Numerical Mathematics: A Textbook by Quarteroni, Sacco and Saleri
Numerical Mathematics: A Textbook by Quarteroni, Sacco and Saleri
Numerical mathematics is the branch of mathematics that 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 demonstrates 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.
Quarteroni Sacco Saleri Matematica Numerica.pdf
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.
The book is divided into three parts: around matrices and linear systems; around functions and functionals; and transforms, differentiation and problem discretization. Each part covers topics such as matrix analysis, numerical linear algebra, rootfinding for nonlinear equations, polynomial interpolation, numerical integration, orthogonal polynomials in approximation theory, numerical solution of ordinary and partial differential equations, and more.
The book is based on the Italian version "Matematica Numerica" by Alfio Quarteroni, Riccardo Sacco, Fausto Saleri and Paola Gervasio. It is available in PDF format from SpringerLink[^1^] [^2^]. The book also includes supplementary material such as MATLAB codes and exercises[^1^] [^2^].
The book is a comprehensive and rigorous introduction to numerical mathematics, with a strong emphasis on the theoretical aspects and the practical implementation of the methods. The authors provide a clear and concise exposition of the main concepts and techniques, as well as numerous examples and exercises that illustrate the applications and the limitations of the methods. The book also offers a glimpse into the current research topics and challenges in numerical mathematics, such as adaptive methods, multiscale methods, parallel computing, uncertainty quantification, and inverse problems.
The book is suitable for undergraduate and graduate students who want to learn the fundamentals of numerical mathematics and how to use MATLAB for solving scientific problems. It can also serve as a reference for researchers and practitioners who need to apply numerical methods in their work. The book assumes some basic knowledge of calculus, linear algebra, and differential equations, but it also reviews some of the necessary background material in the appendices.
The book is organized into 13 chapters, each covering a specific topic in numerical mathematics. The first chapter introduces the basic notions and principles of numerical mathematics, such as round-off errors, stability, accuracy, and complexity. The second chapter reviews some of the essential results of matrix analysis, such as norms, condition numbers, eigenvalues, and singular values. The third chapter presents the direct methods for solving linear systems, such as Gaussian elimination, LU factorization, Cholesky factorization, and QR factorization. The fourth chapter discusses the iterative methods for solving linear systems, such as Jacobi method, Gauss-Seidel method, conjugate gradient method, and GMRES method.
The fifth chapter deals with the approximation of eigenvalues and eigenvectors of matrices, such as power method, inverse iteration method, Rayleigh quotient iteration method, and QR algorithm. The sixth chapter focuses on the rootfinding for nonlinear equations and systems, such as bisection method, Newton's method, secant method, and fixed-point iteration method. The seventh chapter introduces the polynomial interpolation of functions, such as Lagrange interpolation, Newton interpolation, Chebyshev interpolation, and spline interpolation. The eighth chapter explains the numerical integration of functions, such as trapezoidal rule, Simpson's rule, Gaussian quadrature, and adaptive quadrature. e0e6b7cb5c