SRC/cbdsqr.f(3)            Library Functions Manual            SRC/cbdsqr.f(3)

NAME
       SRC/cbdsqr.f

SYNOPSIS
   Functions/Subroutines
       subroutine cbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
           ldc, rwork, info)
           CBDSQR

Function/Subroutine Documentation
   subroutine cbdsqr (character uplo, integer n, integer ncvt, integer nru,
       integer ncc, real, dimension( * ) d, real, dimension( * ) e, complex,
       dimension( ldvt, * ) vt, integer ldvt, complex, dimension( ldu, * ) u,
       integer ldu, complex, dimension( ldc, * ) c, integer ldc, real,
       dimension( * ) rwork, integer info)
       CBDSQR

       Purpose:


            CBDSQR 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 CGEBRD, 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.

       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 COMPLEX 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

                     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 COMPLEX 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 COMPLEX 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

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

           RWORK

                     RWORK 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:  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.

       Note:


             Bug report from Cezary Dendek.
             On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is
             removed since it can overflow pretty easily (for N larger or equal
             than 18,919). We instead use MAXITDIVN = MAXITR*N.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Definition at line 231 of file cbdsqr.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

LAPACK                          Version 3.12.0                 SRC/cbdsqr.f(3)