SRC/sbdsvdx.f(3)

NAME
SRC/sbdsvdx.f

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

  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:

    !>
    !>  SBDSVDX computes the singular value decomposition (SVD) of a real
    !>  N-by-N (upper or lower) bidiagonal matrix B, 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.
    !>
    !>  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
    !>          = 'U':  B is upper bidiagonal;
    !>          = 'L':  B is lower bidiagonal.
    !>

    JOBZ

    !>          JOBZ is CHARACTER*1
    !>          = 'N':  Compute singular values only;
    !>          = 'V':  Compute singular values and singular vectors.
    !>

    RANGE

    !>          RANGE is CHARACTER*1
    !>          = '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
    !>          = 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

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