Inverse function, Taylor's expansion and extended Schröder's processes

Abstract

Based on the Taylor's expansion of an inverse function, we extend the Schröder's process to find and improve high-order fixed-point iteration functions (IFs) for solving a nonlinear equation. We illustrate the extended processes by using them to find better iterative methods to compute the nth root and the logarithm of a strictly positive real number. IFs for inverse trigonometric function evaluations are also considered.

Keywords:

• Schröder's process
• inverse function
• Taylor's expansion
• high-order method
• nth root
• logarithm
• arcsin
• arctan

2010 AMS Subject Classifications:

• 65-01
• 11B37
• 65B99
• 65D99
• 65H05

Acknowledgements

This work has been financially supported by an individual discovery grant from NSERC (Natural Sciences and Engineering Research Council of Canada).