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NAME

        DBDSDC - compute the singular value decomposition (SVD) of a real N-by-
        N (upper or lower) bidiagonal matrix B
 

SYNOPSIS

        SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ, WORK,
                           IWORK, INFO )
 
            CHARACTER      COMPQ, UPLO
 
            INTEGER        INFO, LDU, LDVT, N
 
            INTEGER        IQ( * ), IWORK( * )
 
            DOUBLE         PRECISION  D(  * ), E( * ), Q( * ), U( LDU, * ), VT(
                           LDVT, * ), WORK( * )
 

PURPOSE

        DBDSDC computes the singular value decomposition (SVD) of a real N-by-N
        (upper  or  lower)  bidiagonal matrix B: B = U * S * VT, using a divide
        and conquer method, where S is  a  diagonal  matrix  with  non-negative
        diagonal elements (the singular values of B), and U and VT are orthogo‐
        nal matrices of left and right singular vectors,  respectively.  DBDSDC
        can  be  used  to compute all singular values, and optionally, singular
        vectors or singular vectors in compact form.
 
        This code makes very mild assumptions about floating point  arithmetic.
        It  will  work  on  machines  with a guard digit in add/subtract, or on
        those binary machines without guard digits which subtract like the Cray
        X-MP,  Cray  Y-MP,  Cray C-90, or Cray-2.  It could conceivably fail on
        hexadecimal or decimal machines without guard digits, but  we  know  of
        none.  See DLASD3 for details.
 
        The  code  currently  call  DLASDQ if singular values only are desired.
        However, it can be slightly modified to compute singular  values  using
        the divide and conquer method.
 

ARGUMENTS

        UPLO    (input) CHARACTER*1
                = ’U’:  B is upper bidiagonal.
                = ’L’:  B is lower bidiagonal.
 
        COMPQ   (input) CHARACTER*1
                Specifies  whether  singular vectors are to be computed as fol‐
                lows:
                = ’N’:  Compute singular values only;
                = ’P’:  Compute singular values and compute singular vectors in
                compact form; = ’I’:  Compute singular values and singular vec‐
                tors.
 
        N       (input) INTEGER
                The order of the matrix B.  N >= 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.
 
        E       (input/output) DOUBLE PRECISION array, dimension (N)
                On entry, the elements of E contain the offdiagonal elements of
                the bidiagonal matrix whose SVD is desired.   On  exit,  E  has
                been destroyed.
 
        U       (output) DOUBLE PRECISION array, dimension (LDU,N)
                If   COMPQ  =  ’I’,  then: On exit, if INFO = 0, U contains the
                left singular vectors of the bidiagonal matrix.  For other val‐
                ues of COMPQ, U is not referenced.
 
        LDU     (input) INTEGER
                The  leading  dimension of the array U.  LDU >= 1.  If singular
                vectors are desired, then LDU >= max( 1, N ).
 
        VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
                If  COMPQ = ’I’, then: On exit, if INFO = 0, VT’  contains  the
                right  singular  vectors  of  the bidiagonal matrix.  For other
                values of COMPQ, VT is not referenced.
 
        LDVT    (input) INTEGER
                The leading dimension of the array VT.  LDVT >= 1.  If singular
                vectors are desired, then LDVT >= max( 1, N ).
 
        Q       (output) DOUBLE PRECISION array, dimension (LDQ)
                If   COMPQ  = ’P’, then: On exit, if INFO = 0, Q and IQ contain
                the left and right singular vectors in a compact form,  requir‐
                ing  O(N log N) space instead of 2*N**2.  In particular, Q con‐
                tains all the DOUBLE PRECISION data in LDQ >= N*(11 +  2*SMLSIZ
                +  8*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is
                returned by ILAENV and is equal to the maximum size of the sub‐
                problems  at  the bottom of the computation tree (usually about
                25).  For other values of COMPQ, Q is not referenced.
 
        IQ      (output) INTEGER array, dimension (LDIQ)
                If  COMPQ = ’P’, then: On exit, if INFO = 0, Q and  IQ  contain
                the  left and right singular vectors in a compact form, requir‐
                ing O(N log N) space instead of 2*N**2.  In particular, IQ con‐
                tains  all  INTEGER  data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SML‐
                SIZ+1)))) words of memory, where SMLSIZ is returned  by  ILAENV
                and is equal to the maximum size of the subproblems at the bot‐
                tom of the computation tree (usually about 25).  For other val‐
                ues of COMPQ, IQ is not referenced.
 
        WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK)
                If  COMPQ  =  ’N’  then  LWORK >= (4 * N).  If COMPQ = ’P’ then
                LWORK >= (6 * N).  If COMPQ = ’I’ then LWORK >= (3 * N**2 + 4 *
                N).
 
        IWORK   (workspace) INTEGER array, dimension (8*N)
 
        INFO    (output) INTEGER
                = 0:  successful exit.
                < 0:  if INFO = -i, the i-th argument had an illegal value.
                >  0:   The algorithm failed to compute an singular value.  The
                update process of divide and conquer failed.
        Based on contributions by
           Ming Gu and Huan Ren, Computer Science Division, University of
           California at Berkeley, USA
 

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