SRC/sbdsvdx.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

💖 NAME

SRC/sbdsvdx.f — bestie, this page is from the Library Functions Manual, and it's all about the LAPACK subroutine sbdsvdx. 💅

✨ SYNOPSIS

🧚 Functions/Subroutines

subroutine sbdsvdx (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s,
    z, ldz, work, iwork, info)
    SBDSVDX

🛠️ Function/Subroutine Documentation

This is the whole glow-up for SBDSVDX:

subroutine sbdsvdx (character uplo, character jobz, character range, integer
    n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu,
    integer il, integer iu, integer ns, real, dimension( * ) s, real,
    dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer,
    dimension( * ) iwork, integer info)
    SBDSVDX

🌸 Purpose

Okay so SBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, like:

B = U * S * VT

where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. No cap. 💖

Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix:

       |  0  d_1                |
       | d_1  0  e_1            |
TGK = |     e_1  0  d_2        |
       |         d_2  .   .     |
       |              .   .   . |

If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. 🦋

Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder the values (and corresponding vectors). SBDSVDX implements a) by calling SSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.) ✨

🎀 Parameters

UPLO

UPLO is CHARACTER*1 — it tells us which bidiagonal flavor B is:

  • = 'U': B is upper bidiagonal;
  • = 'L': B is lower bidiagonal.

JOBZ

JOBZ is CHARACTER*1 — the “do we want vectors?” switch:

  • = 'N': Compute singular values only;
  • = 'V': Compute singular values and singular vectors.

RANGE

RANGE is CHARACTER*1 — picks which singular values get found:

  • = 'A': all singular values will be found.
  • = 'V': all singular values in the half-open interval [VL,VU) will be found.
  • = 'I': the IL-th through IU-th singular values will be found.

N

N is INTEGER — the order of the bidiagonal matrix. N >= 0.

D

D is REAL array, dimension (N) — the n diagonal elements of the bidiagonal matrix B.

E

E is REAL array, dimension (max(1,N-1)) — the (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1.

VL

VL is REAL — if RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

VU

VU is REAL — if RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

IL

IL is INTEGER — if RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

IU

IU is INTEGER — if RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

NS

NS is INTEGER — the total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

S

S is REAL array, dimension (N) — the first NS elements contain the selected singular values in ascending order.

Z

Z is REAL array, dimension (2*N,K) — if JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i.e.

Z = [ U ]
    [ V ]

If JOBZ = 'N', then Z is not referenced.

Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used. 🫶

LDZ

LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2).

WORK

WORK is REAL array, dimension (14*N).

IWORK

IWORK is INTEGER array, dimension (12*N) — if JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX.

INFO

INFO is INTEGER — the status report:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, then i eigenvectors failed to converge in SSTEVX. The indices of the eigenvectors (as returned by SSTEVX) are stored in the array IWORK. if INFO = N*2 + 1, an internal error occurred.

👑 Author

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

Definition at line 224 of file sbdsvdx.f.

🌈 Author

Generated automatically by Doxygen for LAPACK from the source code.

Okay bestie, that's the whole page! 💖 Source: SRC/sbdsvdx.f(3). Version: LAPACK 3.12.0. Date: the original footer didn't give a separate date, so I'm not inventing one — periodt. ✨🎀

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