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2006 | Buch

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Scientific Computing with MATLAB and Octave

Second Edition

verfasst von: Alfio Quarteroni, Fausto Saleri

Verlag: Springer Berlin Heidelberg

Buchreihe : Texts in Computational Science and Engineering

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

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Über dieses Buch

Preface to the First Edition This textbook is an introduction to Scienti?c Computing. We will illustrate several numerical methods for the computer solution of c- tain classes of mathematical problems that cannot be faced by paper and pencil. We will show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of di?erential equations. With this aim, in Chapter 1 we will illustrate the rules of the game that computers adopt when storing and operating with realand complex numbers, vectors and matrices. In order to make our presentation concrete and appealing we will 1 adopt the programming environment MATLAB as a faithful c- panion. We will gradually discover its principal commands, statements and constructs. We will show how to execute all the algorithms that we introduce throughout the book. This will enable us to furnish an - mediate quantitative assessment of their theoretical properties such as stability, accuracy and complexity. We will solve several problems that will be raised through exercises and examples, often stemming from s- ci?c applications.

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Inhaltsverzeichnis

Frontmatter

1. What can’t be ignored

Abstract

In this book we will systematically use elementary mathematical concepts which the reader should know already, yet he or she might not recall them immediately.

Alfio Quarteroni, Fausto Saleri

2. Nonlinear equations

Abstract

Computing the zeros of a real function f (equivalently, the roots of the equation f(x) = 0) is a problem that we encounter quite often in scienti fic computing. In general, this task cannot be accomplished in a finite number of operations. For instance, we have already seen in Section 1.4.1 that when f is a generic polynomial of degree greater than four, there do not exist explicit formulae for the zeros. The situation is even more difficult when f is not a polynomial.

Alfio Quarteroni, Fausto Saleri

3. Approximation of functions and data

Abstract

Approximating a function f consists of replacing it by another function f of simpler form that may be used as its surrogate. This strategy is used frequently in numerical integration where, instead of computing ∫ ^b_a f(x)dx, one carries out the exact computation of ∫ ^b_a f(x)dx, f being a function simple to integrate (e.g. a polynomial), as we will see in the next chapter. In other instances the function f may be available only partially through its values at some selected points. In these cases we aim at constructing a continuous function f that could represent the empirical law which is behind the finite set of data. We provide some examples which illustrate this kind of approach.

Alfio Quarteroni, Fausto Saleri

4. Numerical differentiation and integration

Abstract

In this chapter we propose methods for the numerical approximation of derivatives and integrals of functions. Concerning integration, quite often for a generic function it is not possible to find a primitive in an explicit form. Even when a primitive is known, its use might not be easy.

Alfio Quarteroni, Fausto Saleri

5. Linear systems

Abstract

In applied sciences, one is quite often led to face a linear system of the form

Alfio Quarteroni, Fausto Saleri

6. Eigenvalues and eigenvectors

Abstract

Given a square matrix A ∈ℂ^n×n, the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector x such that

Alfio Quarteroni, Fausto Saleri

7. Ordinary differential equations

Abstract

A differential equation is an equation involving one or more derivatives of an unknown function. If all derivatives are taken with respect to a single independent variable we call it an ordinary differential equation, whereas we have a partial differential equation when partial derivatives are present.

Alfio Quarteroni, Fausto Saleri

8. Numerical methods for (initial-)boundary-value problems

Abstract

Boundary-value problems are differential problems set in an interval (a, b) of the real line or in an open multidimensional region Ω ⊂ ℝ^d (d = 2, 3) for which the value of the unknown solution (or its derivatives) is prescribed at the end-points a and b of the interval, or on the boundary ∂Ω of the multidimensional region.

Alfio Quarteroni, Fausto Saleri

9. Solutions of the exercises

Abstract

Solution 1.1 Only the numbers of the form ±0.1a₂ · 2^e with a₂ = 0,1 and e = ±2,±1,0 belong to the set F(2, 2,−2, 2). For a given exponent, we can represent in this set only the two numbers 0.10 and 0.11, and their opposites. Consequently, the number of elements belonging to F(2, 2,−2, 2) is 20. Finally, ∈M = ½.

Alfio Quarteroni, Fausto Saleri

Backmatter

Titel: Scientific Computing with MATLAB and Octave
verfasst von: Alfio Quarteroni
Fausto Saleri
Copyright-Jahr: 2006
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-32613-7
Print ISBN: 978-3-540-32612-0
DOI: https://doi.org/10.1007/3-540-32613-8

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