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Über dieses Buch

"It is important to prove, is more but it important to improve." This textbook is an introduction to Scientific Computing. We will illustrate several numerical methods for the computer solution of cer­ 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 differential equations. With this aim, in Chapter 1 we will illustrate the rules of the game that computers adopt when storing and operating with real and complex numbers, vectors and matrices. In order to make our presentation concrete and appealing we will adopt the programming environment MATLAB ® 1 as a faithful companion. We will gradually discover its principal commands, statements and con­ structs. We will show how to execute all the algorithms that we intro­ duce throughout the book. This will enable us to furnish an immediate 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 specific ap­ plications.

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 function f (equivalently, the roots of the equation f(x) = 0) is a problem that we encounter quite often in scientific 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 4, there do not exist closed 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 \( \tilde f \) of simpler form that may be used as its surrogate. This strategy is used frequently in numerical integration where, instead of computing a b f(x)dx one carries out the exact computation of \( \int_{a}^{b} {\tilde{f}\left( x \right)dx} \)\( \tilde f \) 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 \( \tilde f \) that could represent the empirical law which is behind the finite set of data. We provide a couple of 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, it is known that for a generic function it is not always possible to find a primitive in an explicit form. In other cases, it could be hard to evaluate a primitive as, for instance, in the example
$$\frac{1}{\pi }\int\limits_{0}^{\pi } {\cos \left( {4x} \right)\cos \left( {3\sin \left( x \right)} \right)} dx = {{\left( {\frac{3}{2}} \right)}^{4}}\sum\limits_{{k = 0}}^{\infty } {\frac{{{{{\left( { - 9/4} \right)}}^{k}}}}{{k!\left( {k + 4} \right)!}},} $$
where the problem of computing an integral is transformed into the equally troublesome one of summing a series. It is also worth mentioning that sometimes the function that we want to integrate or differentiate could only be known on a set of nodes (for instance, when the latter represent the results of an experimental measurement), exactly as happens in the case of the approximation of functions, which was discussed in chapter 3.
Alfio Quarteroni, Fausto Saleri

5. Linear systems

Abstract
In applied sciences, one is quite often led to face a linear system of the form
$$Ax = b$$
(5.1)
for the solution of complex problems, where A is a square matrix of dimension n x n whose elements a ij are either real or complex, while x and b are column vectors of dimension n with x representing the unknown solution and b is a given vector. Component-wise, (5.1) can be written as
$$\begin{gathered} {a_{11}}{x_1} + {a_{12}}{x_2} + \ldots + {a_{1n}}{x_n} = {b_1}, \hfill \\ {a_{21}}{x_1} + {a_{22}}{x_2} + \ldots + {a_{2n}}{x_n} = {b_2}, \hfill \\ \vdots \vdots \vdots \hfill \\ {a_{n1}}{x_1} + {a_{n2}}{x_2} + \ldots + {a_{nn}}{x_n} = {b_n}, \hfill \\ \end{gathered} $$
.
Alfio Quarteroni, Fausto Saleri

6. Eigenvalues and eigenvectors

Abstract
We consider the following problem: given a square matrix A ∈ ℝnxn, find a scalar λ (real or complex) and a non-null vector x such that
$$Ax = \lambda x$$
(6.1)
.
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 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

Without Abstract
Alfio Quarteroni, Fausto Saleri

Backmatter

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