SRC/dlals0.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💅 NAME

SRC/dlals0.f — bestie, this is the Fortran source file for the dlals0 subroutine, and it's literally the back-multiplying baddie of the divide-and-conquer SVD world 👑

✨ SYNOPSIS

🌸 Functions/Subroutines

subroutine dlals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm,
    givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c,
    s, work, info)

So what does it do? DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd. That's the whole vibe, no cap 🫶

🦋 Function/Subroutine Documentation

Full signature, character for character, because we do NOT play with the source 💖

subroutine dlals0 (integer icompq, integer nl, integer nr, integer sqre,
    integer nrhs, double precision, dimension( ldb, * ) b, integer ldb,
    double precision, dimension( ldbx, * ) bx, integer ldbx, integer,
    dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * )
    givcol, integer ldgcol, double precision, dimension( ldgnum, * ) givnum,
    integer ldgnum, double precision, dimension( ldgnum, * ) poles, double
    precision, dimension( * ) difl, double precision, dimension( ldgnum, * )
    difr, double precision, dimension( * ) z, integer k, double precision c,
    double precision s, double precision, dimension( * ) work, integer info)

DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd. (Yes I said it twice, it's that iconic ✨)

🎀 Purpose

Okay so, the actual purpose, in all its glory:

DLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.
For the left singular vector matrix, three types of orthogonal matrices are involved:
(1L) Givens rotations: the number of such rotations is GIVPTR; the pairs of columns/rows they were applied to are stored in GIVCOL; and the C- and S-values of these rotations are stored in GIVNUM.
(2L) Permutation. The (NL+1)-st row of B is to be moved to the first row, and for J=2:N, PERM(J)-th row of B is to be moved to the J-th row.
(3L) The left singular vector matrix of the remaining matrix.
For the right singular vector matrix, four types of orthogonal matrices are involved:
(1R) The right singular vector matrix of the remaining matrix.
(2R) If SQRE = 1, one extra Givens rotation to generate the right null space.
(3R) The inverse transformation of (2L).
(4R) The inverse transformation of (1L).

✨ Parameters

Get comfy, there's a lot of them, but every single one matters 💅

  • ICOMPQ — INTEGER. Specifies whether singular vectors are to be computed in factored form:
    • = 0: Left singular vector matrix.
    • = 1: Right singular vector matrix.
  • NL — INTEGER. The row dimension of the upper block. NL >= 1.
  • NR — INTEGER. The row dimension of the lower block. NR >= 1.
  • SQRE — INTEGER.
    • = 0: the lower block is an NR-by-NR square matrix.
    • = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    • The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.
  • NRHS — INTEGER. The number of columns of B and BX. NRHS must be at least 1.
  • B — DOUBLE PRECISION array, dimension ( LDB, NRHS ). On input, B contains the right hand sides of the least squares problem in rows 1 through M. On output, B contains the solution X in rows 1 through N. Glow-up moment ✨
  • LDB — INTEGER. The leading dimension of B. LDB must be at least max(1,MAX( M, N )).
  • BX — DOUBLE PRECISION array, dimension ( LDBX, NRHS ).
  • LDBX — INTEGER. The leading dimension of BX.
  • PERM — INTEGER array, dimension ( N ). The permutations (from deflation and sorting) applied to the two blocks.
  • GIVPTR — INTEGER. The number of Givens rotations which took place in this subproblem.
  • GIVCOL — INTEGER array, dimension ( LDGCOL, 2 ). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.
  • LDGCOL — INTEGER. The leading dimension of GIVCOL, must be at least N.
  • GIVNUM — DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). Each number indicates the C or S value used in the corresponding Givens rotation.
  • LDGNUM — INTEGER. The leading dimension of arrays DIFR, POLES and GIVNUM, must be at least K.
  • POLES — DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). On entry, POLES(1:K, 1) contains the new singular values obtained from solving the secular equation, and POLES(1:K, 2) is an array containing the poles in the secular equation.
  • DIFL — DOUBLE PRECISION array, dimension ( K ). On entry, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.
  • DIFR — DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). On entry, DIFR(I, 1) contains the distances between I-th updated (undeflated) singular value and the I+1-th (undeflated) old singular value. And DIFR(I, 2) is the normalizing factor for the I-th right singular vector.
  • Z — DOUBLE PRECISION array, dimension ( K ). Contain the components of the deflation-adjusted updating row vector.
  • K — INTEGER. Contains the dimension of the non-deflated matrix. This is the order of the related secular equation. 1 <= K <=N. (yes, the original wrote <=N with no space, we keep it exactly like that, bestie 🎀)
  • C — DOUBLE PRECISION. C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1. Literally garbage, not even joking 💀
  • S — DOUBLE PRECISION. S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.
  • WORK — DOUBLE PRECISION array, dimension ( K ).
  • INFO — INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
🫶 Quick vibe check: C and S are straight-up garbage when SQRE = 0. Don't be reading them, don't be trusting them, they're just there. Periodt.

👑 Author

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

💖 Contributors:

  • Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
  • Osni Marques, LBNL/NERSC, USA

⭐ Definition at line 265 of file dlals0.f.

🌈 Author

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✨💖 and that's the whole page, bestie 💖✨ — shoutout to LAPACK Version 3.12.0, generated with love by Doxygen from the source code, SRC/dlals0.f(3), slay 🦋🎀💅

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