Checking properties of polynomials

Abstract

In this paper we show how to construct efficient checkers for programs that supposedly compute properties of polynomials. The properties we consider are roots, norms, and other analytic/algebraic functions of polynomials. In our model, both the program II and the polynomial p are available to the checker each as a black box. We show how to check programs that compute a specific root (e.g., the largest) or a subset of roots of the given polynomial.

The checkers, in addition to never computing the root(s) themselves, strive to minimize both the running time (preferably o(deg² p)) and the number of black box evaluations of p (preferably o(degp)). We obtain deterministic checkers when a separation bound between the roots is known and probabilistic checkers when the roots can be arbitrarily close. We then extend the checkers to handle the situations when the program II returns an approximation to the root and when the evaluation of the polynomial p is approximate. Our results translate into efficient checkers for matrix spectra computations both in the exact and approximate settings, operating in the library model of [BLR93]. Next we show that the usual characterization of norms using the triangle inequality is not suited for self-testing in the exact case, but surprisingly, could be used in the approximate case.

Our results are complementary to most of the existing results on testing polynomials. The testers in the latter have the goal of determining whether a program computes a polynomial of given degree, whereas we are interested in checking the properties of a given polynomial.

This work was done while the first, second, and fourth authors were visiting Sandia National Labs. The second and fourth authors are also supported by the NSF Career Award CCR-9624552, the Alfred P. Sloan Research Award, and the NSF grant DMI-91157199.