Acceleration of multiple precision solver for ill-conditioned algebraic equations with lower precision eigensolver

TL;DR

This paper compares direct iterative methods and eigensolvers for solving ill-conditioned algebraic equations, demonstrating that iterative methods can outperform eigensolvers in certain cases despite higher precision requirements.

Contribution

It provides a numerical comparison showing that direct iterative methods can be more efficient than eigensolvers for specific ill-conditioned problems.

Findings

Iterative methods outperform eigensolvers in Wilkinson's example.

Iterative methods are more efficient for Chebyshev quadrature problems.

Numerical results support the advantage of iterative methods in ill-conditioned cases.

Abstract

There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the problem solved via eigensolver (eigenvalue method) is well-conditioned if the corresponding companion matrix has its small condition number. However, directly solving them with Newton or simultaneous iteration methods (direct iterative method for short) should be considered as ill-conditioned because of increasing density of its root distribution. Although a greater number of mantissa of floating-point arithmetic is necessary in the direct iterative method than eigenvalue method, the total computational costs cannot obviously be determined. In this study, we target Wilkinson's example and Chebyshev quadrature problem as examples of ill-conditioned algebraic equations, and demonstrate some concrete numerical results to prove that the direct iterative method can perform better than standard eigensolver.