SRC/zbdsqr.f(3)

NAME 💅

okay so this is SRC/zbdsqr.f, bestie — straight out of the Library Functions Manual, and it's giving iconic LAPACK subroutine energy. Periodt. ✨

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SYNOPSIS 🎀

Functions/Subroutines

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

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Function/Subroutine Documentation 🦋

The main character is literally ZBDSQR:

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)
     ZBDSQR

Purpose 👑

okay so 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.

See 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 by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.

Parameters 🌸

• UPLO — UPLO is CHARACTER*1
  • = 'U': B is upper bidiagonal;
  • = 'L': B is lower bidiagonal.
• N — N is INTEGER; the order of the matrix B. N >= 0.
• NCVT — NCVT is INTEGER; the number of columns of the matrix VT. NCVT >= 0.
• NRU — NRU is INTEGER; the number of rows of the matrix U. NRU >= 0.
• NCC — NCC is INTEGER; the number of columns of the matrix C. NCC >= 0.
• D — 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 — 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 — 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 — LDVT is INTEGER; the leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
• U — 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 — LDU is INTEGER; the leading dimension of the array U. LDU >= max(1,NRU).
• C — 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 — LDC is INTEGER; the leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
• RWORK — RWORK is DOUBLE PRECISION array, dimension (LRWORK). LRWORK = 4*N, if NCVT = NRU = NCC = 0, and LRWORK = 4*(N-1), otherwise.
• INFO — INFO is INTEGER
  • = 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.

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.

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.

Author 👑

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

Definition at line 233 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 — final footer energy: LAPACK, Version 3.12.0, file SRC/zbdsqr.f(3). Byeeee ✨💖🎀