bdsvdx - Man Page

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

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.

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.