sbdsqr.f(3)

NAME
sbdsqr.f -

SYNOPSIS

  Functions/Subroutines
subroutine SBDSQR (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
SBDSQR

Function/Subroutine Documentation

  subroutine SBDSQR (CHARACTERUPLO, INTEGERN, INTEGERNCVT, INTEGERNRU, INTEGERNCC, REAL, dimension( * )D, REAL, dimension( * )E, REAL, dimension( ldvt, * )VT, INTEGERLDVT, REAL, dimension( ldu, * )U, INTEGERLDU, REAL, dimension( ldc, * )C, INTEGERLDC, REAL, dimension( * )WORK, INTEGERINFO)

SBDSQR

Purpose:

     SBDSQR computes the singular values and, optionally, the right and/or
     left singular vectors from the singular value decomposition (SVD) of
     a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
     zero-shift QR algorithm.  The SVD of B has the form

        B = Q * S * P**T

     where S is the diagonal matrix of singular values, Q is an orthogonal
     matrix of left singular vectors, and P is an orthogonal matrix of
     right singular vectors.  If left singular vectors are requested, this
     subroutine actually returns U*Q instead of Q, and, if right singular
     vectors are requested, this subroutine returns P**T*VT instead of
     P**T, for given real input matrices U and VT.  When U and VT are the
     orthogonal matrices that reduce a general matrix A to bidiagonal
     form:  A = U*B*VT, as computed by SGEBRD, then

        A = (U*Q) * S * (P**T*VT)

     is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
     for a given real input matrix C.
     See "Computing  Small Singular Values of Bidiagonal Matrices With
     Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
     LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
     no. 5, pp. 873-912, Sept 1990) and
     "Accurate singular values and differential qd algorithms," by
     B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
     Department, University of California at Berkeley, July 1992
     for a detailed description of the algorithm.

Parameters:
    UPLO

              UPLO is CHARACTER*1
              = 'U':  B is upper bidiagonal;
              = 'L':  B is lower bidiagonal.

    N

              N is INTEGER
              The order of the matrix B.  N >= 0.

    NCVT

              NCVT is INTEGER
              The number of columns of the matrix VT. NCVT >= 0.

    NRU

              NRU is INTEGER
              The number of rows of the matrix U. NRU >= 0.

    NCC

              NCC is INTEGER
              The number of columns of the matrix C. NCC >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the bidiagonal matrix B.
              On exit, if INFO=0, the singular values of B in decreasing
              order.

    E

              E is REAL array, dimension (N-1)
              On entry, the N-1 offdiagonal elements of the bidiagonal
              matrix B.
              On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
              will contain the diagonal and superdiagonal elements of a
              bidiagonal matrix orthogonally equivalent to the one given
              as input.

    VT

              VT is REAL array, dimension (LDVT, NCVT)
              On entry, an N-by-NCVT matrix VT.
              On exit, VT is overwritten by P**T * VT.
              Not referenced if NCVT = 0.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.
              LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

    U

              U is REAL array, dimension (LDU, N)
              On entry, an NRU-by-N matrix U.
              On exit, U is overwritten by U * Q.
              Not referenced if NRU = 0.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= max(1,NRU).

    C

              C is REAL array, dimension (LDC, NCC)
              On entry, an N-by-NCC matrix C.
              On exit, C is overwritten by Q**T * C.
              Not referenced if NCC = 0.

    LDC

              LDC is INTEGER
              The leading dimension of the array C.
              LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

    WORK

              WORK is REAL array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  If INFO = -i, the i-th argument had an illegal value
              > 0:
                 if NCVT = NRU = NCC = 0,
                    = 1, a split was marked by a positive value in E
                    = 2, current block of Z not diagonalized after 30*N
                         iterations (in inner while loop)
                    = 3, termination criterion of outer while loop not met
                         (program created more than N unreduced blocks)
                 else NCVT = NRU = NCC = 0,
                       the algorithm did not converge; D and E contain the
                       elements of a bidiagonal matrix which is orthogonally
                       similar to the input matrix B;  if INFO = i, i
                       elements of E have not converged to zero.

Internal Parameters:

      TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
              TOLMUL controls the convergence criterion of the QR loop.
              If it is positive, TOLMUL*EPS is the desired relative
                 precision in the computed singular values.
              If it is negative, abs(TOLMUL*EPS*sigma_max) is the
                 desired absolute accuracy in the computed singular
                 values (corresponds to relative accuracy
                 abs(TOLMUL*EPS) in the largest singular value.
              abs(TOLMUL) should be between 1 and 1/EPS, and preferably
                 between 10 (for fast convergence) and .1/EPS
                 (for there to be some accuracy in the results).
              Default is to lose at either one eighth or 2 of the
                 available decimal digits in each computed singular value
                 (whichever is smaller).
      MAXITR  INTEGER, default = 6
              MAXITR controls the maximum number of passes of the
              algorithm through its inner loop. The algorithms stops
              (and so fails to converge) if the number of passes
              through the inner loop exceeds MAXITR*N**2.

Author:
    Univ'. of Tennessee

    Univ'. of California Berkeley

    Univ'. of Colorado Denver

    NAG Ltd'.

Date:
    November 2011

Definition at line 230 of file sbdsqr.f'.

Author

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