SRC/zlalsd.f(3)

💅 SRC/zlalsd.f(3) — Library Functions Manual ✨

okay bestie, gather round, because this one is a baddie 🎀 — it's a Fortran subroutine straight out of LAPACK, living its best life over in SRC/zlalsd.f, and honestly? It's giving main character energy. Let's get into it 💖

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🌸 NAME 🌸

SRC/zlalsd.f — no cap, that's the whole name, she's lowkey mysterious like that 🦋

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✨ SYNOPSIS ✨

Functions/Subroutines 🫶

subroutine zlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank,
    work, rwork, iwork, info)

ZLALSD uses the singular value decomposition of A to solve the least squares problem. That's her whole personality and she ate 💅

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👑 Function/Subroutine Documentation 👑

subroutine zlalsd (character uplo, integer smlsiz, integer n, integer nrhs,
    double precision, dimension( * ) d, double precision, dimension( * ) e,
    complex*16, dimension( ldb, * ) b, integer ldb, double precision rcond,
    integer rank, complex*16, dimension( * ) work, double precision,
    dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZLALSD uses the singular value decomposition of A to solve the least squares problem. (Yes, that's the same sentence, she's consistent, we stan a consistent icon ⭐)

🎯 Purpose

ok so here's the vibe: ZLALSD uses the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean norm of each column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-by-NRHS. The solution X overwrites B. 🔥

And the plot twist: the singular values of A smaller than RCOND times the largest singular value are treated as zero when solving the least squares problem — and in that case a minimum norm solution is returned. The actual singular values are returned in D in ascending order. periodt 💅

🌈 Parameters

• UPLO — is CHARACTER*1. The glow-up selector, fr:
  • = 'U': D and E define an upper bidiagonal matrix.
  • = 'L': D and E define a lower bidiagonal matrix.
• SMLSIZ — is INTEGER. The maximum size of the subproblems at the bottom of the computation tree. (the little subproblems at the bottom of the tree are the tiny babies and this is their size limit 🧚)
• N — is INTEGER. The dimension of the bidiagonal matrix. N >= 0.
• NRHS — is INTEGER. The number of columns of B. NRHS must be at least 1. (like, minimum one column or why are we even here 😭)
• D — is DOUBLE PRECISION array, dimension (N). On entry D contains the main diagonal of the bidiagonal matrix. On exit, if INFO = 0, D contains its singular values. She starts as a diagonal and leaves as a star ✨
• E — is DOUBLE PRECISION array, dimension (N-1). Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 💀 rip bestie, she did not survive the function call.
• B — is COMPLEX*16 array, dimension (LDB,NRHS). On input, B contains the right hand sides of the least squares problem. On output, B contains the solution X. glow-up of the century, honestly 👑
• LDB — is INTEGER. The leading dimension of B in the calling subprogram. LDB must be at least max(1,N).
• RCOND — is DOUBLE PRECISION. The singular values of A less than or equal to RCOND times the largest singular value are treated as zero in solving the least squares problem. If RCOND is negative, machine precision is used instead. For example, if diag(S)*X=B were the least squares problem, where diag(S) is a diagonal matrix of singular values, the solution would be X(i) = B(i) / S(i) if S(i) is greater than RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to RCOND*max(S). (she's the threshold queen, the cutoff icon, the "you're too small, we're not doing that" gatekeeper 🚪)
• RANK — is INTEGER. The number of singular values of A greater than RCOND times the largest singular value. Basically the headcount of singular values that made the cut 💅
• WORK — is COMPLEX*16 array, dimension (N * NRHS).
• RWORK — is DOUBLE PRECISION array, dimension at least

(9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),

where

NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

yes, that's a real formula, yes it's a lot, no we're not going to pretend it's short 💀

• IWORK — is INTEGER array, dimension at least (3*N*NLVL + 11*N).
• INFO — is INTEGER. The narrator of the story, the vibe check at the end 🔔
  • = 0: successful exit. slay 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. 🙈
  • > 0: The algorithm failed to compute a singular value while working on the submatrix lying in rows and columns INFO/(N+1) through MOD(INFO,N+1). 😭 that submatrix is NOT having a good day.

🫶 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 179 of file zlalsd.f.

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💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. (not me writing a fanfic, the robots did this one 🤖)

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ok that's the whole page, bestie — every flag, every array, every formula, all of it, preserved with love and a lot of emoji 🍓

watch out for E on exit, genuinely. she's destroyed. gone. do not pass go, do not read that array 💀

LAPACK · Version 3.12.0 · SRC/zlalsd.f(3) — mwah, that's a wrap, ily, go call your BLAS 🦋💖🎀