DGEES man page on IRIX

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DGEES(3F)							     DGEES(3F)

NAME
     DGEES - 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 DGEES( 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( * )

	 DOUBLE	       PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( *
		       ), WR( * )

	 LOGICAL       SELECT

	 EXTERNAL      SELECT

PURPOSE
     DGEES 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).

									Page 1

DGEES(3F)							     DGEES(3F)

     SELECT  (input) LOGICAL FUNCTION of two DOUBLE PRECISION 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 (especially 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) DOUBLE PRECISION 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 sorting) for which SELECT is true. (Complex
	     conjugate pairs for which SELECT is true for either eigenvalue
	     count as 2.)

     WR	     (output) DOUBLE PRECISION array, dimension (N)
	     WI	     (output) DOUBLE PRECISION array, dimension (N) WR and WI
	     contain the real and imaginary 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
	     eigenvalues will appear consecutively with the eigenvalue having
	     the positive imaginary part first.

     VS	     (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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.

									Page 2

DGEES(3F)							     DGEES(3F)

     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-conditioned); = N+2: after reordering, roundoff
	     changed values of some complex eigenvalues so that leading
	     eigenvalues in the Schur form no longer satisfy SELECT=.TRUE.
	     This could also be caused by underflow due to scaling.

									Page 3

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