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2014 | Buch

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Scientific Computing with MATLAB and Octave

verfasst von: Alfio Quarteroni, Fausto Saleri, Paola Gervasio

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

This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer-based solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros, the extrema, and the integrals of continuous functions, solve linear systems, approximate functions using polynomials and construct accurate approximations for the solution of ordinary and partial differential equations. To make the format concrete and appealing, the programming environments Matlab and Octave are adopted as faithful companions. The book contains the solutions to several problems posed in exercises and examples, often originating from important applications. At the end of each chapter, a specific section is devoted to subjects which were not addressed in the book and contains bibliographical references for a more comprehensive treatment of the material.

From the review:

".... This carefully written textbook, the third English edition, contains substantial new developments on the numerical solution of differential equations. It is typeset in a two-color design and is written in a style suited for readers who have mathematics, natural sciences, computer sciences or economics as a background and who are interested in a well-organized introduction to the subject." Roberto Plato (Siegen), Zentralblatt MATH 1205.65002.

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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, Paola Gervasio

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 Scien- tific Computing. In general, this task cannot be accomplished in a finite number of operations. For instance, we have already seen in Section 1.5.1 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, Paola Gervasio

3. Approximation of functions and data

Abstract

Approximating a function f consists of replacing it by another function \(\tilde{f}\) of simpler form that may be used as its surrogate.

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

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, Paola Gervasio

5. Linear systems

Abstract

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

$$\mathrm{A}\mathrm{x}=\mathrm{b}$$

, where A is a square matrix of dimension \(n\times n\) whose elements aij are either real or complex, while x and b are column vectors of dimension n: x represents the unknown solution while b is a given vector.

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

6. Eigenvalues and eigenvectors

Abstract

Given a square matrix \(\mathrm{A}\in\mathbb{C}^{n\times n}\), the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector x such that

$$\mathrm{A}\mathrm{x}=\lambda\mathrm{x}$$

Any such λ is called an eigenvalue of A, while x is the associated eigenvector.

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

7. Numerical optimization

Abstract

Let \(f:\mathbb{R}^n\rightarrow\mathbb{R},n\geq1\), be a function that we call cost function or objective function.

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

8. 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, Paola Gervasio

9. Numerical approximation of 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 \(\it\Omega\subset\mathbb{R}^d(d=\mathrm{2,3})\) 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 \(\partial\it\Omega\) of the multidimensional region.

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

10. Solutions of the exercises

Abstract

In this chapter we will provide solutions of the exercises that we have proposed at the end of the previous eight chapters. The expression “Solution n.m” is an abridged notation for “Solution of Exercise n.m” (mth Exercise of the nth Chapter).

Alfio Quarteroni, Fausto Saleri, Paola Gervasio

Backmatter

Titel: Scientific Computing with MATLAB and Octave
verfasst von: Alfio Quarteroni
Fausto Saleri
Paola Gervasio
Copyright-Jahr: 2014
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-45367-0
Print ISBN: 978-3-642-45366-3
DOI: https://doi.org/10.1007/978-3-642-45367-0

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