Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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

SYNOPSIS

       SUBROUTINE ZBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU

           DOUBLE         PRECISION D( * ), E( * ), RWORK( * )

           COMPLEX*16     C( LDC, * ), U( LDU, * ), VT( LDVT, * )

PURPOSE

       ZBDSQR  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**H
        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**H*VT instead of
        P**H, for given complex input matrices U and VT.  When U and VT are
        the unitary matrices that reduce a general matrix A to bidiagonal
        form: A = U*B*VT, as computed by ZGEBRD, then
           A = (U*Q) * S * (P**H*VT)
        is the SVD of A.  Optionally, the subroutine may also compute Q**H*C
        for a given complex 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.

ARGUMENTS

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

        N       (input) INTEGER
                The order of the matrix B.  N >= 0.

        NCVT    (input) INTEGER
                The number of columns of the matrix VT. NCVT >= 0.

        NRU     (input) INTEGER
                The number of rows of the matrix U. NRU >= 0.

        NCC     (input) INTEGER
                The number of columns of the matrix C. NCC >= 0.

        D       (input/output) DOUBLE PRECISION 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       (input/output) DOUBLE PRECISION 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      (input/output) COMPLEX*16 array, dimension (LDVT, NCVT)
                On entry, an N-by-NCVT matrix VT.
                On exit, VT is overwritten by P**H * VT.
                Not referenced if NCVT = 0.

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

        U       (input/output) COMPLEX*16 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     (input) INTEGER
                The leading dimension of the array U.  LDU >= max(1,NRU).

        C       (input/output) COMPLEX*16 array, dimension (LDC, NCC)
                On entry, an N-by-NCC matrix C.
                On exit, C is overwritten by Q**H * C.
                Not referenced if NCC = 0.

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

        RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
                if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise

        INFO    (output) INTEGER
                = 0:  successful exit
                < 0:  If INFO = -i, the i-th argument had an illegal value
                > 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.

PARAMETERS

        TOLMUL  DOUBLE PRECISION, 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.

 LAPACK routine (version 3.2)               April 2011                            ZBDSQR(3lapack)