Numerical Analysis

verfasst von: Walter Gautschi

Verlag: Birkhäuser Boston

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Revised and updated, this second edition of Walter Gautschi's successful Numerical Analysis explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Topics included in the book are presented with a view toward stressing basic principles and maintaining simplicity and teachability as far as possible, while subjects requiring a higher level of technicality are referenced in detailed bibliographic notes at the end of each chapter. Readers are thus given the guidance and opportunity to pursue advanced modern topics in more depth.

Along with updated references, new biographical notes, and enhanced notational clarity, this second edition includes the expansion of an already large collection of exercises and assignments, both the kind that deal with theoretical and practical aspects of the subject and those requiring machine computation and the use of mathematical software. Perhaps most notably, the edition also comes with a complete solutions manual, carefully developed and polished by the author, which will serve as an exceptionally valuable resource for instructors.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Machine Arithmetic and Related Matters

Abstract

The questions addressed in this introductory chapter are fundamental in the sense that they are relevant in any situation that involves numerical machine computation, regardless of the kind of problem that gave rise to these computations. In the first place, one has to be aware of the rather primitive type of number system available on computers.

Walter Gautschi

Chapter 2. Approximation and Interpolation

Abstract

The present chapter is basically concerned with the approximation of functions. The functions in question may be functions defined on a continuum – typically a finite interval –or functions defined only on a finite set of points. The first instance arises, for example, in the context of special functions (elementary or transcendental) that one wishes to evaluate as a part of a subroutine. Since any such evaluation must be reduced to a finite number of arithmetic operations, we must ultimately approximate the function by means of a polynomial or a rational function. The second instance is frequently encountered in the physical sciences when measurements are taken of a certain physical quantity as a function of some other physical quantity (such as time). In either case one wants to approximate the given function “as well as possible” in terms of other simpler functions.

Walter Gautschi

Chapter 3. Numerical Differentiation and Integration

Abstract

Differentiation and integration are infinitary concepts of calculus; that is, they are defined by means of a limit process – the limit of the difference quotient in the first instance, the limit of Riemann sums in the second. Since limit processes cannot be carried out on the computer, we must replace them by finite processes. The tools to do so come from the theory of polynomial interpolation (Chap. 2, Sect. 2.2). They not only provide us with approximate formulae for the limits in question, but also permit us to estimate the errors committed and discuss convergence.

Walter Gautschi

Chapter 4. Nonlinear Equations

Abstract

The problems discussed in this chapter may be written generically in the form but allow different interpretations depending on the meaning of x and f. The simplest case is a single equation in a single unknown, in which case f is a given function of a real or complex variable, and we are trying to find values of this variable for which f vanishes. Such values are called roots of the equation (4.1), or zeros of the function f.

Walter Gautschi

Chapter 5. Initial Value Problems for ODEs: One-Step Methods

Abstract

Initial value problems for ordinary differential equations (ODEs) occur in almost all the sciences, notably in mechanics (including celestial mechanics), where the motion of particles (resp., planets) is governed by Newton’s second law – a system of second-order differential equations.

Walter Gautschi

Chapter 6. Initial Value Problems for ODEs: Multistep Methods

Abstract

We saw in Chap. 5 that (explicit) one–step methods are increasingly difficult to construct as one upgrades the order requirement. This is no longer true for multistep methods, where an increase in order is straightforward but comes with a price: a potential danger of instability. In addition, there are other complications such as the need for an initialization procedure and considerably more complicated procedures for changing the grid length. Yet, in terms of work involved, multistep methods are still among the most attractive methods.We discuss them along lines similar to one– step methods, beginning with a local description and examples and proceeding to the global description and problems of stiffness. By the very nature of multistep methods, the discussion of stability is now more extensive.

Walter Gautschi

Chapter 7. Two-Point Boundary Value Problems for ODEs

Abstract

Many problems in applied mathematics require solutions of differential equations specified by conditions at more than one point of the independent variable. These are called boundary value problems; they are considerably more difficult to deal with than initial value problems, largely because of their global nature. Unlike (local) existence and uniqueness theorems known for initial value problems (cf. Theorem 5.0.1), there are no comparably general theorems for boundary value problems. Neither existence nor uniqueness is, in general, guaranteed.

Walter Gautschi

Backmatter

Titel: Numerical Analysis
verfasst von: Walter Gautschi
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-8259-0
Print ISBN: 978-0-8176-8258-3
DOI: https://doi.org/10.1007/978-0-8176-8259-0