SRC/sbdsqr.f(3)

Okay bestie 💖 this is the SRC/sbdsqr.f(3) page from the Library Functions Manual, and it is literally serving pure LAPACK subroutine energy. We are talking SBDSQR, the Fortran routine for singular values and optionally singular vectors from the SVD of a real bidiagonal matrix. Let’s get into it 🎀

🌸 NAME 🌸

SRC/sbdsqr.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

       subroutine sbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
           ldc, work, info)
           SBDSQR

🎀 Function/Subroutine Documentation 🎀

       subroutine sbdsqr (character uplo, integer n, integer ncvt, integer nru,
           integer ncc, real, dimension( * ) d, real, dimension( * ) e, real,
           dimension( ldvt, * ) vt, integer ldvt, real, dimension( ldu, * ) u,
           integer ldu, real, dimension( ldc, * ) c, integer ldc, real, dimension(
           * ) work, integer info)
           SBDSQR

✨ Purpose ✨

SBDSQR 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**T

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**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by SGEBRD, then

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

is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real 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 REAL 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 REAL 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 REAL array, dimension (LDVT, NCVT) On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**T * 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 REAL 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 REAL array, dimension (LDC, NCC) On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * 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.
• WORK — WORK is REAL array, dimension (LWORK) LWORK = 4*N, if NCVT = NRU = NCC = 0, and LWORK = 4*(N-1), otherwise
• INFO — INFO is INTEGER
  • = 0: successful exit
  • < 0: If INFO = -i, the i-th argument had an illegal value
  • > 0:
    • if NCVT = NRU = NCC = 0,
      • = 1, a split was marked by a positive value in E
      • = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop)
      • = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
    • else NCVT = NRU = NCC = 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 — REAL, 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 algorithms 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 March 23rd 2017, 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 240 of file sbdsqr.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the whole page, bestie — LAPACK Version 3.12.0, page SRC/sbdsqr.f(3), xoxo 💅✨