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1991 | Buch

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Numerical Mathematics

verfasst von: Günther Hämmerlin, Karl-Heinz Hoffman

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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"In truth, it is not knowledge, but learning, not possessing, but production, not being there, but travelling there, which provides the greatest pleasure. When I have completely understood something, then I turn away and move on into the dark; indeed, so curious is the insatiable man, that when he has completed one house, rather than living in it peacefully, he starts to build another. " Letter from C. F. Gauss to W. Bolyai on Sept. 2, 1808 This textbook adds a book devoted to applied mathematics to the series "Grundwissen Mathematik. " Our goals, like those of the other books in the series, are to explain connections and common viewpoints between various mathematical areas, to emphasize the motivation for studying certain prob lem areas, and to present the historical development of our subject. Our aim in this book is to discuss some of the central problems which arise in applications of mathematics, to develop constructive methods for the numerical solution of these problems, and to study the associated questions of accuracy. In doing so, we also present some theoretical results needed for our development, especially when they involve material which is beyond the scope of the usual beginning courses in calculus and linear algebra. This book is based on lectures given over many years at the Universities of Freiburg, Munich, Berlin and Augsburg.

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Inhaltsverzeichnis

Frontmatter

1. Computing

Abstract

As we have already mentioned in the preface to this book, we consider numerical analysis to be the mathematics of constructive methods which can be realized numerically. Thus, one of the problems of numerical analysis is to design computer algorithms for either exactly or approximately solving problems in mathematics itself, or in its applications in the natural sciences, technology, economics, and so forth. Our aim is to design algorithms which can be programmed and run on a calculator or digital computer. The key to this approach is to have an appropriate way of representing numbers which is compatible with the physical properties of the memory of the computer. In a practical computer, each number can only be stored to a finite number of digits, and thus some way of rounding off numbers is needed. This in turn implies that for complicated algorithms, there may be an accumulation of errors, and hence it is essential to have a way of performing an error analysis of our methods. In this connection there are several different kinds of error types, which in addition to the roundoff error mentioned above, include data error and method error.

Günther Hämmerlin, Karl-Heinz Hoffman

2. Linear Systems of Equations

Abstract

Many problems in mathematics lead to linear systems of equations. In fact, in using computers to solve such problems, we frequently encounter very large linear systems. Thus, the development of efficient algorithms to solve such systems is of central importance in numerical analysis. We differentiate between two types of methods. Direct methods solve the problem in a finite number of steps, and so are not subject to method error, although, of course, the results can be very badly affected by roundoff error. Indirect methods seek to find the solution by iteration, and thus usually lead only to approximate solutions since the iteration has to be stopped at some point. Although in this case we have both method and roundoff errors, iterative methods have their advantages. In this chapter we will primarily discuss direct methods. Iterative methods for linear system of equations will be discussed in Chapter 8.

Günther Hämmerlin, Karl-Heinz Hoffman

3. Eigenvalues

Abstract

In Chapter 2 we have seen that in order to compute a singular-value decomposition of a matrix A, we need to have the eigenvalues of A^T A. This process was illustrated in Example 2.6.3, where because of the small size of the problem, we were able to find the necessary eigenvalues by hand calculation. For larger problems, however, this is no longer possible, and we need to use a computer to find eigenvalues. Such problems arise, for example, in the study of oscillations, where the eigenfrequences are to be determined by discretizing the associated differential equation. In this chapter we discuss various methods for computing eigenvalues of matrices.

Günther Hämmerlin, Karl-Heinz Hoffman

4. Approximation

Abstract

In Chapters 2 and 3 we have discussed methods of numerical linear algebra. In this chapter we turn to another central question in numerical analysis, the approximation of mathematical objects. The methods presented here have a wide variety of applications.

