Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

Preface

      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,

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: