## AB09JX

### Check stability/antistability of finite eigenvalues

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To check stability/antistability of finite eigenvalues with
respect to a given stability domain.

```
Specification
```      SUBROUTINE AB09JX( DICO, STDOM, EVTYPE, N, ALPHA, ER, EI, ED,
\$                   TOLINF, INFO )
C     .. Scalar Arguments ..
CHARACTER        DICO, EVTYPE, STDOM
INTEGER          INFO, N
DOUBLE PRECISION ALPHA, TOLINF
C     .. Array Arguments ..
DOUBLE PRECISION ED(*), EI(*), ER(*)

```
Arguments

Mode Parameters

```  DICO    CHARACTER*1
Specifies the type of the stability domain as follows:
= 'C':  for a continuous-time system;
= 'D':  for a discrete-time system.

STDOM   CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S':  stability type domain;
= 'U':  instability type domain.

EVTYPE  CHARACTER*1
Specifies whether the eigenvalues arise from a standard
or a generalized eigenvalue problem as follows:
= 'S':  standard eigenvalue problem;
= 'G':  generalized eigenvalue problem;
= 'R':  reciprocal generalized eigenvalue problem.

```
Input/Output Parameters
```  N       (input) INTEGER
The dimension of vectors ER, EI and ED.  N >= 0.

ALPHA   (input) DOUBLE PRECISION
Specifies the boundary of the domain of interest for the
eigenvalues. For a continuous-time system
(DICO = 'C'), ALPHA is the boundary value for the real
parts of eigenvalues, while for a discrete-time system
(DICO = 'D'), ALPHA >= 0 represents the boundary value for
the moduli of eigenvalues.

ER, EI, (input) DOUBLE PRECISION arrays, dimension (N)
ED      If EVTYPE = 'S', ER(j) + EI(j)*i, j = 1,...,N, are
the eigenvalues of a real matrix.
ED is not referenced and is implicitly considered as
a vector having all elements equal to one.
If EVTYPE = 'G' or EVTYPE = 'R', (ER(j) + EI(j)*i)/ED(j),
j = 1,...,N, are the generalized eigenvalues of a pair of
real matrices. If ED(j) is zero, then the j-th generalized
eigenvalue is infinite.
Complex conjugate pairs of eigenvalues must appear
consecutively.

```
Tolerances
```  TOLINF  DOUBLE PRECISION
If EVTYPE = 'G' or 'R', TOLINF contains the tolerance for
detecting infinite generalized eigenvalues.
0 <= TOLINF < 1.

```
Error Indicator
```  INFO    INTEGER
=  0:  successful exit, i.e., all eigenvalues lie within
the domain of interest defined by DICO, STDOM
and ALPHA;
<  0:  if INFO = -i, the i-th argument had an illegal
value;
=  1:  some eigenvalues lie outside the domain of interest
defined by DICO, STDOM and ALPHA.
```
Method
```  The domain of interest for an eigenvalue lambda is defined by the
parameters ALPHA, DICO and STDOM as follows:
- for a continuous-time system (DICO = 'C'):
Real(lambda) < ALPHA if STDOM = 'S';
Real(lambda) > ALPHA if STDOM = 'U';
- for a discrete-time system (DICO = 'D'):
Abs(lambda) < ALPHA if STDOM = 'S';
Abs(lambda) > ALPHA if STDOM = 'U'.
If EVTYPE = 'R', the same conditions apply for 1/lambda.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```