Computing the determinant of a matrix with polynomial entries by approximation

TL;DR

This paper introduces a hybrid symbolic-numerical algorithm for efficiently computing determinants of matrices with polynomial entries, utilizing Newton's interpolation, degree estimation, and degree homomorphism for dimension reduction, with natural parallelization.

Contribution

It presents a novel hybrid approach combining symbolic and numerical methods, including degree estimation and homomorphism techniques, for determinant computation of polynomial matrices.

Findings

Effective algorithm with error control for polynomial determinants

Utilizes Newton's interpolation and degree homomorphism methods

Supports natural parallelization for efficiency

Abstract

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the determinant of a matrix with polynomial entries using hybrid symbolic and numerical computation. The algorithm relies on the Newton's interpolation method with error control for solving Vandermonde systems. It is also based on a novel approach for estimating the degree of variables, and the degree homomorphism method for dimension reduction. Furthermore, the parallelization of the method arises naturally.