We propose an algorithm to compute the spectral set of a polytope of polynomials. The proposed algorithm offers several key guarantees that are not available with existing techniques. It guarantees that the generated spectral set: (i) contains all the actual points, (ii) is computed to a prescribed accuracy, (iii) is computed reliably in face of all kinds of computational errors, and (iv) is computed in a finite number of algorithmic iterations. A further merit is that the computational complexity of the proposed algorithm is $O(n)$ in contrast to $O(n2)$ for existing techniques, where $n$ is the degree of the polynomial. The algorithm is demonstrated on a few examples.

1.
Bhattacharyya
,
S. P.
,
Chapellat
,
H.
, and
Keel
,
L. H.
, 1995,
Robust Control—The Parametric Approach
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
2.
Barmish
,
B. R.
, and
Tempo
,
R.
, 1991, “
On the Spectral Set for a Family of Polynomials
,”
IEEE Trans. Autom. Control
0018-9286,
36
, pp.
111
115
.
3.
Cerone
,
V.
, 1997, “
A Fast Technique for the Generation of the Spectral Set of a Polytope of Polynomials
,”
Automatica
0005-1098,
33
(
2
), pp.
277
280
.
4.
Hwang
,
C.
, and
Chen
,
J.-J.
, 1999, “
Plotting Robust Root Loci for Linear Systems With Multilinearly Parametric Uncertainties
,”
Int. J. Control
0020-7179,
72
(
6
), pp.
501
511
.
5.
Yang
,
S.-F.
, and
Hwang
,
C.
, 2001, “
Generation of Robust Root Loci for Linear Systems With Parametric Uncertainties in an Ellipsoid
,”
Int. J. Control
0020-7179,
74
(
15
),
1483
1491
.
6.
Moore
,
R. E.
, 1979,
Methods and Applications of Interval Analysis
,
SIAM
,
.
7.
Moore
,
R. E.
, 1966,
Interval Analysis
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
8.
Klatte
,
R.
,
Kulisch
,
U.
,
Neaga
,
M.
,
Ratz
,
D.
, and
Ullrich
,
Ch.
, 1993,
PASCAL-XSC Language Reference With Examples
,
Springer-Verlag
,
Berlin
.
9.
Nataraj
,
P. S. V.
, and
Sheela
,
S.
, 2002, “
A New Subdivision Strategy for Range Computations
,”
Reliable Comput.
,
8
, pp.
1
10
.
10.
Cormen
,
T. H.
,
Leiserson
,
C. E.
, and
Rivest
,
R. L.
, 2001,
Introduction to Algorithms
,
Prentice-Hall of India
,
New Delhi
.
11.
Henrici
,
P.
, 1974,
Applied and Computational Complex Analysis
,
Wiley
,
New York
, Vol.
1
.
12.
Barve
,
J. J.
, 2003, “
Interval Methods for Analysis of Linear and Nonlinear Control Systems
,” Ph.D. thesis, IIT, Mumbai.
13.
Rump
,
S. M.
, 1999, “
INTLAB—Interval Laboratory
,”
Developments in Reliable Computing
,
T.
Csendes
, ed.
Kluwer
,
Dordrecht
.
14.
Ratschek
,
H.
, 1985, “
Inclusion Functions and Global Optimization
,”
Math. Program.
0025-5610,
33
, pp.
300
317
.
15.
Alefeld
,
G.
, and
Herzberger
,
J.
, 1983,
Introduction to Interval Computations
,