Günther Hämmerlin, Karl-Heinz Hoffman

5. Interpolation

Abstract

The process of constructing a function which takes on given data values at given data points is called interpolation. In a certain sense, interpolation is a special case of discrete approximation, but the subject deserves a separate and detailed treatment. The results of the theory of interpolation are a basic part of the constructive theory of functions, and moreover, provide the basis for a wide variety of methods for numerical integration, numerical treatment of differential equations, and the discretization of general operator equations.

Günther Hämmerlin, Karl-Heinz Hoffman

6. Splines

Abstract

A spline is a function which is piecewise defined on intervals such that the pieces are joined together smoothly. The terminology was introduced by I. J. Schoenberg [1946], although these kinds of functions had been used earlier by several other authors. For example, the Euler method for constructing a piecewise polynomial approximation to the solution of an initial-value problem for ordinary differential equations (and which is often used to establish the Peano Theorem on the existence of solutions of such problems) can be regarded as a simple application of splines. In this regard we should also mention the papers of C. Runge [1901], W. Quade and L. Collatz [1938], J. Favard [1940] and R. Courant [1943], among others. The theory of splines is a good example of an area in mathematics which was developed in response to practical needs. One of the early problems which gave impetus to the development of splines was the need for usuable methods for constructing smooth approximations on the basis of tabulated data arising in ballistics. The subject has steadily developed over the past thirty years, and at present there are several thousand research papers on splines and their applications. In view of this large literature, it is clear that within the framework of this book, we will only be able to give an introduction to a part of the theory. Our discussion will focus on splines constructed from polynomial pieces.

Günther Hämmerlin, Karl-Heinz Hoffman

7. Integration

Abstract

The numerical computation of definite integrals is one of the oldest problems in mathematics. The problem, which in its earliest form involved finding the area of regions bounded by curved lines, has been around for thousands of years, long before the concept of integrals in the framework of analysis was developed in the 17th and 18th century. Certainly the best-known example of this problem was that of computing the area contained in a circle, which in turn led to a study of the number π and its computation. Using a numerical method involving the approximation of a circle by inscribed and circumscribed polygons, Archimedes (287–212 B.C.) was able to give the astonishingly good bounds \(3\frac{{10}}{{71}} < \pi < 3\frac{1}{7}\). For more on this, see Chap. 5 of the book Numbers by H.-D. Ebbinghaus, et.al. [1990].

Günther Hämmerlin, Karl-Heinz Hoffman

8. Iteration

Abstract

The problem of solving an equation or a system of equations is one of the basic problems in mathematics and its applications. This problem can be formulated as follows: Find a solution x of the operator equation Fx = 0 in a given normed linear space (X, || · ||). Here the operator F is a mapping F: D → X, D ⊂ X. An element ξ ∈ D for which F ξ = 0 holds is called a zero of F.

Günther Hämmerlin, Karl-Heinz Hoffman

9. Linear Optimization

Abstract

“Since everything in the entire world is best possible, and since it is all the work of the wisest of creators, there is nothing in this world which is not blessed with either a maximum or minimum property. Thus, there can be no doubt that all of our worldly processes can as easily be derived by the method of maxima and minima as from their basic properties themselves.” This observation of Leonhard Euler — freely translated from an article in Commentationes Mechanicae — makes crystal clear the key role that the problem of maximizing or minimizing a function plays in mathematics and its applications. In this chapter we will restrict our attention primarily to the special case of linear functions with linear side conditions. Nevertheless, the theory presented here has many applications, since in fact there are a huge number of natural problems which are linear, and, moreover, nonlinear problems can often be linearized. A central role in our considerations will be played by the simplex method, one of the most used methods in all of numerical analysis.

Günther Hämmerlin, Karl-Heinz Hoffman

Backmatter

Titel: Numerical Mathematics
verfasst von: Günther Hämmerlin
Karl-Heinz Hoffman
Copyright-Jahr: 1991
Verlag: Springer New York
Electronic ISBN: 978-1-4612-4442-4
Print ISBN: 978-0-387-97494-1
DOI: https://doi.org/10.1007/978-1-4612-4442-4

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