SRC/clalsd.f(3)

💅 SRC/clalsd.f — like, the Library Functions Manual page, but make it make sense

okay so bestie, welcome to the glow-up of SRC/clalsd.f(3) from the Library Functions Manual. this is one of those LAPACK girlies and she is lowkey doing the most — she solves least squares problems via SVD and honestly? she ate. let's get into it 🎀

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✨ SYNOPSIS ✨

Functions/Subroutines

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

CLALSD uses the singular value decomposition of A to solve the least squares problem. periodt. 💅

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🌸 Function/Subroutine Documentation 🌸

subroutine clalsd (character uplo, integer smlsiz, integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldb, * ) b, integer ldb, real rcond, integer rank, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CLALSD uses the singular value decomposition of A to solve the least squares problem. no cap.

Purpose:

so here's the vibe 💖 — CLALSD uses the SVD 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. iconic behavior tbh.

the singular values of A smaller than RCOND times the largest singular value are treated as zero in solving the least squares problem; in this case a minimum norm solution is returned. the actual singular values are returned in D in ascending order. slay. 🦋

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💜 Parameters 💜

• UPLO — is CHARACTER*1
  • = '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. she's the floor of the recursion, your honor 🫶
• 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. (so, like, at least one column, bestie, don't be weird about it 🙈)
• D — is REAL array, dimension (N)
  • on entry D contains the main diagonal of the bidiagonal matrix. on exit, if INFO = 0, D contains its singular values. main character transformation ✨
• E — is REAL array, dimension (N-1)
  • contains the super-diagonal entries of the bidiagonal matrix. on exit, E has been destroyed 💀 — yes, for real, she does not survive the function. do not come looking for her afterwards.
• B — is COMPLEX array, dimension (LDB,NRHS)
  • on input, B contains the right hand sides of the least squares problem. on output, B contains the solution X. full send. 🔥
• LDB — is INTEGER
  • the leading dimension of B in the calling subprogram. LDB must be at least max(1,N).
• RCOND — is REAL
  • 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.
  • example time, because this one deserves it 🍓: 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).
• RANK — is INTEGER
  • the number of singular values of A greater than RCOND times the largest singular value. she's counting the keepers 👑
• WORK — is COMPLEX array, dimension (N * NRHS).
• RWORK — is REAL 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 lot of math, no I will not be simplifying it, the formula is the formula 💀

• IWORK — is INTEGER array, dimension (3*N*NLVL + 11*N).
• INFO — is INTEGER
  • = 0: successful exit. we love a clean win 🌈
  • < 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). 🫠 at least she tells you exactly where it went wrong, which is honestly so considerate of her.

⚠️ bestie warning: E gets destroyed on exit and B gets overwritten with the solution. if you needed either of those original values, back them up before you call. iykyk 😭

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👑 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

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📍 Definition at line 178 of file clalsd.f.

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💫 Author (the meta one) 💫

generated automatically by Doxygen for LAPACK from the source code. not a human, just vibes and a parser. 🤖

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🧚 sign-off: this page is from LAPACK, Version 3.12.0. mwah, go solve something, bestie — you got this 💖✨🎀