Singular algebraic equations with empirical data

TL;DR

This paper introduces a low-rank Newton's iteration method that effectively regularizes and solves singular algebraic equations with empirical data, maintaining accuracy within data error bounds.

Contribution

It presents a novel low-rank Newton's iteration technique for solving singular algebraic equations from empirical data, addressing challenges of nonisolated solutions and data inaccuracies.

Findings

Effective regularization of singular equations

Accurate solutions within data error bounds

Applications to linear systems, GCD, eigenvalues

Abstract

Singular equations with rank-deficient Jacobians arise frequently in algebraic computing applications. As shown in case studies in this paper, direct and intuitive modeling of algebraic problems often results in nonisolated singular solutions. The challenges become formidable when the problems need to be solved from empirical data of limited accuracy. A newly discovered low-rank Newton's iteration emerges as an effective regularization mechanism that enables solving singular equations accurately with an error bound in the same order as the data error. This paper elaborates applications of new methods on solving singular algebraic equations such as singular linear systems, polynomial GCD and factorizations as well as matrix defective eigenvalue problems.