Principles for Testing Polynomial Zerofinding Programs

Abstract

The state-of-the-art in polynomial zero finding algorithms and programs is briefly summarized.Our focus in this paper is on principles for testing such programs. We view testing as requiring The results of part of this testing are described in this paper.

Bounds on a polynomial

“…Following [8], empirical analysis of the algorithm was done by generating the power coefficients with a random number generator, a different distribution being used for the mantissa and exponent, respectively. The Cauchy criterion [9] was used to obtain a root bounding interval, which was then mapped to [0,1] followed by a change of basis to the Bernstein form.…”

“…As in Cargo and Shisha [1] and Rivlin [2], the algorithm makes use of the Bernstein form of the polynomial, and like Collins and Akritas [10], the method utilizes the variation-diminishing properties of polynomials combined with a bisection technique. A bound on the complexity is derived and analysis of the algorithm performed as suggested in [8].…”

An Efficient Reliable Algorithm for the Approximation of All Polynomial Roots Based on the Method of D. Bernoulli

“…The essential theoretical results were published in [1]. But many of the facts which let the corresponding rootfinding program fulfill the criteria of [2] have been developed and tested by D. Gunsthövel who planned to write a dissertation in this field. For example, in Theorem 1 he has found the explicit a priori lower bound for the number of steps needed to get the desired accuracy depending on a fixed upper bound of | 1 / 2 |.…”

“…[3]). Therefore, only some supplementary remarks shall help to compare the SPA with efficient reliable methods already known.…”

Pseudozeros of polynomials and pseudospectra of companion matrices

“…The idea of considering polynomials with random coefficients of this form to test the quality of zerofinding algorithms was proposed by Jenkins and Traub in 1974 [9]. We should note that the conditions of the balanced companion matrix eigenvalue problem and the coefficientwise perturbed zerofinding problem are closer for some polynomials than others.…”