Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer


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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.

       A. Meseguer

      Barcelona, September 2019

      Many people read early drafts at various stages of the evolution of this book and contributed many corrections and constructive comments. I am particularly grateful to Daniel Loghin, Mark Embree, and Francisco (Paco) Marqués for their careful reading of some parts of the manuscript. Other people who helped with proofreading include Ercília Sousa, Juan M. López, and Julia Amorós. I would also like to thank Toni Castillo for efficiently adapting Wiley LaTeX template to different Debian and Ubuntu's Linux environments I have been working with during the writing. Finally, I wish to thank Kathleen Santoloci, Mindy Okura‐Marszycki, Elisha Benjamin, and Devi Ignasi of John Wiley & Sons, for their splendid editorial work.

       A. Meseguer

Part I

      1.1 Nonlinear Equations in Physics

, whose exact zeros or roots are well known:

. The answer is yes, and such expression is usually termed as Cardano's formula.1 We will not detail here the explicit expression of Cardano's formula but, as an example, if we apply such formulas to solve the equation

      we can obtain one of its roots:

      There are similar formulas to solve arbitrary quartic equations in terms of radicals, but not for quintic or higher degree polynomials, as proposed by the Italian mathematician Paolo Ruffini in 1799 but eventually proved by the Norwegian mathematician Niels Henrik Abel around 1824.


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