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## 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

Download SBDSVDX + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  UPLO

      !>          UPLO is CHARACTER*1
      !>          = 'U':  B is upper bidiagonal;
      !>          = 'L':  B is lower bidiagonal.
      !>

  [in]

  JOBZ

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

  [in]

  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.
      !>

  [in]

  N

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

  [in]

  D

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

  [in]

  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.
      !>

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [out]

  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.
      !>

  [out]

  S

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

  [out]

  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.
      !>

  [in]

  LDZ

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

  [out]

  WORK

      !>          WORK is REAL array, dimension (14*N)
      !>

  [out]

  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.
      !>

  [out]

  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.