SRC/dlamtsqr.f(3)

💖 NAME 💖

SRC/dlamtsqr.f — hey bestie, this page is from the Library Functions Manual and it's literally the DLAMTSQR routine. Main character energy for blocked QR, no cap. 💅

🌸 SYNOPSIS 🌸

🦋 Functions/Subroutines 🦋

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

DLAMTSQR — periodt. ✨

✨ Function/Subroutine Documentation ✨

🧮 subroutine dlamtsqr (...) — DLAMTSQR

okay so here's the full signature, copied exactly because we do not play with details:

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 — that's the name, bestie. 🫶

🎀 Purpose 🎀

Lowkey the whole plot: 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). It's giving factorization but make it blocked. 💅

🌈 Parameters 🌈

• SIDE — CHARACTER*1. = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right. 💖
• TRANS — CHARACTER*1. = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.
• M — INTEGER. The number of rows of the matrix A. M >=0.
• N — INTEGER. The number of columns of the matrix C. N >= 0.
• K — INTEGER. The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0;
• MB — INTEGER. The block size to be used in the blocked QR. MB > N. (must be the same as DLATSQR)
• NB — INTEGER. The column block size to be used in the blocked QR. N >= NB >= 1.
• A — DOUBLE PRECISION 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 DLATSQR in the first k columns of its array argument A.
• LDA — INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N).
• T — 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 — INTEGER. The leading dimension of the array T. LDT >= NB.
• C — 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 — 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 — 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 — 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 🔥

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 dlamtsqr.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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xoxo, bestie 💖 — Doxygen made this for LAPACK from the source code, and the footer screams LAPACK Version 3.12.0 + SRC/dlamtsqr.f(3). No extra date listed, fr fr. Slay. 🌈👑