LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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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.