bdsvdx(3)

NAME
bdsvdx - bdsvdx: bidiagonal SVD, bisection

SYNOPSIS

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

Detailed Description

Function Documentation

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

DBDSVDX

Purpose:

      DBDSVDX 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 ], DBDSVDX 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 DGESVD/DGESDD) or b) compute s,v and reorder
      the values (and corresponding vectors). DBDSVDX implements a) by
      calling DSTEVX (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 DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the bidiagonal matrix B.

    E

              E is DOUBLE PRECISION 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 DOUBLE PRECISION
              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 DOUBLE PRECISION
              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 DOUBLE PRECISION array, dimension (N)
              The first NS elements contain the selected singular values in
              ascending order.

    Z

              Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSTEVX. The indices of the eigenvectors
                       (as returned by DSTEVX) 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 dbdsvdx.f.

  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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