Overview of Numerical Methods

Abstract

This chapter describes a number of topics that are full-fledged research subjects in and of themselves and to do them justice in a just a few pages is not possible. The hope of the author is to present the material in such a way as to wet the readers’ appetite. The importance of numerical methods in solving various kinds of problems is paramount in modern product engineering and scientific research. Both the engineering and scientific communities are heavily involved in the development and use of these methodologies. In an effort to contain these vast subject matters, the focus will be on methods the reader is likely to encounter in an electrical engineering context and as such certain types of approximations to differential equations will be highlighted. The same thing goes for matrix equations where we here only use simple examples to highlight the fundamental ideas. The more advanced iterative methods that have been so successful in recent decades are mentioned briefly with an accompanying Python code example. Nonlinear equations and how to solve them efficiently is likewise another intense field of study, and over the years many methods have been developed that are in wide use in the scientific/engineering community today. Here we describe a method that is perhaps the most important to know due to its relative ease of implementation attributed to Isaac Newton, although other researchers such as Joseph Raphson were also involved over the years. Even though the presentation is held at a fundamental level, some basic familiarity with numerical methods, corresponding to an introductory class on the subject, will be helpful since we will be rather brief. We will start the chapter discussing differential equations and how one might implement them numerically. We will discuss implementations of what is called initial value problems where the state is known at a certain moment in time, and from then on, the system develops according to the governing equations. We will present implementations commonly used in circuit simulators. The chapter continues with nonlinear solution methods, and we wrap up the presentation with a description of matrix solvers. Rather than going through the mathematical theories behind these methods, we choose to present the basic ideas using examples, and for the interested reader a much more in-depth discussion of these issues can be found in Chap. 7 and the references at the end of the chapter. The importance of the subject matter presented in this chapter cannot be overstated, and it is the hope of the author the reader will explore the topic more deeply on his or her own.