slasd3(3S)

NAME
     SLASD3 - find all the square roots of the roots of the secular equation,
     as defined by the values in D and Z

SYNOPSIS
     SUBROUTINE SLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2,
                        VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO )

         INTEGER        INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE

         INTEGER        CTOT( * ), IDXC( * )

         REAL           D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ), U2(
                        LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )

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
     SLASD3 finds all the square roots of the roots of the secular equation,
     as defined by the values in D and Z. It makes the appropriate calls to
     SLASD4 and then updates the singular vectors by matrix multiplication.

     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 XMP,
     Cray YMP, Cray C 90, or Cray 2.  It could conceivably fail on hexadecimal
     or decimal machines without guard digits, but we know of none.

     SLASD3 is called from SLASD1.


ARGUMENTS
     NL     (input) INTEGER
            The row dimension of the upper block.  NL >= 1.

     NR     (input) INTEGER
            The row dimension of the lower block.  NR >= 1.

     SQRE   (input) INTEGER
            = 0: the lower block is an NR-by-NR square matrix.
            = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
            The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >=
            N columns.

     K      (input) INTEGER
            The size of the secular equation, 1 =< K = < N.

     D      (output) REAL array, dimension(K)
            On exit the square roots of the roots of the secular equation, in
            ascending order.

     Q      (workspace) REAL array,
            dimension at least (LDQ,K).

     LDQ    (input) INTEGER
            The leading dimension of the array Q.  LDQ >= K.

            DSIGMA (input) REAL array, dimension(K) The first K elements of
            this array contain the old roots of the deflated updating problem.
            These are the poles of the secular equation.

     U      (input) REAL array, dimension (LDU, N)
            The last N - K columns of this matrix contain the deflated left
            singular vectors.

     LDU    (input) INTEGER
            The leading dimension of the array U.  LDU >= N.

     U2     (input) REAL array, dimension (LDU2, N)
            The first K columns of this matrix contain the non-deflated left
            singular vectors for the split problem.

     LDU2   (input) INTEGER
            The leading dimension of the array U2.  LDU2 >= N.

     VT     (input) REAL array, dimension (LDVT, M)
            The last M - K columns of VT' contain the deflated right singular
            vectors.

     LDVT   (input) INTEGER
            The leading dimension of the array VT.  LDVT >= N.

     VT2    (input) REAL array, dimension (LDVT2, N)
            The first K columns of VT2' contain the non-deflated right
            singular vectors for the split problem.

     LDVT2  (input) INTEGER
            The leading dimension of the array VT2.  LDVT2 >= N.

     IDXC   (input) INTEGER array, dimension ( N )
            The permutation used to arrange the columns of U (and rows of VT)
            into three groups:  the first group contains non-zero entries only
            at and above (or before) NL +1; the second contains non-zero
            entries only at and below (or after) NL+2; and the third is dense.
            The first column of U and the row of VT are treated separately,
            however.

            The rows of the singular vectors found by SLASD4 must be likewise
            permuted before the matrix multiplies can take place.

     CTOT   (input) INTEGER array, dimension ( 4 )
            A count of the total number of the various types of columns in U
            (or rows in VT), as described in IDXC. The fourth column type is
            any column which has been deflated.

     Z      (input) REAL array, dimension (K)
            The first K elements of this array contain the components of the
            deflation-adjusted updating row vector.

     INFO   (output) INTEGER
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  if INFO = 1, an singular value did not converge

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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