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 programming language, we will also introduce the MATLAB programming 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 programming. Many textbooks, monographs, and online references on numerical analysis are available that provide greater depth on these topics (e.g., [1,2]).

In this chapter, we will explore some of the basic techniques used for numerical integration. In computational science and engineering, evaluation of integrals numerically is a fundamental problem with many applications. Computational electromagneticscodes 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. Davis and Rabinowitz is a standard reference, but many other papers and books are available on the topic.

Solving large linear systems is a key computational task that forms the core of many numerical methods. Algorithms from all 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 Ax = b, where A is a matrix, b is a given vector, and the vector x is unknown. A linear system is also produced by the finite element method to be discussed in Chapter 8. The field of study that deals with algorithms for solving linear systems and performing other types of matrix operations is referred to as numerical linear algebra.

In this chapter, we will first develop the finite element method for a simple one-dimensional Laplace problem. FEM algorithms will then be developed for two-dimensional (2-D) Helmholtz problems, including modal eigenvalue problems and radiation and scattering. FEM will then be combined with a surface integral equation to develop the FEM-BEM algorithm.

The most basic feature of an inverse problem is that the parameters of a structure, model, or system must be determined from observed data. For the purposes of this chapter, the inverse problem deals with an electromagnetic structure, such as an antenna, microwave component, or underground geophysical layer. We will refer to the structure to be identified or imaged as a scatterer. The forward problem is to compute scattered fields for a known scatterer, whereas the inverse problem considers the scatterer to be unknown, and seeks to determine characteristics of the object from forward scattered field data.