solution time (by direct methods) is *O*(*N*^{3}) and dominates for large *N*. Recently, Canning (1991) has pointed out that this solution time can be reduced by a factor of 2 for symmetric matrices. Also, Cohoon (1980) has shown that group theory can be used to exploit problem symmetries and reduce computational expense.

The main limitation of the MM is that *N* is proportional to some power of frequency, and thus the required computer storage and CPU time increase dramatically as the frequency (and thus the electrical size of the body) increases. For this reason, the MM has always been viewed as a "low-frequency" method applicable when the radiating or scattering body is not too large in terms of the operating wavelength. Over the last several years, a number of methods have been studied to improve the efficiency of the MM and thus allow it to be applicable to electrically larger bodies. This paper will present a non-all-inclusive review of some of these techniques, including fast iterative methods, recursive methods, sparse matrix methods, and hybrid methods.

The method of moments (MM) is a numerical technique for solving a linear operator equation by transforming it into a system of simultaneous linear algebraic equations, i.e., a matrix equation (Harrington, 1982, 1987; Hansen, 1990; Miller, 1988; Miller et al., 1991; Wang, 1991). In electromagnetics, the linear operator equation is almost always a linear integral equation for the actual or equivalent current on or in a radiating or scattering body. Once these currents are known, most parameters of engineering interest, such as input impedance, efficiency and radiated (or scattered) fields, can be evaluated in a straightforward manner and with relatively small computer CPU time and storage (as opposed to that required to obtain the currents).

The first step in the MM solution of the integral equation is to approximate the unknown current **J** as

where **J**^{N} is an *N* term approximation to the true current **J**, the **J**_{n} are a series of *N* known linearly independent expansion functions, and the *I*_{n} are a series of *N* unknown coefficients, *n* = 1,2,3,...,*N*. The next step is to select a series of *N* linearly independent weighting functions **w**_{m}, *m*=1,2,...,*N*, and enforce *N* weighted averages of the integral equation to be valid. This procedure will reduce the integral equation to a system of *N* simultaneous linear algebraic equations that can be compactly written as the order *N* matrix equation

In analogy with Ohm's Law, [*Z*] is the *N* x *N* impedance matrix, *V* is the length *N* right-hand side or voltage vector, and I is the length *N* solution or current vector that contains the *N* unknown coefficients, *I*_{n}, from (10.1). Typical elements of the [*Z*] matrix and V vector are given by