Читать книгу Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer - Страница 10

Much of the material in this book is derived from lecture notes for two courses on numerical methods taught over many years to undergraduate students in Engineering Physics at the Universitat Politècnica de Catalunya (UPC) BarcelonaTech. Its volume is scaled to a one‐year course, that is, a two‐semester course. Accordingly, the book has two parts. Part I is addressed to first or second year undergraduate students who have a solid foundation in differential and integral calculus in one real variable (including Taylor series, notation, and improper integrals), along with elementary linear algebra (including polynomials and systems of linear equations). Part II is addressed to slightly more advanced undergraduate or first‐year graduate students with a broader mathematical background, including multivariate calculus, ordinary differential equations, functions of a complex variable, and Fourier series. In both cases, it is assumed that the students are familiar with basic Matlab commands and functions.

The book has been written thinking not only of the student but also of the instructor (or instructors) that is supposed to teach the material following an academic calendar. Each chapter contains mathematical topics to be addressed in the lectures, along with Matlab codes and computer hands‐on practicals. These practicals are problem‐solving tutorials where the students, always supervised and guided by an instructor, use Matlab on a local computer to solve a given exercise that is focused on the topic previously seen in the lectures. From my point of view, teaching numerical methods should encompass not only theoretical lectures, addressing the underlying mathematics on a blackboard, but also practical computations, where the student learns the actual implementation of those mathematical concepts. There are certain aspects of numerical mathematics, such as conditioning or order of convergence, that can only be properly illustrated by experimentation on a computer. These hands‐on practicals may also help the instructor to efficiently assess the performance of a student. This can be easily carried out by using Matlab's publish function, for example. The end of each chapter also includes a short list of problems and exercises of theoretical (labeled with an A) and/or computational (labeled with an N) nature. The solutions to many of the exercises (and practicals) can be found at the end of the book. Finally, each chapter includes a Complementary Reading section, where the student may find suitable bibliography to broaden his or her knowledge on different aspects of numerical mathematics. Complementary lists of exercises can also be found in many of these recommended references.

This book is mainly written for mathematically inclined scientists and engineers, although applied mathematicians may also find many of the topics addressed in this book interesting. My intention is not simply to give a set of recipes for solving problems, but rather to present the underlying mathematical concepts involved in every numerical method. Throughout the eight chapters, I have tried to write a readable book, always looking for an equilibrium between practicality and mathematical rigor. Clarity in presenting major points often requires the supression of minor ones. A trained mathematician may find certain statements incomplete. In those passages where I think this may be the case, I always refer the rigorous reader to suitable bibliography where the key theorem and its corresponding proof can be found.

Whenever it has been possible, I have tried to illustrate how to apply certain numerical methodologies to solve problems arising in the physical sciences or in engineering. For example, Part I includes some practicals involving very basic Newtonian mechanics. Part II includes practicals and examples that illustrate how to solve problems in electrical networks (Kirchhof's laws), classical thermodynamics (van der Waals equation of state), or quantum mechanics (Schrödinger equation for univariate potentials). In all the previous examples, the mathematical equations have already been derived, so that those readers who are not necessarily familiar with any of those areas of physics should be able to address the problem without any difficulty.

Many of the topics covered throughout the eight chapters are fairly standard and can easily be found in many other textbooks, although probably in a different order. For example, Chapter 1 introduces topics such as nonlinear scalar equations, root‐finding method, convergence, or conditioning. This chapter also shows how to measure in practice the order of convergence of a root‐finding method, and how ill‐conditioning may affect that order. Chapter 2 is devoted to one of the most important methods to approximate functions: interpolation. I have addressed three different interpolatory formulas, namely, monomial, Lagrange, and barycentric, the last one being the most computationally efficient. I devote a few pages to introduce the concept of Lebesgue constant or condition number of a set of interpolatory nodes. This chapter clearly illustrates that global interpolation, performed on a suitable set of nodes, provides unbeatable accuracy. Chapter 3 is devoted to numerical differentiation, introducing the concept of differentiation matrix, which is often omitted in other textbooks. From my point of view, working with differentiation matrices has two major advantages. On the one hand, it is a simple and systematic way to obtain and understand the origin of classical finite difference formulas. On the other hand, differentiation matrices, understood as discretizations of differential operators, will be very important in Part II, when solving boundary value problems numerically. Chapter 4 is devoted to numerical integration or quadratures. This chapter addresses the classical Newton–Cotes quadrature formulas, along with Clenshaw–Curtis and Fejér rules, whose accuracy is known to be comparable to that of Gaussian formulas, but much simpler and easier to implement. This chapter is also devoted to the numerical approximation of integrals with periodic integrands, emphasizing the outstanding accuracy provided by the trapezoidal rule, which will be exploited in Part II, in the numerical approximation of Fourier series. Finally, Chapter 4 briefly addresses the numerical approximation of improper integrals.

Part II starts with Chapter 5, which is an introduction to numerical linear algebra, henceforth referred to as NLA. Some readers may find it unusual not to find NLA in Part I. Certain topics such as matrix norms or condition number of a matrix implicitly involve multivariate calculus, and are therefore unsuitable for Part I, addressed to first or second year undergraduates. Chapter 5 exclusively focuses on just one topic: solving linear systems of equations. The chapter first addresses direct solvers by introducing LU and QR factorizations. A very brief introduction to iterative matrix‐free Krylov solvers can be found at the end of the chapter. I think that the concept of matrix‐free iterative solver is of paramount importance for scientists and engineers. I have tried to introduce the concept of Krylov subspace in a simple, but also unconventional, way. Owing to the limited scale of the book, it has been impossible to address many other important topics in NLA such as eigenvalue computation, the singular value decomposition, or preconditioning. Chapter 6 is devoted to the solution of multidimensional nonlinear systems of equations, also including parameters. This chapter introduces the concept of continuation, also known as homotopy, a very powerful technique that has been shown to be very useful in different areas of nonlinear science. Chapter 7 is a very brief introduction to numerical Fourier analysis, where I introduce the discrete Fourier transform (DFT), and the phenomenon of aliasing. In this chapter, I also introduce Fourier differentiation matrices, to be used later in the numerical solution of boundary value problems. Finally, Chapter 8 is devoted to numerical discretization techniques for ordinary differential equations (ODE). The chapter first addresses how to solve boundary value problems (BVP) within bounded, periodic, and unbounded domains. In this first part of the chapter, I exploit the concept of global differentiation matrix seen previously in the book. Local differentiation formulas are used in the second part of the chapter, exclusively devoted to the solution of initial value problems (IVP). The limited scale of the book has only allowed including a few families of time integrators. The chapter ends with a brief introduction to the concept of stability of a time‐stepper, where I have certainly oversimplified concepts such as consistency or 0‐stability, along with Dahlquist's equivalence theorem. However, I have emphasized other, in my point of view, also important but more practical aspects such as stiffness and A‐stability.

A. Meseguer

Barcelona, September 2019