DLAED3(3)      LAPACK routine of NEC Numeric Library Collection      DLAED3(3)



NAME
       DLAED3

SYNOPSIS
       SUBROUTINE DLAED3 (K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W,
           S, INFO)



PURPOSE
            DLAED3 finds the roots of the secular equation, as defined by the
            values in D, W, and RHO, between 1 and K.  It makes the
            appropriate calls to DLAED4 and then updates the eigenvectors by
            multiplying the matrix of eigenvectors of the pair of eigensystems
            being combined by the matrix of eigenvectors of the K-by-K system
            which is solved here.

            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.




ARGUMENTS
           K         (input)
                     K is INTEGER
                     The number of terms in the rational function to be solved by
                     DLAED4.  K >= 0.

           N         (input)
                     N is INTEGER
                     The number of rows and columns in the Q matrix.
                     N >= K (deflation may result in N>K).

           N1        (input)
                     N1 is INTEGER
                     The location of the last eigenvalue in the leading submatrix.
                     min(1,N) <= N1 <= N/2.

           D         (output)
                     D is DOUBLE PRECISION array, dimension (N)
                     D(I) contains the updated eigenvalues for
                     1 <= I <= K.

           Q         (output)
                     Q is DOUBLE PRECISION array, dimension (LDQ,N)
                     Initially the first K columns are used as workspace.
                     On output the columns 1 to K contain
                     the updated eigenvectors.

           LDQ       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q.  LDQ >= max(1,N).

           RHO       (input)
                     RHO is DOUBLE PRECISION
                     The value of the parameter in the rank one update equation.
                     RHO >= 0 required.

           DLAMDA    (input/output)
                     DLAMDA 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. May be changed on output by
                     having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
                     Cray-2, or Cray C-90, as described above.

           Q2        (input)
                     Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
                     The first K columns of this matrix contain the non-deflated
                     eigenvectors for the split problem.

           INDX      (input)
                     INDX is INTEGER array, dimension (N)
                     The permutation used to arrange the columns of the deflated
                     Q matrix into three groups (see DLAED2).
                     The rows of the eigenvectors found by DLAED4 must be likewise
                     permuted before the matrix multiply can take place.

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

           W         (input/output)
                     W is DOUBLE PRECISION array, dimension (K)
                     The first K elements of this array contain the components
                     of the deflation-adjusted updating vector. Destroyed on
                     output.

           S         (output)
                     S is DOUBLE PRECISION array, dimension (N1 + 1)*K
                     Will contain the eigenvectors of the repaired matrix which
                     will be multiplied by the previously accumulated eigenvectors
                     to update the system.

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



LAPACK routine                  31 October 2017                      DLAED3(3)