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A graduate introduction to numerical methods. From the viewpoint of backward error analysis. (English) Zbl 1295.65001

New York, NY: Springer (ISBN 978-1-4614-8452-3/hbk; 978-1-4614-8453-0/ebook). xxxix, 868 p. (2013).
The book consists of four parts. Part I addresses fundamental issues of numerical mathematics and is divided into three chapters. Chapter 1 provides the three concepts of error (forward error, backward error, and residual) and the concept of conditioning. In addition, the basic numerical properties of algorithms are defined, including stability and computational complexity. Floating point arithmetic is also considered. Chapter 2 introduces the basics of polynomials. Algorithms to evaluate polynomials with respect to different bases are considered, including the Chebyshev basis, the Lagrange basis, and monomials. Other topics in this chapter are, e.g., different condition numbers, and the numerical computation of zeros. Chapter 3 is devoted to the evaluation of functions and the determination of roots.
The second part of the text focuses on numerical linear algebra. It starts with Chapter 4 which addresses methods for the solution of linear systems of equations, including the main factorizations, e.g., LU factorization, QR factorization and the singular value decomposition. The two subsequent chapters introduce the reader to the numerical treatment of eigenvalue problems and structured linear systems of equations, e.g., sparse systems or systems with correlated entries. In Chapter 7, iterative methods for solving linear systems of equations are considered. In addition, the authors look at iterative methods for eigenvalue problems that benefit from sparsity or structure.
Part III treats interpolation, differentiation and numerical integration and is divided into the Chapters 8–11. First, Lagrange interpolation and Hermite interpolation and their barycentric forms are studied. Other topics in this chapter are rational interpolation and piecewise interpolation. In Chapter 9, the discrete Fourier transform is considered, and Chapter 10 introduces the reader to basic schemes for the numerical integration, e.g., the trapezoidal rule, Simpson’s rule, extrapolation methods, adaptive methods, and Gaussian quadrature. In Chapter 11, finite difference schemes for the numerical differentiation are considered. Other topics in this chapter are regularization, smoothing and automatic differentiation.
Part IV of the text is devoted to differential equations. It starts with a chapter on initial value problems (IVPs) for ordinary differential equations (ODEs), with residuals, stiff problems and singular problems as special topics. Chapter 13 gives a brief survey of numerical methods for solving IVPs for ODEs. This includes Runge-Kutta methods, multistep methods and Taylor series methods. Adaptive step-size control is also considered in this chapter. The subject of the subsequent chapter is the numerical solution of boundary value problems for ODEs and delay differential equations. The final chapter in Part IV deals with the numerical solution of partial differential equations by the method of lines, spectral methods and compact finite differences. The text concludes with appendices on floating-point arithmetic, complex numbers and basics from linear algebra.
This textbook provides a very readable and comprehensive graduate-level introduction to numerical methods and their analysis. Special emphasis in all chapters is drawn on the examination of backward error analysis and condition numbers. It is suited for readers who have mathematics, natural sciences, computer sciences or engineering as a background. Basic knowledge of calculus, linear algebra, complex analysis, differential equations and programming is required. The textbook contains many references, exercises, Matlab codes, and numerical illustrations. It is written in a style that is both informal and informative which makes this book unique in fact.

MSC:

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Gxx Error analysis and interval analysis
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Fxx Numerical linear algebra
65T50 Numerical methods for discrete and fast Fourier transforms
65Lxx Numerical methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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