sbdsdc(3S)

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

SYNOPSIS
     SUBROUTINE SBDSDC( 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( * )

         REAL           D( * ), E( * ), Q( * ), U( LDU, * ), VT( LDVT, * ),
                        WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SBDSDC 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 orthogonal matrices
     of left and right singular vectors, respectively. SBDSDC 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 SLASD3 for details.

     The code currently call SLASDQ 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 follows:
             = 'N':  Compute singular values only;
             = 'P':  Compute singular values and compute singular vectors in
             compact form; = 'I':  Compute singular values and singular
             vectors.

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

     D       (input/output) 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.

     E       (input/output) REAL 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) REAL 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 values 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) REAL 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) REAL 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, requiring
             O(N log N) space instead of 2*N**2.  In particular, Q contains
             all the REAL 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
             subproblems 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, requiring
             O(N log N) space instead of 2*N**2.  In particular, IQ contains
             all INTEGER data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and is equal
             to the maximum size of the subproblems at the bottom of the
             computation tree (usually about 25).  For other values of COMPQ,
             IQ is not referenced.

     WORK    (workspace) REAL 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.

FURTHER DETAILS
     Based on contributions by
        Ming Gu and Huan Ren, Computer Science Division, University of
        California at Berkeley, USA


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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