DLAED7(3)      LAPACK routine of NEC Numeric Library Collection      DLAED7(3)



NAME
       DLAED7

SYNOPSIS
       SUBROUTINE DLAED7 (ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ,
           INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL,
           GIVNUM, WORK, IWORK, INFO)



PURPOSE
            DLAED7 computes the updated eigensystem of a diagonal
            matrix after modification by a rank-one symmetric matrix. This
            routine is used only for the eigenproblem which requires all
            eigenvalues and optionally eigenvectors of a dense symmetric matrix
            that has been reduced to tridiagonal form.  DLAED1 handles
            the case in which all eigenvalues and eigenvectors of a symmetric
            tridiagonal matrix are desired.

              T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

               where Z = Q**Tu, u is a vector of length N with ones in the
               CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

               The eigenvectors of the original matrix are stored in Q, and the
               eigenvalues are in D.  The algorithm consists of three stages:

                  The first stage consists of deflating the size of the problem
                  when there are multiple eigenvalues or if there is a zero in
                  the Z vector.  For each such occurence the dimension of the
                  secular equation problem is reduced by one.  This stage is
                  performed by the routine DLAED8.

                  The second stage consists of calculating the updated
                  eigenvalues. This is done by finding the roots of the secular
                  equation via the routine DLAED4 (as called by DLAED9).
                  This routine also calculates the eigenvectors of the current
                  problem.

                  The final stage consists of computing the updated eigenvectors
                  directly using the updated eigenvalues.  The eigenvectors for
                  the current problem are multiplied with the eigenvectors from
                  the overall problem.




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.

           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.

           TLVLS     (input)
                     TLVLS is INTEGER
                    The total number of merging levels in the overall divide and
                    conquer tree.

           CURLVL    (input)
                     CURLVL is INTEGER
                    The current level in the overall merge routine,
                    0 <= CURLVL <= TLVLS.

           CURPBM    (input)
                     CURPBM is INTEGER
                    The current problem in the current level in the overall
                    merge routine (counting from upper left to lower right).

           D         (input/output)
                     D is DOUBLE PRECISION array, dimension (N)
                    On entry, the eigenvalues of the rank-1-perturbed matrix.
                    On exit, the eigenvalues of the repaired matrix.

           Q         (input/output)
                     Q is DOUBLE PRECISION array, dimension (LDQ, N)
                    On entry, the eigenvectors of the rank-1-perturbed matrix.
                    On exit, the eigenvectors of the repaired tridiagonal matrix.

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

           INDXQ     (output)
                     INDXQ is INTEGER array, dimension (N)
                    The permutation which will reintegrate the subproblem just
                    solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
                    will be in ascending order.

           RHO       (input)
                     RHO is DOUBLE PRECISION
                    The subdiagonal element used to create the rank-1
                    modification.

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

           QSTORE    (input/output)
                     QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
                    Stores eigenvectors of submatrices encountered during
                    divide and conquer, packed together. QPTR points to
                    beginning of the submatrices.

           QPTR      (input/output)
                     QPTR is INTEGER array, dimension (N+2)
                    List of indices pointing to beginning of submatrices stored
                    in QSTORE. The submatrices are numbered starting at the
                    bottom left of the divide and conquer tree, from left to
                    right and bottom to top.

           PRMPTR    (input)
                     PRMPTR is INTEGER array, dimension (N lg N)
                    Contains a list of pointers which indicate where in PERM a
                    level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
                    indicates the size of the permutation and also the size of
                    the full, non-deflated problem.

           PERM      (input)
                     PERM is INTEGER array, dimension (N lg N)
                    Contains the permutations (from deflation and sorting) to be
                    applied to each eigenblock.

           GIVPTR    (input)
                     GIVPTR is INTEGER array, dimension (N lg N)
                    Contains a list of pointers which indicate where in GIVCOL a
                    level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
                    indicates the number of Givens rotations.

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

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

           WORK      (output)
                     WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (4*N)

           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                      DLAED7(3)