lamswlq(3)

📦

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💅 lamswlq(3) — Library Functions Manual 💅

✨ NAME ✨

lamswlq — lamswlq: multiply by Q from laswlq. 💖

🎀 SYNOPSIS 🎀

💖 Functions 💖

subroutine clamswlq (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — CLAMSWLQ
subroutine dlamswlq (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — DLAMSWLQ
subroutine slamswlq (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — SLAMSWLQ
subroutine zlamswlq (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — ZLAMSWLQ

🌈 Detailed Description 🌈

Bestie, the source just drops this heading and then immediately moves into the function docs. No extra lore here. 💅

👑 Function Documentation 👑

⭐ subroutine clamswlq ⭐

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

AKA CLAMSWLQ.

💖 Purpose 💖

Okay so CLAMSWLQ overwrites the general complex M-by-N matrix C with:

                   SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'T':      Q**H * C       C * Q**H

where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization (CLASWLQ). 🦋

🎀 Parameters 🎀

SIDE — SIDE is CHARACTER*1. = 'L': apply Q or QH from the Left; = 'R': apply Q or QH from the Right.
TRANS — TRANS is CHARACTER*1. = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H.
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 COMPLEX 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 CLASWLQ 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 COMPLEX 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 COMPLEX array, dimension (LDC,N). On entry, the M-by-N matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M).
WORK — (workspace) COMPLEX 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 unitary transformations, representing Q as a product of other unitary 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 clamswlq.f.

⭐ subroutine dlamswlq ⭐

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

AKA DLAMSWLQ.

💖 Purpose 💖

DLAMSWLQ 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 (DLASWLQ). ✨

🎀 Parameters 🎀

SIDE — SIDE is CHARACTER*1. = 'L': apply Q or QT from the Left; = 'R': apply Q or QT 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 DOUBLE PRECISION 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 DLASWLQ 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) DOUBLE PRECISION 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

…and the rest of the author squad, carrying on:

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

🌸 Further Details 🌸

Okay so bestie, this is the part where it all clicks 💖 Short-Wide LQ (SWLQ) does LQ by a whole sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices, like:

Q = Q(1) * Q(2) * . . . * Q(k)

where each Q(i) zeros out upper diagonal entries of a block of NB rows of A. Watch:

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
. . .

So Q(1) gets computed by GELQT — it represents Q(1) with Householder vectors stashed under the diagonal of rows 1:MB of A, plus upper triangular block reflectors living in the array T(1:LDT,1:N). For more info, peep Further Details over in GELQT 👀

Then Q(i) for i>1 is computed by TPLQT — it represents Q(i) with Householder vectors stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and with upper triangular block reflectors stored in the array T(1:LDT,(i-1)*M+1:i*M). Heads up: the last Q(k) may use fewer rows. For more info, see Further Details in TPLQT ✨

And for the full tea on the overall algorithm, go read 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 dlamswlq.f.

👑 `subroutine slamswlq` — the real-precision one, she's here 👑

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) 💅

🌈 Parameters 🌈

SIDE — SIDE is CHARACTER*1, and it's literally just picking a side, bestie:
- = 'L': apply Q or Q**T from the Left;
- = 'R': apply Q or Q**T from the Right.
TRANS — TRANS is CHARACTER*1, the vibe check on transposition:
- = 'N': No transpose, apply Q;
- = 'T': Transpose, apply Q**T.
M — M is INTEGER, the number of rows of the matrix C. Constraint: M >=0. Zero is allowed, she's chill like that.
N — N is INTEGER, the number of columns of the matrix C. Constraint: N >= 0. Same energy.
K — K is INTEGER, the number of elementary reflectors whose product defines the matrix Q. Constraint: M >= K >= 0; — yes, there are two > signs in the original, it's giving dramatic emphasis 💀
MB — MB is INTEGER, the row block size to be used in the blocked LQ. Constraint: M >= MB >= 1 — so no zero blocks, she needs at least one.
NB — NB is INTEGER, the column block size to be used in the blocked LQ. Constraint: NB > M. — strictly bigger than M, no cap, not negotiable 🚨
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. Iconic handoff tbh 🫶

LDA — LDA is INTEGER, the leading dimension of the array A. Constraint: 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 — and yes, "see below" is doing a lot of heavy lifting there 💅

LDT — LDT is INTEGER, the leading dimension of the array T. Constraint: 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. So yeah, your baby gets replaced, that's the whole bit ✨
LDC — LDC is INTEGER, the leading dimension of the array C. Constraint: LDC >= max(1,M).
WORK — (workspace) REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the minimal LWORK. She's a reporter too 💖
LWORK — LWORK is INTEGER, the dimension of the array WORK. Now the rules:
- 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. Sneaky but legal 🕵️‍♀️
INFO — INFO is INTEGER, the receipts:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value — so go count your arguments, bestie 😭

👑 Author 👑

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

🌸 Further Details 🌸

(Yeah it's the same gorgeous explanation again — copy-paste queen behavior, but like, correct.)

