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.

          NL is INTEGER
         The row dimension of the upper block.  NL >= 1.

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

          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 is INTEGER
         The size of the secular equation, 1 =< K = < N.

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

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

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

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

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

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

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

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

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

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

          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