In this book, we will survey general numerical methods as well as specialized algorithms, with a focus on examples from computational electromagnetics. General numerical tools and techniques to be covered include numerical integration, differentiation, the solution of large linear systems, and methods for discretizing and solving partial differential equations and integral equations. Finite difference methods, the method of moments, and the finite element method will be treated in detail. While the emphasis is on wave propagation and electromagnetics, the principles and techniques of numerical analysis that are illustrated throughout the book can be applied to many branches of engineering.

As a warmup before developing more sophisticated numerical methods, we will first develop in this chapter some of the basic routines that are commonly used in scientific computation and numerical modeling. These include numerical differentiation, integration, interpolation, and curve fitting. Since numerical differentiation and integration are building blocks for algorithms like the finite difference method and the method of moments, we will return to these two key topics in later chapters with more rigorous treatments. Since the algorithm examples given throughout this book are based on the MATLAB® programing language, we will also introduce the MATLAB programing environment and some of the basic commands that will be used throughout the book. The simple algorithms introduced in this chapter are intended as an easy way to learn MATLAB programing. Many textbooks, monographs, and online references on numerical analysis are available that provide greater depth on these topics (e.g., [1,2]).

In computational science and engineering, evaluation of integrals numerically is a fundamental problem with many applications. Computational electromagnetics codes often use integration routines that are evaluated thousands of times to fill a large matrix, so efficient methods for numerical integration are very important. From the point of view of numerical analysis, the theory of numerical integration is mathematically rich, with interesting theoretical connections to other areas such as interpolation, orthogonal functions, and matrix theory. In this chapter, we will explore some of the basic techniques used for numerical integration.

Solving large linear systems is a key computational task that forms the core of many numerical methods. Algorithms from all the areas of scientific computations lead to large linear systems that must be solved for a vector of unknowns, or to the closely related problem of finding eigenvalues and eigenvectors of a large matrix. We have already seen that the finite difference method and the method of moments reduce a boundary value problem to a linear system of the form

So far, we have considered analysis problems, in the sense that a structure is given, and the goal is to solve Maxwell's equations, an integral equation, or a partial differential equation to find an unknown field or current solution. In engineering, analysis is typically only a preparatory step or a building block for tools that can be used to solve design synthesis problems. A synthesis problem typically involves one or more performance goals and constraints, which must be met by the designed structure, circuit, or system. Synthesis problems can be solved using a variety of techniques, including analytical formulas, trial and error, and empirical methods. As the cost of computational power has decreased, it has become feasible to solve synthesis problems using algorithms for numerical optimization.