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sgeesx (3)
  • >> sgeesx (3) ( Solaris man: Библиотечные вызовы )
  • 
    NAME
         sgeesx - compute for an N-by-N real nonsymmetric  matrix  A,
         the eigenvalues, the real Schur form T, and, optionally, the
         matrix of Schur vectors Z
    
    SYNOPSIS
         SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE,  N,  A,  LDA,
                   SDIM,  WR,  WI,  VS,  LDVS,  RCONDE, RCONDV, WORK,
                   LWORK, IWORK, LIWORK, BWORK, INFO )
    
         CHARACTER JOBVS, SENSE, SORT
    
         INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
    
         REAL RCONDE, RCONDV
    
         LOGICAL BWORK( * )
    
         INTEGER IWORK( * )
    
         REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )
    
         LOGICAL SELECT
    
         EXTERNAL SELECT
    
    
    
         #include <sunperf.h>
    
         void sgeesx(char jobvs, char  sort,  int  (*select)(),  char
                   sense, int n, float *sa, int lda, int *sdim, float
                   *wr,  float  *wi,  float  *vs,  int  ldvs,   float
                   *rconde, float *rcondv, int *info) ;
    
    PURPOSE
         SGEESX computes for an N-by-N real  nonsymmetric  matrix  A,
         the eigenvalues, the real Schur form T, and, optionally, the
         matrix of Schur vectors Z.  This gives the Schur  factoriza-
         tion A = Z*T*(Z**T).
    
         Optionally, it also orders the eigenvalues on  the  diagonal
         of  the  real Schur form so that selected eigenvalues are at
         the top left; computes a reciprocal condition number for the
         average of the selected eigenvalues (RCONDE); and computes a
         reciprocal condition number for the right invariant subspace
         corresponding  to  the  selected  eigenvalues (RCONDV).  The
         leading columns of Z form  an  orthonormal  basis  for  this
         invariant subspace.
    
         For further explanation of the reciprocal condition  numbers
         RCONDE  and  RCONDV,  see  Section 4.10 of the LAPACK Users'
         Guide (where these quantities are called s and  sep  respec-
         tively).
    
         A real matrix is in real Schur form if it  is  upper  quasi-
         triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will
         be standardized in the form
                   [  a  b  ]
                   [  c  a  ]
    
         where b*c < 0. The eigenvalues of such  a  block  are  a  +-
         sqrt(bc).
    
    
    ARGUMENTS
         JOBVS     (input) CHARACTER*1
                   = 'N': Schur vectors are not computed;
                   = 'V': Schur vectors are computed.
    
         SORT      (input) CHARACTER*1
                   Specifies whether or not to order the  eigenvalues
                   on  the diagonal of the Schur form.  = 'N': Eigen-
                   values are not ordered;
                   = 'S': Eigenvalues are ordered (see SELECT).
    
         SELECT    (input) LOGICAL FUNCTION of two REAL arguments
                   SELECT must be declared EXTERNAL  in  the  calling
                   subroutine.   If  SORT  =  'S',  SELECT is used to
                   select eigenvalues to sort to the top left of  the
                   Schur  form.   If SORT = 'N', SELECT is not refer-
                   enced.   An  eigenvalue  WR(j)+sqrt(-1)*WI(j)   is
                   selected  if SELECT(WR(j),WI(j)) is true; i.e., if
                   either one of a complex conjugate pair  of  eigen-
                   values  is  selected,  then both are.  Note that a
                   selected complex eigenvalue may no longer  satisfy
                   SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
                   ordering may change the value  of  complex  eigen-
                   values  (especially  if  the  eigenvalue  is  ill-
                   conditioned); in this case INFO may be set to  N+3
                   (see INFO below).
    
         SENSE     (input) CHARACTER*1
                   Determines which reciprocal condition numbers  are
                   computed.  = 'N': None are computed;
                   = 'E': Computed for  average  of  selected  eigen-
                   values only;
                   = 'V': Computed for selected right invariant  sub-
                   space only;
                   = 'B': Computed for both.  If SENSE = 'E', 'V'  or
                   'B', SORT must equal 'S'.
    
