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Numerical mathematics proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations.

This book provides the mathematical foundations of numerical methods and demonstrate their performance on examples, exercises and real-life applications. This is done using the MATLAB software environment, which allows an easy implementation and testing of the algorithms for any specific class of problems.

The book is addressed to students in Engineering, Mathematics, Physics and Computer Sciences. The attention to applications and software development makes it valuable also for users in a wide variety of professional fields.

In this second edition, the readability of pictures, tables and program headings has been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

Inhaltsverzeichnis

Frontmatter

Getting Started

1. Foundations of Matrix Analysis

In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text. For most of the proofs as well as for the details, the reader is referred to [Bra75], [Nob69] [Ha158]. Further results on eigenvalues can be found in [Hou75] and [Wi165].

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

2. Principles of Numerical Mathematics

The basic concepts of consistency, stability and convergence of a numerical method will be introduced in a very general context in the first part of the chapter: they provide the common framework for the analysis of any method considered henceforth. The second part of the chapter deals with the computer finite representation of real numbers and the analysis of error propagation in machine operations.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Numerical Linear Algebra

3. Direct Methods for the Solution of Linear Systems

A system of m linear equations in n unknowns consists of a set of algebraic relations of the form 3.1% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca % WGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadIhadaWgaaWcbaGa % amOAaaqabaGccqGH9aqpcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaai % ilaiaadMgacqGH9aqpcaaIXaGaaiilaiabl+UimjaacYcacaWGTbaa % leaacaWGQbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa!4BA4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\sum\limits_{j = 1}^n {{a_{ij}}{x_j} = {b_i},i = 1, \cdots ,m} $$ where x j are the unknowns, aij are the coefficients of the system and bi are the components of the right hand side. System (3.1) can be more conveniently written in matrix form as 3.2% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadI % hacqGH9aqpcaWGIbaaaa!39A4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$Ax = b$$ where we have denoted by A = (aij) ∈ ℂm×n the coefficient matrix, by b=(bi) ∈ (ℂm the right side vector and by x=(xi) ∈ ℂm the unknown vector, respectively. We call a solution of (3.2) any n-tuple of values xi which satisfies (3.1).

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

4. Iterative Methods for Solving Linear Systems

Iterative methods formally yield the solution x of a linear system after an infinite number of steps. At each step they require the computation of the residual of the system. In the case of a full matrix, their computational cost is therefore of the order of n2 operations for each iteration, to be compared with an overall cost of the order of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIYaaabaGaaG4maaaaaaa!377D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{2}{3}$$n3 operations needed by direct methods. Iterative methods can therefore become competitive with direct methods provided the number of iterations that are required to converge (within a prescribed tolerance) is either independent of n or scales sublinearly with respect to n.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

5. Approximation of Eigenvalues and Eigenvectors

In this chapter we deal with approximations of the eigenvalues and eigen-vectors of a matrix A ∈ ℂn×n Two main classes of numerical methods exist to this purpose, partial methods, which compute the extremal eigen-values of A (that is, those having maximum and minimum module), or global methods, which approximate the whole spectrum of A.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Around Functions and Functionals

6. Rootfinding for Nonlinear Equations

This chapter deals with the numerical approximation of the zeros of a real-valued function of one variable, that is 6.1% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaadM % gacaWG2bGaamyzaiaad6gacaaMc8UaamOzaiaaykW7caGG6aGaamys % aiabg2da9iaacIcacaWGHbGaaiilaiaadkgacaGGPaGaeyOHI0SaeS % yhHeQaeyOKH4QaeSyhHeQaaiilaiaaykW7caWGMbGaamyAaiaad6ga % caWGKbGaaGPaVlabeg7aHjaaykW7cqGHiiIZcaaMc8UaeSOaHmQaaG % PaVlaadohacaWG1bGaam4yaiaadIgacaaMc8UaamiDaiaadIgacaWG % HbGaamiDaiaaykW7caWGMbGaaiikaiabeg7aHjaacMcacqGH9aqpca % aIWaaaaa!6CB8!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$given\,f\,:I = (a,b) \subseteq \mathbb{R} \to \mathbb{R},\,find\,\alpha \, \in \,\mathbb{C}\,such\,that\,f(\alpha ) = 0$$.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

7. Nonlinear Systems and Numerical Optimization

In this chapter we address the numerical solution of systems of nonlinear equations and the minimization of a function of several variables.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

8. Polynomial Interpolation

This chapter is addressed to the approximation of a function which is known through its nodal values.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

9. Numerical Integration

In this chapter we present the most commonly used methods for numerical integration. We will mainly consider one-dimensional integrals over bounded intervals, although in Sections 9.8 and 9.9 an extension of the techniques to integration over unbounded intervals (or integration of functions with singularities) and to the multidimensional case will be considered.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Transforms, Differentiation and Problem Discretization

10. Orthogonal Polynomials in Approximation Theory

Trigonometric polynomials, as well as other orthogonal polynomials like Legendre’s and Chebyshev’s, are widely employed in approximation theory.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

11. Numerical Solution of Ordinary Differential Equations

In this chapter we deal with the numerical solutions of the Cauchy problem for ordinary differential equations (henceforth abbreviated by ODEs). After a brief review of basic notions about ODEs, we introduce the most widely used techniques for the numerical approximation of scalar equations. The concepts of consistency, convergence, zero-stability and absolute stability will be addressed. Then, we extend our analysis to systems of ODEs, with emphasis on stiff problems.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

12. Two-Point Boundary Value Problems

This chapter is devoted to the analysis of approximation methods for two-point boundary value problems for differential equations of elliptic type. Finite differences, finite elements and spectral methods will be considered. A short account is also given on the extension to elliptic boundary value problems in two-dimensional regions.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

13. Parabolic and Hyperbolic Initial Boundary Value Problems

The final chapter of this book is devoted to the approximation of time-dependent partial differential equations. Parabolic and hyperbolic initial-boundary value problems will be addressed and either finite differences and finite elements will be considered for their discretization.

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Backmatter

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