dgees man page on Scientific

```DGEES(1)	      LAPACK driver routine (version 3.2)	      DGEES(1)

NAME
DGEES  -	 computes for an N-by-N real nonsymmetric matrix A, the eigen‐
values, 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 eigenval‐
ues, the real Schur form T, and, optionally, the matrix of  Schur  vec‐
tors  Z.	  This	gives the Schur factorization A = Z*T*(Z**T).  Option‐
ally, 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 diago‐
nal of the Schur form.  = 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).

SELECT  (external procedure) 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 ref‐
erenced.	 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 eigen‐
value 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 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  ei‐
genvalues  (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
(MAX(1,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 calcu‐
lates 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 con‐
tains 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	 prob‐
lem is very ill-conditioned); = N+2: after reordering, roundoff
changed values of some complex eigenvalues so that leading  ei‐
genvalues  in  the  Schur  form no longer satisfy SELECT=.TRUE.
This could also be caused by underflow due to scaling.

LAPACK driver routine (version 3November 2008			      DGEES(1)
```
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