slaed1(3S)

NAME
     SLAED1 - compute the updated eigensystem of a diagonal matrix after
     modification by a rank-one symmetric matrix

SYNOPSIS
     SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO )

         INTEGER        CUTPNT, INFO, LDQ, N

         REAL           RHO

         INTEGER        INDXQ( * ), IWORK( * )

         REAL           D( * ), Q( LDQ, * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SLAED1 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 eigenvectors of a
     tridiagonal matrix.  SLAED7 handles the case in which eigenvalues only or
     eigenvalues and eigenvectors of a full symmetric matrix (which was
     reduced to tridiagonal form) are desired.

       T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)

        where Z = Q'u, 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 SLAED2.

           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine SLAED4 (as called by SLAED3).
           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
     N      (input) INTEGER
            The dimension of the symmetric tridiagonal matrix.  N >= 0.

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

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

     INDXQ  (input/output) INTEGER array, dimension (N)
            On entry, the permutation which separately sorts the two
            subproblems in D into ascending order.  On exit, the permutation
            which will reintegrate the subproblems back into sorted order,
            i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

     RHO    (input) REAL
            The subdiagonal entry used to create the rank-1 modification.

            CUTPNT (input) INTEGER The location of the last eigenvalue in the
            leading sub-matrix.  min(1,N) <= CUTPNT <= N/2.

     WORK   (workspace) REAL array, dimension (4*N + N**2)

     IWORK  (workspace) INTEGER array, dimension (4*N)

     INFO   (output) 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

FURTHER DETAILS
     Based on contributions by
        Jeff Rutter, Computer Science Division, University of California
        at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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