On the convergence of two-step modified Newton method for nonsymmetric algebraic Riccati equations from transport theory

TL;DR

This paper analyzes the convergence of a two-step modified Newton method for solving nonsymmetric algebraic Riccati equations from transport theory, demonstrating its effectiveness and efficiency in large-scale, nearly singular cases.

Contribution

It provides a convergence analysis of the two-step modified Newton method, including for singular Jacobians, and shows its computational advantages over existing methods.

Findings

Monotonic convergence under mild assumptions

Effective for nearly singular, large-scale problems

Reduces computation time significantly

Abstract

This paper is concerned with the convergence of a two-step modified Newton method for solving the nonlinear system arising from the minimal nonnegative solution of nonsymmetric algebraic Riccati equations from neutron transport theory. We show the monotonic convergence of the two-step modified Newton method under mild assumptions. When the Jacobian of the nonlinear operator at the minimal positive solution is singular, we present a convergence analysis of the two-step modified Newton method in this context. Numerical experiments are conducted to demonstrate that the proposed method yields comparable results to several existing Newton-type methods and that it brings a significant reduction in computation time for nearly singular and large-scale problems.