slaed3(3S)

NAME
     SLAED3 - find the roots of the secular equation, as defined by the values
     in D, W, and RHO, between 1 and K

SYNOPSIS
     SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W,
                        S, INFO )

         INTEGER        INFO, K, LDQ, N, N1

         REAL           RHO

         INTEGER        CTOT( * ), INDX( * )

         REAL           D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), S( * ), W(
                        * )

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
     SLAED3 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
     SLAED4 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) INTEGER
             The number of terms in the rational function to be solved by
             SLAED4.  K >= 0.

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

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

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

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

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

     DLAMDA  (input/output) REAL 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) REAL array, dimension (LDQ2, N)
             The first K columns of this matrix contain the non-deflated
             eigenvectors for the split problem.

     INDX    (input) INTEGER array, dimension (N)
             The permutation used to arrange the columns of the deflated Q
             matrix into three groups (see SLAED2).  The rows of the
             eigenvectors found by SLAED4 must be likewise permuted before the
             matrix multiply can take place.

     CTOT    (input) 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) REAL array, dimension (K)
             The first K elements of this array contain the components of the
             deflation-adjusted updating vector. Destroyed on output.

     S       (workspace) REAL 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.
     LDS     (input) INTEGER
             The leading dimension of S.  LDS >= max(1,K).

     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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