual. Page file: /home/abui.../zlaswlq.f(3). okay bestie, this is the LAPACK lore for ZLASWLQ, and we’re keeping every single deet ✨

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlaswlq.f

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

• subroutine ZLASWLQ (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) — literally ZLASWLQ.

🧚 Function/Subroutine Documentation 🧚

🪄 subroutine ZLASWLQ

   subroutine ZLASWLQ (integer m, integer n, integer mb, integer nb,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldt, *) t, integer ldt, complex*16, dimension( * ) work, integer lwork,
       integer info)

ZLASWLQ (yes, again — the docs are extra like that 💅)

✨ Purpose: ✨

Here’s the tea: ZLASWLQ computes a blocked Tall-Skinny LQ factorization of a complexx M-by-N matrix A for M <= N:

   A = ( L 0 ) *  Q,

where:

• Q is a n-by-N orthogonal matrix, stored on exit in an implicit form in the elements above the diagonal of the array A and in the elements of the array T;
• L is a lower-triangular M-by-M matrix stored on exit in the elements on and below the diagonal of the array A.
• 0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

🎀 Parameters 🎀

• M — M is INTEGER. The number of rows of the matrix A. M >= 0.
• N — N is INTEGER. The number of columns of the matrix A. N >= M >= 0.
• MB — MB is INTEGER. The row block size to be used in the blocked QR. M >= MB >= 1.
• NB — NB is INTEGER. The column block size to be used in the blocked QR. NB > 0.
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix 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 (see Further Details).
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• T — T is COMPLEX*16 array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((N-M)/(NB-M)). The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= MB.
• WORK — (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)).
• LWORK — LWORK is 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.
• 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 TPQRT.

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 165 of file zlaswlq.f.

🌸 Author 🌸

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