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RedHat 9 (Linux i386) - man page for sgees (redhat section l)

SGEES(l)					)					 SGEES(l)

NAME
       SGEES  - 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 SGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,	VS,  LDVS,  WORK,  LWORK,
			 BWORK, INFO )

	   CHARACTER	 JOBVS, SORT

	   INTEGER	 INFO, LDA, LDVS, LWORK, N, SDIM

	   LOGICAL	 BWORK( * )

	   REAL 	 A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )

	   LOGICAL	 SELECT

	   EXTERNAL	 SELECT

PURPOSE
       SGEES  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 factorization
       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.  The  leading  columns  of	Z
       then  form  an  orthonormal basis for the invariant subspace corresponding to the selected
       eigenvalues.

       A 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': Eigenvalues 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 referenced.  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
	       eigenvalues is selected, then both complex eigenvalues are selected.  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 eigenvalues  (espe-
	       cially if the eigenvalue is ill-conditioned); in this case INFO is set to N+2 (see
	       INFO below).

       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 has been 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 sort-
	       ing) 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 imagi-
	       nary parts, respectively, of the computed eigenvalues in the same order that  they
	       appear on the diagonal of the output Schur form T.  Complex conjugate pairs of ei-
	       genvalues will appear consecutively with the eigenvalue having the positive imagi-
	       nary 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; if JOBVS = 'V', LDVS >= N.

       WORK    (workspace/output) REAL array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK.  LWORK	>=  max(1,3*N).   For  good  performance,
	       LWORK must generally be larger.

	       If  LWORK = -1, then a workspace query is assumed; the routine only calculates the
	       optimal size of the WORK array, returns this value as the first entry of the  WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       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 illegal 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 matrix which  reduces  A  to
	       its partially converged Schur form.  = N+1: the eigenvalues could not be reordered
	       because some eigenvalues were too close to separate (the problem is very  ill-con-
	       ditioned); = N+2: after reordering, roundoff changed values of some complex eigen-
	       values  so  that  leading  eigenvalues  in  the	Schur  form  no  longer   satisfy
	       SELECT=.TRUE.  This could also be caused by underflow due to scaling.

LAPACK version 3.0			   15 June 2000 				 SGEES(l)


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