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Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros and analytic fuctions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computational complex analysis that lead to a rich blend of mathematics and numerical analysis. This book treats these four problems in a unified way. It contains not only theoretical results (based on formal orthogonal polynomials or rational interpolation) but also numerical analysis and algorithmic aspects, implementation heuristics, and polished software (the package ZEAL) that is available via the CPC Program Library. Graduate studets and researchers in numerical mathematics will find this book very readable.
List of contents
Zeros of analytic functions: Introduction.- Formal orthogonal polynomials.- An accurate algorithm to compute zeros of FOPs.- Numerical examples.- The software package ZEAL.- A derivative-free approach.-Clusters of zeros of analytic functions: How to abtain the centre of a cluster and its weight.- A numerical example.- Rational interpolation at roots of unity.- More numerical examples.-Zeros and ples of meromorphic functions: Introduction.- Theoretical considerations and numerical algorithm.- A numerical example.-Systems of analytic equations: Introduction.- A multidimensional logarithmic residue formula.- The algorithm.- Numerical examples.
