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Zbl 0911.65038
Petković, Miodrag S.; Petković, Ljiljana D.
Complex interval arithmetic and its applications. (English)
[B] Mathematical Research. 105. Weinheim: Wiley-VCH. 284 p. DM 148,00; öS 1.080,00; sFr. 131,00 (1998). ISBN 3-527-40134-2/pbk

The aim of this monography is to present formulas and methods for complex interval arithmetic and its applications to complex analysis, i.e., techniques for obtaining outer estimates for the range of complex functions over certain domains. Thus, the two main streams of complex interval arithmetic, that is, the circular and the rectangular form, are introduced in Chapter 1. In Chapter 2, methods to obtain outer approximations of the range of a complex function over a disk are developed.\par Basic tool for these methods is the already legendary concept of a centered form as proposed by {\it R. E. Moore} [Interval Analysis (1966; Zbl 0176.13301)]. The isotony property of disk-valued functions is investigated extensively as its importance to all kinds of outer approximations is far-reaching. Outer circular approximations with minimum diameter are constructed for some standard transcendental functions explicitly in Chapter 3.\par In Chapter 4, procedures for enclosing single zeros of polynomials by disks or rectangles which converge to the zeros are described. Iterative methods for obtaining inclusions of all polynomial zeros simultaneously are developed in Chapters 5 and 6. Techniques how to implement these methods on parallel computers are then discussed in Chapter 7. In the next chapter, it is shown how circular interval arithmetic can be used as a means for an error analysis of iterative methods for solving single complex equations. Finally, the computation of curvilinear integrals in the presence of errors or uncertain data is considered in Chapter 9.\par The monography gives an excellent systematic survey of complex interval analysis. It can be used as textbook as well as a reference source. The book is appropriate for mathematicians, engineers, physicists and computer scientists. The mathematical background which is necessary for an understanding of the book is settled between the undergraduate and graduate level. The book is well written and organized, it contains many helpful explanations, and one can find numerous historical remarks and good motivations. The standard of the print quality is very high. Alltogether, it is a pleasure to read this book.
[H.Ratschek (Düsseldorf)]

MSC 2000:: *65G30 Interval and finite arithmetic
65H05 Single nonlinear equations (numerical methods)
12Y05 Computational aspects of field theory and polynomials
30C15 Zeros of polynomials, etc. (one complex variable)
65E05 Numerical methods in complex analysis
65Y05 Parallel computation (numerical methods)
30E20 Integration (one complex variable)
65-02 Research monographs (numerical analysis)

Keywords: textbook; parallel computation; circular interval arithmetic; rectangular interval arithmetic; iterative methods; complex interval arithmetic; range of complex functions; centered form; zeros of polynomials; error analysis; curvilinear integrals

Citations: Zbl 0176.13301

Cited in: Zbl 0987.65027

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