ZLALSD(3)

💖 ZLALSD(3) — the LAPACK bidiagonal least-squares solver 💖

okay so bestie, this page is from the Library Functions Manual, and it is literally about ZLALSD(3). It’s giving “linear algebra main character” energy, no cap. 🎀

📛 NAME 📛

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlalsd.f — yep, that’s the source file path this page is about, and the subroutine is ZLALSD. 💅

🌀 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. ✨

🧚 Function/Subroutine Documentation 🧚

✨ subroutine ZLALSD ✨

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 bestie, you’re gonna see that sentence again — it’s the whole point. 🫶

🎀 Purpose 🎀

Okay so the actual glow-up: 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. 💥

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. So yes, D is absolutely serving sorted singular values on exit. 🌸

🌸 Parameters 🌸

• UPLO — UPLO is CHARACTER*1.
  • = 'U': D and E define an upper bidiagonal matrix.
  • = 'L': D and E define a lower bidiagonal matrix.
• SMLSIZ — SMLSIZ is INTEGER. The maximum size of the subproblems at the bottom of the computation tree. 🌳
• N — N is INTEGER. The dimension of the bidiagonal matrix. N >= 0.
• NRHS — NRHS is INTEGER. The number of columns of B. NRHS must be at least 1.
• D — 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.
• E — E is DOUBLE PRECISION array, dimension (N-1). Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 💀 not literally bestie, it’s just overwritten.
• B — 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.
• LDB — LDB is INTEGER. The leading dimension of B in the calling subprogram. LDB must be at least max(1,N).
• RCOND — 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).
• RANK — RANK is INTEGER. The number of singular values of A greater than RCOND times the largest singular value.
• WORK — WORK is COMPLEX*16 array, dimension (N * NRHS).
• RWORK — 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 )

• IWORK — IWORK is INTEGER array, dimension at least (3*N*NLVL + 11*N).
• INFO — INFO is INTEGER.
  • = 0: successful exit.
  • < 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).

🫶 quick vibe check: on exit, E has been destroyed, and INFO > 0 means the algorithm failed to compute a singular value while working on that submatrix. Not a drill. 💀

👑 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. 📌

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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and that’s the whole page, bestie. big cute sign-off: shoutout to LAPACK Version 3.12.0, and the footer path is /home/abui...RC/zlalsd.f(3). love that for us. 💖✨🎀