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Numerical mathematics. II: Methods of analysis. (Numerische Mathematik. II: Methoden der Analysis.) (German) Zbl 0911.65001
This book presents an introduction to numerical analysis, thus completing the author’s earlier book on numerical linear algebra. The subjects of the book are those usually found in a basic lecture (but the author emphasizes the necessity of practical exercises to be added). The selection of the content, made by the author, follows some fundamental analytic principles.
Hence, in the beginning the book treats the ideas of error analysis by the discussion of differential error analysis. The following chapters deal with techniques of interpolation (up to cubic splines), and of extrapolation, including examples for the summation with the help of the Euler-Maclaurin formulas. The listing of some methods of numerical integration is followed by an investigation of convergence by the theorem of Banach-Steinhaus. The fixed point theorems (carefully discussed in their relation to various classes of nonlinear operators) are the starting point to study the solution of nonlinear systems, especially by certain Newton type methods.
The last chapters are concerned with the numerics of initial value and boundary value problems of ordinary differential equations. Coming from a general theory of discretization (with the famous high-light: stability and consistency give rise to convergence) Euler and general Runge-Kutta methods are discussed, including a strategy of choosing stepsizes. Shooting procedures form a transition to (also nonlinear) boundary value problems, mainly treated by difference and projection methods.
Evidently, problems arising in special situations (e.g. stiff differential equations) cannot be included in an introductory presentation. Altogether, this is a competent, transparent description of the numerical methods of analysis and - especially - the recent ideas supporting them. (However, the Chapter 4.5. on improper integrals contains a lot of interesting facts, but mostly not concerned strictly with improper integrals. Unfortunately, Chapter 4.5.5. gives only a single hint to valuable methods for such integrals).

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65B15 Euler-Maclaurin formula in numerical analysis
65H10 Numerical computation of solutions to systems of equations
65Lxx Numerical methods for ordinary differential equations