Newton Method on Riemannian Manifolds: Covariant Alpha-Theory

TL;DR

The paper extends Smale's $\alpha$-theory to intrinsic Newton methods on complete analytic Riemannian manifolds, for zeros of analytic maps $f:\mathbb{M}_n\to\mathbb{R}^n$ and vector fields $X:\mathbb{M}_n\to T\mathbb{M}_n$. By introducing invariants $\beta$, $\gamma$ (with $\alpha=\beta\gamma$), a curvature measure $K_\zeta$, and the injectivity radius ${\bf r}_\zeta$, it derives R-$\gamma$ and R-$\alpha$ theorems that guarantee quadratic convergence of Newton iterations within explicitly computable basins. The results specialize to familiar manifolds (e.g., $\mathbb{S}^n$, $\mathbb{O}_n$, $\mathbb{P}_n(\mathbb{R})$, Hermitian manifolds) with concrete radii, and also yield zero-separation and singular-locus bounds. An alternative formulation and comparisons are discussed, along with implementation considerations and future research directions for geometry-respecting Newton methods.

Abstract

In this paper we study quantitative aspects of Newton method for finding zeros of mappings f: M_n -> R^n and vector fields X: M_x -> TM_n

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Remark 1.1