Taylor Models and Computaions in the Complex Plane

Abstract

In this note we show how Taylor model methods can successfully alleviate the problems of conventional interval arithmetic in the complex plane being susceptible to significant overestimations caused by the dependency problems in the elementary operations. The use of Taylor models on the other hand results in self-validated methods that fully utilize the rich structure of the complex numbers.

To show how high order methods can be used for self-validated computations in the complex plane, we extend real-valued Taylor model methods to complex ones. The extension provides the tools to compute enclosures of the results of elementary operations and standard functions. We show how the new methods provide sharp and validated descriptions of image sets of analytic functions even after extended computations.

A. Ovsyannikov, M. Berz, Bulletin of St. Petersburg University, 10,4 (2009) 172-185

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