A Reliable Algorithm to Compute the Spectral Set of a Polytope of Polynomials to Prescribed Accuracy

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.