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Complex interval arithmetic and its applications. (English) Zbl 0911.65038
Mathematical Research. 105. Weinheim: Wiley-VCH. 284 p. DM 148,00; öS 1.080,00; sFr. 131,00 (1998).

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.

Basic tool for these methods is the already legendary concept of a centered form as proposed by 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.

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.

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.

65G30Interval and finite arithmetic
65H05Single nonlinear equations (numerical methods)
12Y05Computational aspects of field theory and polynomials
30C15Zeros of polynomials, etc. (one complex variable)
65E05Numerical methods in complex analysis
65Y05Parallel computation (numerical methods)
30E20Integration, integrals of Cauchy type, etc. (one complex variable)
65-02Research monographs (numerical analysis)