Damped Newton's Method on Riemannian Manifolds

TL;DR

This paper introduces a damped Newton's method for finding singularities of vector fields on Riemannian manifolds, demonstrating its global convergence and superior performance over classical Newton's method through theoretical analysis and numerical experiments.

Contribution

It presents a novel damped Newton's method for Riemannian manifolds with proven global convergence and improved efficiency compared to traditional methods.

Findings

The method converges globally and superlinearly.

After finite steps, it aligns with the classical Newton's method.

Numerical experiments show fewer iterations and less computational time.

Abstract

A damped Newton's method to find a singularity of a vector field in Riemannian setting is presented with global convergence study. It is ensured that the sequence generated by the proposed method reduces to a sequence generated by the Riemannian version of the classical Newton's method after a finite number of iterations, consequently its convergence rate is superlinear/quadratic. Moreover, numerical experiments illustrate that the damped Newton's method has better performance than Newton's method in number of iteration and computational time.