         N         (input) INTEGER
                   The order of the matrix A. N >= 0.
    
         A         (input/output) REAL array, dimension (LDA, N)
                   On entry, the N-by-N matrix  A.   On  exit,  A  is
                   overwritten by its real Schur form T.
    
         LDA       (input) INTEGER
                   The leading dimension of  the  array  A.   LDA  >=
                   max(1,N).
    
         SDIM      (output) INTEGER
                   If SORT = 'N', SDIM = 0.  If SORT =  'S',  SDIM  =
                   number  of  eigenvalues  (after sorting) for which
                   SELECT is true. (Complex conjugate pairs for which
                   SELECT is true for either eigenvalue count as 2.)
    
         WR        (output) REAL array, dimension (N)
                   WI      (output) REAL array, dimension (N) WR  and
                   WI  contain  the real and imaginary parts, respec-
                   tively, of the computed eigenvalues, in  the  same
                   order that they appear on the diagonal of the out-
                   put Schur form  T.   Complex  conjugate  pairs  of
                   eigenvalues  appear  consecutively with the eigen-
                   value having the positive imaginary part first.
    
         VS        (output) REAL array, dimension (LDVS,N)
                   If JOBVS = 'V', VS contains the orthogonal  matrix
                   Z  of  Schur  vectors.   If JOBVS = 'N', VS is not
                   referenced.
    
         LDVS      (input) INTEGER
                   The leading dimension of the array VS.  LDVS >= 1,
                   and if JOBVS = 'V', LDVS >= N.
    
         RCONDE    (output) REAL
                   If  SENSE  =  'E'  or  'B',  RCONDE  contains  the
                   reciprocal condition number for the average of the
                   selected eigenvalues.  Not referenced if  SENSE  =
                   'N' or 'V'.
    
         RCONDV    (output) REAL
                   If  SENSE  =  'V'  or  'B',  RCONDV  contains  the
                   reciprocal condition number for the selected right
                   invariant subspace.  Not referenced if SENSE = 'N'
                   or 'E'.
    
         WORK      (workspace/output) REAL array, dimension (LWORK)
                   On exit, if INFO = 0, WORK(1) returns the  optimal
                   LWORK.
    
         LWORK     (input) INTEGER
                   The  dimension  of  the  array  WORK.   LWORK   >=
                   max(1,3*N).   Also,  if SENSE = 'E' or 'V' or 'B',
                   LWORK >=  N+2*SDIM*(N-SDIM),  where  SDIM  is  the
                   number  of  selected  eigenvalues computed by this
                   routine.  Note that N+2*SDIM*(N-SDIM) <=  N+N*N/2.
                   For  good  performance,  LWORK  must  generally be
                   larger.
    
         IWORK     (workspace) INTEGER array, dimension (LIWORK)
                   Not referenced if SENSE = 'N' or 'E'.
    
         LIWORK    (input) INTEGER
                   The dimension of the array IWORK.  LIWORK >= 1; if
                   SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
    
         BWORK     (workspace) LOGICAL array, dimension (N)
                   Not referenced if SORT = 'N'.
    
         INFO      (output) INTEGER
                   = 0: successful exit
                   < 0: if INFO = -i, the i-th argument had an  ille-
                   gal value.
                   > 0: if INFO = i, and i is
                   <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR  and
                   WI contain those eigenvalues which have converged;
                   if JOBVS = 'V',  VS  contains  the  transformation
                   which  reduces  A to its partially converged Schur
                   form.  = N+1: the eigenvalues could not  be  reor-
                   dered  because  some eigenvalues were too close to
                   separate (the problem is very ill-conditioned);  =
                   N+2:  after reordering, roundoff changed values of
                   some complex eigenvalues so  that  leading  eigen-
                   values   in  the  Schur  form  no  longer  satisfy
                   SELECT=.TRUE.  This could also be caused by under-
                   flow due to scaling.
    
    
    
    


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