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 — she's built different. 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.

💜 `subroutine zlamswlq` — now in complex*16, she's fancy 💜

subroutine zlamswlq (character side, character trans, integer m, integer n,
    integer k, integer mb, integer nb, complex*16, dimension( lda, * ) a,
    integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16,
    dimension( ldc, * ) c, integer ldc, complex*16, dimension( * ) work,
    integer lwork, integer info)
    ZLAMSWLQ

🎀 Purpose 🎀

ZLAMSWLQ overwrites the general complex M-by-N matrix C with

                 SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'C':      Q**H * C       C * Q**H

where Q is a complex unitary matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization (ZLASWLQ) 👑

🌈 Parameters 🌈

SIDE — SIDE is CHARACTER*1:
- = 'L': apply Q or Q**H from the Left;
- = 'R': apply Q or Q**H from the Right.
TRANS — TRANS is CHARACTER*1, and note she said 'C' not 'T', that's the complex-girl signature move:
- = 'N': No transpose, apply Q;
- = 'C': Conjugate Transpose, apply Q**H.
M — M is INTEGER, the number of rows of the matrix C. Constraint: M >=0.
N — N is INTEGER, the number of columns of the matrix C. Constraint: N >= 0.
K — K is INTEGER, the number of elementary reflectors whose product defines the matrix Q. Constraint: M >= K >= 0; — again with the double >=, she's consistent 💀
MB — MB is INTEGER, the row block size to be used in the blocked LQ. Constraint: M >= MB >= 1
NB — NB is INTEGER, the column block size to be used in the blocked LQ. Constraint: NB > M. — still strictly greater, no exceptions 🚨
A — A is COMPLEX*16 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 ZLASWLQ in the first k rows of its array argument A. Same deal, fancier type ✨

LDA — LDA is INTEGER, the leading dimension of the array A. Constraint: LDA >= MAX(1,K). (yes, uppercase MAX here, she switched it up, we love range 🫶)
T — T is COMPLEX*16 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 — that unclosed paren is canon, don't @ me 💅

LDT — LDT is INTEGER, the leading dimension of the array T. Constraint: LDT >= MB.
C — C is COMPLEX*16 array, dimension (LDC,N). On entry, the M-by-N matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. She eats and leaves no crumbs 😭
LDC — LDC is INTEGER, the leading dimension of the array C. Constraint: LDC >= max(1,M).
WORK — 🫧

Back to WORK (workspace), bestie: it’s a COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) returns the minimal LWORK — iconic workspace behavior, no cap. ✨

LWORK — LWORK is INTEGER, and it’s the dimension of the array WORK.
- If MIN(M,N,K) = 0, then LWORK >= 1.
- If SIDE = 'L', then LWORK >= max(1,NB*MB).
- If SIDE = 'R', then 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 unitary transformations, representing Q as a product of other unitary 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 zlamswlq.f.

Auto-generated author credit: Generated automatically by Doxygen for LAPACK from the source code.

💖 That’s the whole tea. Source/version: LAPACK Version 3.12.0; page: lamswlq(3). The footer doesn’t print an explicit date, so no invented date — just accurate vibes. 🫶

📦 where this page comes from

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lamswlq(3)

💅 lamswlq(3) — Library Functions Manual 💅

✨ NAME ✨

🎀 SYNOPSIS 🎀

💖 Functions 💖

🌈 Detailed Description 🌈

👑 Function Documentation 👑

⭐ subroutine clamswlq ⭐

💖 Purpose 💖

🎀 Parameters 🎀

🌸 Author 🌸

🧚 Further Details 🧚

⭐ subroutine dlamswlq ⭐

💖 Purpose 💖

🎀 Parameters 🎀

🌸 Author 🌸

🌸 Further Details 🌸

👑 subroutine slamswlq — the real-precision one, she's here 👑

🎀 Purpose 🎀

🌈 Parameters 🌈

👑 Author 👑

🌸 Further Details 🌸

💜 subroutine zlamswlq — now in complex*16, she's fancy 💜

🎀 Purpose 🎀

🌈 Parameters 🌈

👑 Author

🌈 Further Details

👑 `subroutine slamswlq` — the real-precision one, she's here 👑

💜 `subroutine zlamswlq` — now in complex*16, she's fancy 💜