DLAED8(3)      LAPACK routine of NEC Numeric Library Collection      DLAED8(3)



NAME
       DLAED8

SYNOPSIS
       SUBROUTINE DLAED8 (ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
           Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM, INDXP, INDX,
           INFO)



PURPOSE
            DLAED8 merges the two sets of eigenvalues together into a single
            sorted set.  Then it tries to deflate the size of the problem.
            There are two ways in which deflation can occur:  when two or more
            eigenvalues are close together or if there is a tiny element in the
            Z vector.  For each such occurrence the order of the related secular
            equation problem is reduced by one.




ARGUMENTS
           ICOMPQ    (input)
                     ICOMPQ is INTEGER
                     = 0:  Compute eigenvalues only.
                     = 1:  Compute eigenvectors of original dense symmetric matrix
                           also.  On entry, Q contains the orthogonal matrix used
                           to reduce the original matrix to tridiagonal form.

           K         (output)
                     K is INTEGER
                    The number of non-deflated eigenvalues, and the order of the
                    related secular equation.

           N         (input)
                     N is INTEGER
                    The dimension of the symmetric tridiagonal matrix.  N >= 0.

           QSIZ      (input)
                     QSIZ is INTEGER
                    The dimension of the orthogonal matrix used to reduce
                    the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

           D         (input/output)
                     D is DOUBLE PRECISION array, dimension (N)
                    On entry, the eigenvalues of the two submatrices to be
                    combined.  On exit, the trailing (N-K) updated eigenvalues
                    (those which were deflated) sorted into increasing order.

           Q         (input/output)
                     Q is DOUBLE PRECISION array, dimension (LDQ,N)
                    If ICOMPQ = 0, Q is not referenced.  Otherwise,
                    on entry, Q contains the eigenvectors of the partially solved
                    system which has been previously updated in matrix
                    multiplies with other partially solved eigensystems.
                    On exit, Q contains the trailing (N-K) updated eigenvectors
                    (those which were deflated) in its last N-K columns.

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

           INDXQ     (input)
                     INDXQ is INTEGER array, dimension (N)
                    The permutation which separately sorts the two sub-problems
                    in D into ascending order.  Note that elements in the second
                    half of this permutation must first have CUTPNT added to
                    their values in order to be accurate.

           RHO       (input/output)
                     RHO is DOUBLE PRECISION
                    On entry, the off-diagonal element associated with the rank-1
                    cut which originally split the two submatrices which are now
                    being recombined.
                    On exit, RHO has been modified to the value required by
                    DLAED3.

           CUTPNT    (input)
                     CUTPNT is INTEGER
                    The location of the last eigenvalue in the leading
                    sub-matrix.  min(1,N) <= CUTPNT <= N.

           Z         (input)
                     Z is DOUBLE PRECISION array, dimension (N)
                    On entry, Z contains the updating vector (the last row of
                    the first sub-eigenvector matrix and the first row of the
                    second sub-eigenvector matrix).
                    On exit, the contents of Z are destroyed by the updating
                    process.

           DLAMDA    (output)
                     DLAMDA is DOUBLE PRECISION array, dimension (N)
                    A copy of the first K eigenvalues which will be used by
                    DLAED3 to form the secular equation.

           Q2        (output)
                     Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
                    If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
                    a copy of the first K eigenvectors which will be used by
                    DLAED7 in a matrix multiply (DGEMM) to update the new
                    eigenvectors.

           LDQ2      (input)
                     LDQ2 is INTEGER
                    The leading dimension of the array Q2.  LDQ2 >= max(1,N).

           W         (output)
                     W is DOUBLE PRECISION array, dimension (N)
                    The first k values of the final deflation-altered z-vector and
                    will be passed to DLAED3.

           PERM      (output)
                     PERM is INTEGER array, dimension (N)
                    The permutations (from deflation and sorting) to be applied
                    to each eigenblock.

           GIVPTR    (output)
                     GIVPTR is INTEGER
                    The number of Givens rotations which took place in this
                    subproblem.

           GIVCOL    (output)
                     GIVCOL is INTEGER array, dimension (2, N)
                    Each pair of numbers indicates a pair of columns to take place
                    in a Givens rotation.

           GIVNUM    (output)
                     GIVNUM is DOUBLE PRECISION array, dimension (2, N)
                    Each number indicates the S value to be used in the
                    corresponding Givens rotation.

           INDXP     (output)
                     INDXP is INTEGER array, dimension (N)
                    The permutation used to place deflated values of D at the end
                    of the array.  INDXP(1:K) points to the nondeflated D-values
                    and INDXP(K+1:N) points to the deflated eigenvalues.

           INDX      (output)
                     INDX is INTEGER array, dimension (N)
                    The permutation used to sort the contents of D into ascending
                    order.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.



LAPACK routine                  31 October 2017                      DLAED8(3)