lamtsqr(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 NAME 💅

okay so bestie, welcome to lamtsqr(3) from the Library Functions Manual ✨. the NAME line is literally lamtsqr - lamtsqr: multiply by Q from latsqr, and that's the whole assignment: multiply by Q from latsqr. no cap, it's giving “Q is the main character” energy. 💅

✨ SYNOPSIS ✨

🫶 Functions 🫶

  • subroutine clamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — CLAMTSQR
  • subroutine dlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — DLAMTSQR
  • subroutine slamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — SLAMTSQR
  • subroutine zlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info) — ZLAMTSQR

🦋 Detailed Description 🦋

this is the “Detailed Description” stop on the tour, and the original page keeps it cute and brief here. no extra parameters are hiding in the wings, bestie — the function lore is right below. 🫶

👑 Function Documentation 👑

💅 subroutine clamtsqr

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

CLAMTSQR

💖 Purpose

okay so CLAMTSQR 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 tall skinny QR factorization (CLATSQR). periodt. 🎀

🍓 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. = 'N': no transpose, apply Q; = 'C': Conjugate Transpose, apply Q**H. ✨
  • 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 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 block size to be used in the blocked QR. MB > N. (must be the same as CLATSQR).
  • NB — NB is INTEGER. The column block size to be used in the blocked QR. N >= NB >= 1.
  • A — A is COMPLEX array, dimension (LDA,K). The i-th column must contain the vector which defines the blocked elementary reflector H(i), for i = 1,2,...,k, as returned by CLATSQR in the first k columns of its array argument A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N).
  • T — T is COMPLEX array, dimension ( N * Number of blocks(CEIL(M-K/MB-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 >= NB.
  • 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,N*NB). If SIDE = 'R', LWORK >= max(1,MB*NB). 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. 😭 workspace query is doing its little thing, okay?
  • 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

okay so Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:

Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, 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 GEQRT. ✨

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). 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 199 of file clamtsqr.f.

💅 subroutine dlamtsqr

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

DLAMTSQR

💖 Purpose

okay so DLAMTSQR 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 tall skinny QR factorization (DLATSQR). slay. 🔥

🍓 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 A. 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 block size to be used in the blocked QR. MB > N. (must be the same as DLATSQR).
  • NB — NB is INTEGER. The column block size to be used in the blocked QR. N >= NB >= 1.
  • A — A is DOUBLE PRECISION array, dimension (LDA,K). The i-th column must contain the vector which defines the blocked elementary reflector H(i), for i = 1,2,...,k, as returned by DLATSQR in the first k columns of its array argument A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N).
  • T — T is DOUBLE PRECISION array, dimension ( N * Number of blocks(CEIL(M-K/MB-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 >= NB.
  • 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,N*NB). If SIDE = 'R', LWORK >= max(1,MB*NB). 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. — that's the whole author squad, slay 🫶

💫 Further Details

okay bestie, THIS is the part that actually explains the whole TSQR vibe — Tall-Skinny QR does QR by stringing together a bunch of orthogonal transformations, and it doesn't store Q as one big chonky matrix. Nope. It represents Q as a product of other orthogonal matrices, like:

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

and each Q(i) zeroes out the subdiagonal entries of a block of MB rows of A, one at a time:

Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

so Q(1) is the main character of the first block ✨ — it's computed by GEQRT, which represents Q(1) using Householder vectors stored under the diagonal of rows 1:MB of A, plus upper triangular block reflectors stored in the array T(1:LDT,1:N). Want the full tea? Check Further Details in GEQRT 💅

then Q(i) for i>1 is computed by TPQRT — that one represents Q(i) with Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and with upper triangular block reflectors stored in the array T(1:LDT,(i-1)*N+1:i*N). Also fyi the last Q(k) may use fewer rows, because of course it can 🙈 More deets live in Further Details in TPQRT.

and if you want the whole algorithm spelled out? 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 199 of file dlamtsqr.f.

🦋 subroutine slamtsqr (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)

SLAMTSQR — periodt. 💖

✨ Purpose

okay so SLAMTSQR 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 tall skinny QR factorization (SLATSQR) — iconic behavior, no cap 🔥

🌸 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 A. 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; (yes bestie, that's a semicolon, it's just like that 💀)
  • MB — MB is INTEGER. The block size to be used in the blocked QR. MB > N. (must be the same as SLATSQR) ⚠️ like, actually the same, don't play with it.
  • NB — NB is INTEGER. The column block size to be used in the blocked QR. N >= NB >= 1.
  • A — A is REAL array, dimension (LDA,K). The i-th column must contain the vector which defines the blockedelementary reflector H(i), for i = 1,2,...,k, as returned by SLATSQR in the first k columns of its array argument A. 🎀
  • LDA — LDA is INTEGER. The leading dimension of the array A.
    • If SIDE = 'L', LDA >= max(1,M);
    • if SIDE = 'R', LDA >= max(1,N).
  • T — T is REAL array, dimension ( N * Number of blocks(CEIL(M-K/MB-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 >= NB.
  • 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. bye original C, you had a good run 😭
  • 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,N*NB).
    • If SIDE = 'R', LWORK >= max(1,MB*NB).
    • 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. 🫶 classic query-mode behavior, we stan a non-scolding error path.
  • 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

same TSQR situation as above, bestie — Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:

Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, 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 GEQRT. 💜

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). 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 199 of file slamtsqr.f.

🌈 subroutine zlamtsqr (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)

ZLAMTSQR — the complex one, and she's gorgeous 💅✨

✨ Purpose

okay so ZLAMTSQR 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 tall skinny QR factorization (ZLATSQR). notice the transpose option is 'C' here and it's the conjugate transpose — that's the whole difference, and it eats 🔥

🌸 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
    • = 'N': No transpose, apply Q;
    • = 'C': Conjugate Transpose, apply Q**H.
  • 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 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 block size to be used in the blocked QR. MB > N. (must be the same as ZLATSQR) ⚠️ same rule as always, bestie.
  • NB — NB is INTEGER. The column block size to be used in the blocked QR. N >= NB >= 1.
  • A — A is COMPLEX*16 array, dimension (LDA,K). The i-th column must contain the vector which defines the blockedelementary reflector H(i), for i = 1,2,...,k, as returned by ZLATSQR in the first k columns of its array argument A. 🦋
  • LDA — LDA is INTEGER. The leading dimension of the array A.
    • If SIDE = 'L', LDA >= max(1,M);
    • if SIDE = 'R', LDA >= max(1,N).
  • T — T is COMPLEX*16 array, dimension ( N * Number of blocks(CEIL(M-K/MB-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 >= NB.
  • 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 got rewritten, it's giving main character energy 💀
  • LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M).
  • WORK
  • (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, then WORK(1) returns the minimal LWORK. ✨

🌸 LWORK 🌸

Okay so LWORK is INTEGER — it’s the dimension of the array WORK, and the size rules are very much giving exact math, no cap:

  • If MIN(M,N,K) = 0, then LWORK >= 1.
  • If SIDE = 'L', then LWORK >= max(1,N*NB).
  • If SIDE = 'R', then LWORK >= max(1,MB*NB).

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, and bestie, it’s literally just the vibe check at the end:

  • = 0: successful exit 🌈
  • < 0: if INFO = -i, the i-th argument had an illegal value 🙈

💜 Author 💜

The legends behind this one are:

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

🔥 Further Details: 🔥

Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:

Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, 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 GEQRT. 🫶

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). 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 199 of file zlamtsqr.f.

🤖 Author 🤖

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