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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