sbdsvdx.f - Man Page

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

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.