laswlq(3)

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version 3.12.1 · 9 versions
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💅 laswlq(3) — short-wide LQ factor 💅

okay so picture this: it's section 3, it's the library functions era, and the whole personality of this page is "short-wide LQ factor." that's the name, that's the game. hi bestie, welcome, let's get into it 🫶

🌸 NAME 🌸

laswlq — laswlq: short-wide LQ factor

✨ SYNOPSIS ✨

📞 Functions 📞

  • claswlq — subroutine claswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) → CLASWLQ
  • dlaswlq — subroutine dlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) → DLASWLQ
  • slaswlq — subroutine slaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) → SLASWLQ
  • zlaswlq — subroutine zlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) → ZLASWLQ

like, yes, all four of them, same argument list, different letters in front. the alphabet really said "let me cook" 🔥

subroutine claswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
subroutine dlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
subroutine slaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
subroutine zlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)

🌈 DESCRIPTION 🌈

this is the detailed description zone, and the deets for every routine live right down there in the function documentation. no cap, it's all below 👇

🔍 FUNCTION DOCUMENTATION 🔍

💜 claswlq — CLASWLQ 💜

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)

🎯 Purpose 🎯

bestie, CLASWLQ computes a blocked Tall-Skinny LQ factorization of a complex M-by-N matrix A for M <= N, and it's giving:

   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 — INTEGER. The number of rows of the matrix A. M >= 0.
  • N — INTEGER. The number of columns of the matrix A. N >= M >= 0.
  • MB — INTEGER. The row block size to be used in the blocked QR. M >= MB >= 1
  • NB — INTEGER. The column block size to be used in the blocked QR. NB > 0.
  • A — 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).
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • T — 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.
  • LDT — 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.
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise. 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 — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
little heads up bestie: XERBLA is the one who'd normally read you for the LWORK thing, but on a LWORK = -1 query it stays quiet. no notes 💅

👑 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 167 of file claswlq.f.

💙 dlaswlq — DLASWLQ 💙

subroutine dlaswlq (integer m, integer n, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

🎯 Purpose 🎯

okay same energy, different precision: DLASWLQ computes a blocked Tall-Skinny LQ factorization of a real M-by-N matrix A for M <= N, and it's giving:

   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 — INTEGER. The number of rows of the matrix A. M >= 0.
  • N — INTEGER. The number of columns of the matrix A. N >= M >= 0.
  • MB — INTEGER. The row block size to be used in the blocked QR. M >= MB >= 1
  • NB — INTEGER. The column block size to be used in the blocked QR. NB > 0.
  • A — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • T — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array T. LDT >= MB.
  • WORK — (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise. 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 — 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 168 of file dlaswlq.f.

💅 subroutine slaswlq — the real-valued one, bestie

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

SLASWLQ is literally that girl. Okay so here's the vibe:

✨ Purpose

okay so SLASWLQ computes a blocked Tall-Skinny LQ factorization of a real 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 (INTEGER) — the number of rows of the matrix A. M >= 0.
  • N (INTEGER) — the number of columns of the matrix A. N >= M >= 0.
  • MB (INTEGER) — the row block size to be used in the blocked QR. M >= MB >= 1
  • NB (INTEGER) — the column block size to be used in the blocked QR. NB > 0.
  • A (REAL 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 (INTEGER) — the leading dimension of the array A. LDA >= max(1,M).
  • T (REAL 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 (INTEGER) — the leading dimension of the array T. LDT >= MB.
  • WORK ((workspace) REAL array, dimension (MAX(1,LWORK))) — on exit, if INFO = 0, WORK(1) returns the minimal LWORK.
  • LWORK (INTEGER) — the dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise. 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 (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 167 of file slaswlq.f.

💜 subroutine zlaswlq — the complex one, no cap

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 is the complex sibling and she ate. Here's the tea:

✨ Purpose

okay so ZLASWLQ computes a blocked Tall-Skinny LQ factorization of a complexx M-by-N matrix A for M <= N (yes, "complexx" with two x's — that's literally what the source says, bestie, we don't make the rules):

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 (INTEGER) — the number of rows of the matrix A. M >= 0.
  • N (INTEGER) — the number of columns of the matrix A. N >= M >= 0.
  • MB (INTEGER) — the row block size to be used in the blocked QR. M >= MB >= 1
  • NB (INTEGER) — the column block size to be used in the blocked QR. NB > 0.
  • A (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 (INTEGER) — the leading dimension of the array A. LDA >= max(1,M).
  • T (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 (INTEGER) — the leading dimension of the array T. LDT >= MB.
  • WORK ((workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))) — on exit, if INFO = 0, WORK(1) returns the minimal LWORK.
  • LWORK (INTEGER) — the dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise. 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 (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 166 of file zlaswlq.f.

📜 Author (the page-level one, bestie)

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okay bestie, that's the whole page — LAPACK Version 3.12.0, page laswlq(3), Doxygen did its thing, and we understood it. mwah 💋✨

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