/home/abui.../clatsqr.f(3)

🦋 CLATSQR — the blocked Tall-Skinny QR glow-up (Fortran edition)

okay so bestie, this page is the manual for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/clatsqr.f, a Library Functions Manual situation, and it is literally about a Fortran subroutine that does a blocked TSQR on a complex matrix. main character energy fr fr ✨

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💅 SYNOPSIS

Functions/Subroutines

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

CLATSQR — that's the whole name, no notes, iconic behavior only 👑

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🎀 Function/Subroutine Documentation

subroutine CLATSQR (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)

CLATSQR 🫶

🌸 Purpose:

!>
!> CLATSQR computes a blocked Tall-Skinny QR factorization of
!> a complex 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.
!>
!>

so basically: you hand it a tall skinny complex matrix A (M >= N), and it factorizes it as A = Q * (R ; 0). Q is M-by-M orthogonal but it does not get stored as a big obvious matrix — it lives implicitly below the diagonal of A plus inside T. R is the upper-triangular N-by-N cutie, sitting on and above the diagonal of A. and the zero block, (M-N)-by-N? not stored at all. we simply do not perceive it 💅

⭐ Parameters

• M — M is INTEGER. the number of rows of A. constraint: M >= 0. no negative row era allowed 🙅‍♀️
• N — N is INTEGER. the number of columns of A. constraint: M >= N >= 0. so N can't outgrow M, that's the whole "tall-skinny" thing bestie 🌱
• MB — MB is INTEGER. the row block size used in the blocked QR. constraint: MB > N. strictly greater, not equal, watch out 😭
• NB — NB is INTEGER. the column block size used in the blocked QR. constraint: N >= NB >= 1.
• A — A is COMPLEX array, dimension (LDA,N). on entry, it's 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 (peep Further Details below for the tea 🍵).
• LDA — LDA is INTEGER. the leading dimension of the array A. constraint: LDA >= max(1,M).
• T — T is COMPLEX array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)). the blocked upper triangular block reflectors are stored in compact form as a sequence of upper triangular blocks. more deets down in Further Details 💖
• LDT — LDT is INTEGER. the leading dimension of the array T. constraint: LDT >= NB.
• WORK — (workspace) COMPLEX array, dimension (MAX(1,LWORK)). the scratch space, doing the behind-the-scenes work so A doesn't have to 🫡
• LWORK — LWORK is INTEGER. the dimension of the array WORK. constraint: LWORK >= NB*N. and okay important: if LWORK = -1, then a workspace query is assumed — the routine only calculates the optimal 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. so -1 is the "just tell me how much room I need" mode, no cap ✨
• 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.

(shoutout to the whole committee, they ate 🫶)

🧚 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
!>

so the vibe is: TSQR does QR by stacking a bunch of orthogonal transformations, Q = Q(1) * Q(2) * . . . * Q(k), and each Q(i) zeroes out subdiagonal entries of a block of MB rows of A. Q(1) handles rows 1:MB, Q(2) handles the bottom MB-N rows of rows [1:N,MB+1:2*MB-N], Q(3) handles the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N], and so on forever-ish ✨

Q(1) comes from GEQRT, which stores Householder vectors under the diagonal of rows 1:MB of A and upper triangular block reflectors in T(1:LDT,1:N). Q(i) for i>1 comes from TPQRT, with Householder vectors in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A and block reflectors in T(1:LDT,(i-1)*N+1:i*N). the last Q(k) might use fewer rows, she's flexible like that 💁‍♀️

nerd corner 🎀: for the full algorithm tea, see the description of Sequential TSQR in Section 2.2 of [1], “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” by J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012. iykyk 📚

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🌈 Definition at line 167 of file clatsqr.f.

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💜 sign-off

this whole thing was generated automatically by Doxygen for LAPACK from the source code, brought to you by LAPACK, Version 3.12.0, and the file herself: /home/abui.../clatsqr.f(3). slay, go factorize something tall and skinny 💅✨🦋