Pt\'ak's nondiscrete induction and its application to matrix iterations

TL;DR

This paper explores Pták's nondiscrete induction method, demonstrating its effectiveness in deriving tight convergence estimates for matrix iterative processes like polar decomposition and square root calculations.

Contribution

It applies nondiscrete induction to matrix iterations, providing sharp convergence estimates and illustrating the method's utility in numerical linear algebra.

Findings

Convergence estimates are tight throughout the iteration.

Analytical proof of estimate sharpness for polar decomposition.

Numerical examples confirm theoretical results.

Abstract

Vlastimil Pt\'ak's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence estimates that are tight throughout the iteration rather than just asymptotically. In this paper we motivate and prove a theorem on nondiscrete induction originally due to Potra and Pt\'ak, and we apply it to the Newton iterations for computing the matrix polar decomposition and the matrix square root. Our goal is to illustrate the application of the method of nondiscrete induction in the finite dimensional numerical linear algebra context. We show the sharpness of the resulting convergence estimate analytically for the polar decomposition iteration and for special cases of the square root iteration, as well as on some numerical examples for the square root iteration. We also discuss some of the method's limitations and possible extensions.