SYMPOSIUM MINISYMPOSIA

ABSTRACT

Multivariate polynomial systems pervade engineering design, modeling and simulation. Algorithms for solving these systems form a a fundamental computational kernel in science and engineering. In the last 15 years, new theoretical ideas and computational methodologies have emerged. The current paradigm for polynomial system solving treats symbolic processing as a preparatory step that transforms a multivariate polynomial system into a set of related univariate polynomial equations amenable to numerical methods, which are then passed to a numeric zerofinding process.

The resulting univariate polynomials are often in high-degree, whose zeros are notoriously sensitive to changes in the coefficients, causing problems for available zero-finding software. We will present how this sensitivity depends on the polynomial representation using the notion of pseudozeros of polynomials, and how the algebraic characterization of polynomial pseudozero sets from the power basis is extended to general bases. We will show that for a polynomial, the numerical conditions of its values and zeros are closely related and can be visualized simultaneously by its pseudozero sets. Comparing the pseudozero sets on a set of test polynomials in the power, Taylor, Chebyshev and Bernstein bases reveals that appropriate representation of polynomials gives rise to locally well-conditioned zeros, which then leads to an Iterative Refiment Algorithm. The algorithm has shown a significant reduction of computational errors for polynomial zeros located in the region of interest.

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