ZBDSQR(3)

🦋 hey bestie, welcome to ZBDSQR(3)

okay so first things first — this is a page straight out of the LAPACK manual, Version 3.12.0, and it's documenting ZBDSQR from zbdsqr.f(3). yes it's a Fortran thing. yes it's real. no you can't escape it. let's get into it 💖

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zbdsqr.f

like, that's genuinely the whole "name" — it's just the file path with the build directory showing its whole entire government name 😭 iconic behavior honestly.

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZBDSQR (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
    ldc, rwork, info)

literally her whole vibe in one line: ZBDSQR. that's the description. that's it. slay.

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👑 Function/Subroutine Documentation 👑

subroutine ZBDSQR (character uplo, integer n, integer ncvt, integer nru,
    integer ncc, double precision, dimension( * ) d, double precision,
    dimension( * ) e, complex*16, dimension( ldvt, * ) vt, integer ldvt,
    complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension(
    ldc, * ) c, integer ldc, double precision, dimension( * ) rwork, integer
    info)

💜 Purpose:

so here's the tea, bestie 👇

ZBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. the SVD of B has the form

B = Q * S * P**H

where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. if left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**H*VT instead of P*H*, for given complex input matrices U and VT. when U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form — A = U*B*VT, as computed by ZGEBRD — then

A = (U*Q) * S * (P**H*VT)

is the SVD of A. optionally, the subroutine may also compute Q**H*C for a given complex input matrix C. 🫶

also, for a detailed description of the algorithm, it points you at:

• a paper by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) — and
• a paper by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992

🙈 tiny note: the source doc literally has the titles ripped out — it just says "See by J. Demmel and W. Kahan" with a gap where the title should be. the vibes are giving "citation missing, left as an exercise for the reader." i'm not making that up, that's what's on the page.

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🎀 Parameters 🎀

• UPLO — is CHARACTER*1. the layout check 🌈
  • = 'U': B is upper bidiagonal;
  • = 'L': B is lower bidiagonal.
• N — is INTEGER. the order of the matrix B. N >= 0.
• NCVT — is INTEGER. the number of columns of the matrix VT. NCVT >= 0.
• NRU — is INTEGER. the number of rows of the matrix U. NRU >= 0.
• NCC — is INTEGER. the number of columns of the matrix C. NCC >= 0.
• D — is DOUBLE PRECISION array, dimension (N).
  • on entry: the n diagonal elements of the bidiagonal matrix B.
  • on exit, if INFO=0: the singular values of B in decreasing order. ⭐
• E — is DOUBLE PRECISION array, dimension (N-1).
  • on entry: the N-1 offdiagonal elements of the bidiagonal matrix B.
  • on exit, if INFO = 0: E is destroyed 💀; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.
• VT — is COMPLEX*16 array, dimension (LDVT, NCVT).
  • on entry: an N-by-NCVT matrix VT.
  • on exit: VT is overwritten by P**H * VT.
  • not referenced if NCVT = 0.
• LDVT — is INTEGER. the leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
• U — is COMPLEX*16 array, dimension (LDU, N).
  • on entry: an NRU-by-N matrix U.
  • on exit: U is overwritten by U * Q.
  • not referenced if NRU = 0.
• LDU — is INTEGER. the leading dimension of the array U. LDU >= max(1,NRU).
• C — is COMPLEX*16 array, dimension (LDC, NCC).
  • on entry: an N-by-NCC matrix C.
  • on exit: C is overwritten by Q**H * C.
  • not referenced if NCC = 0.
• LDC — is INTEGER. the leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
• RWORK — is DOUBLE PRECISION array, dimension (4*N).
• INFO — is INTEGER. the receipts 📋
  • = 0: successful exit ✨
  • < 0: if INFO = -i, the i-th argument had an illegal value 😬
  • > 0: the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero. 🫠

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🧚 Internal Parameters 🧚

• TOLMUL — DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))). TOLMUL controls the convergence criterion of the QR loop.
  • if it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values.
  • if it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value).
  • abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results).
  • default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller).
• MAXITR — INTEGER, default = 6. MAXITR controls the maximum number of passes of the algorithm through its inner loop. the algorithm stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.

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💀 Note: 💀

Bug report from Cezary Dendek. On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). we instead use MAXITDIVN = MAXITR*N.

ngl that overflow was lowkey lurking for anyone with a matrix of 18,919 or bigger — not me finding out about it via a bug report from 2023 😭 thank you Cezary, truly the main character.

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🌈 Author 🌈

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

Definition at line 231 of file zbdsqr.f. 📍

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🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie!! ✨ auto-generated by Doxygen, no humans were harmed. you ate that reading it, fr fr. 🫶💅

LAPACK — Version 3.12.0 · footer file: /home/abui...RC/zbdsqr.f(3) · k bye ilysm, go compute some singular values 💖🎀🦋