SRC/claswlq.f(3)

💖 SRC/claswlq.f 💖

okay so this is a page straight outta the Library Functions Manual and it's all about one very specific, very iconic LAPACK subroutine. buckle up bestie, we're doing linear algebra 💅

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✨ NAME ✨

SRC/claswlq.f — that's it, that's the name, no notes 🎀

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🦋 SYNOPSIS 🦋

Functions/Subroutines

• subroutine claswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) — CLASWLQ. the one, the only 💫

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🌸 Function/Subroutine Documentation 🌸

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

CLASWLQ 🫶

🎯 Purpose

okay so here's the whole vibe: CLASWLQ computes a blocked Tall-Skinny LQ factorization of a complex M-by-N matrix A — and this is only for when M <= N, bestie, write that down 📝

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. literally not even invited to the party 🙈

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🌈 Parameters 🌈

• M — M is INTEGER. The number of rows of the matrix A. M >= 0. count 'em 👑
• N — N is INTEGER. The number of columns of the matrix A. N >= M >= 0. and remember N can never be smaller than M, that's the rule fr fr
• 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. strictly greater, no zeroes allowed 💀
• A — A is COMPLEX 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). so yeah she does a full glow-up on exit ✨
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). don't skip the max(1,...) part, that's load-bearing 🫶
• T — T is COMPLEX 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. compact form = it's giving minimalism 💅
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= MB 🎀
• WORK — (workspace) COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the minimal LWORK. the little workspace that could 🌸
• LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise.

🚨 and 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. so -1 = "just tell me how much room you need, no drama" 😌

• INFO — INFO is INTEGER
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 😭

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👑 Author 👑

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

the whole squad showed up for this one 🫶

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🔥 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 📚

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📍 Definition at line 167 of file claswlq.f.

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💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code. yes bestie, a robot wrote the meta part, but the math is all human 🔥

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and that's the whole page! this came to you from LAPACK, Version 3.12.0, out of SRC/claswlq.f(3) — mwah, go factor something tall and skinny 💋🎀🦋