SRC/slatsqr.f(3)

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version 3.12.1 · 5 versions
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🎀 NAME 🎀

SRC/slatsqr.f — bestie, this is the Library Functions Manual page for the SLATSQR subroutine, aka the blocked Tall-Skinny QR factorization icon. 💅✨

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine slatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
    SLATSQR

🌟 Function/Subroutine Documentation 🌟

Okay so here’s the full signature, no cap:

subroutine slatsqr (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)
    SLATSQR

🦋 Purpose 🦋

SLATSQR computes a blocked Tall-Skinny QR factorization of a real M-by-N matrix A for M >= N:

   A = Q * ( R ),
           ( 0 )

where:

  • Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T.
  • R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A.
  • 0 is a (M-N)-by-N zero matrix, and is not stored.

🌸 Parameters 🌸

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 A. M >= N >= 0.

MB

MB is INTEGER. The row block size to be used in the blocked QR. MB > N.

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,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details).

LDA

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

T

T is REAL array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)). The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below.

LDT

LDT is INTEGER. The leading dimension of the array T. LDT >= NB.

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. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, 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

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 💫

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 170 of file slatsqr.f.

🎀 Author 🎀

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💖 that’s the whole page, bestie 💖 — sign-off from LAPACK Version 3.12.0, source page SRC/slatsqr.f(3). The footer doesn’t list a date, so we’re not inventing one, periodt. 💅🎀

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