DLASD3(3)      LAPACK routine of NEC Numeric Library Collection      DLASD3(3)



NAME
       DLASD3

SYNOPSIS
       SUBROUTINE DLASD3 (NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
           LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO)



PURPOSE
            DLASD3 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 DLASD4 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.

            DLASD3 is called from DLASD1.




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

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

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

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

           Q         (output)
                     Q is DOUBLE PRECISION array,
                                dimension at least (LDQ,K).

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

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

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

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

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

           VT        (output)
                     VT is DOUBLE PRECISION array, dimension (LDVT, M)
                    The last M - K columns of VT**T contain the deflated
                    right singular vectors.

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

           VT2       (input/output)
                     VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
                    The first K columns of VT2**T contain the non-deflated
                    right singular vectors for the split problem.

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

           IDXC      (input)
                     IDXC is 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 DLASD4
                    must be likewise permuted before the matrix multiplies can
                    take place.

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

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



LAPACK routine                  31 October 2017                      DLASD3(3)