Newton's Method in Three Precisions

TL;DR

This paper introduces a three-precision variant of Newton's method that combines different precision levels for efficiency, analyzing its convergence and demonstrating its effectiveness through an example.

Contribution

It proposes a novel multi-precision Newton method with in-depth analysis and practical strategies for handling ill-conditioned problems.

Findings

Number of nonlinear iterations remains similar to double precision for well-conditioned Jacobians.

Low precision factorization can serve as an effective preconditioner in ill-conditioned cases.

The method achieves efficient convergence with mixed precision computations.

Abstract

We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with iterative refinement with a factorization in half precision. We analyze the method as an inexact Newton method. This analysis shows that, except for very poorly conditioned Jacobians, the number of nonlinear iterations needed is the same that one would get if one stored and factored the Jacobian in double precision. In many ill-conditioned cases one can use the low precision factorization as a preconditioner for a GMRES iteration. That approach can recover fast convergence of the nonlinear iteration. We present an example to illustrate the results.