All the numerical methods of the preceding chapters involved the solution of systems of linear equations. When these systems are not too large, they can be solved by Gaussian elimination; for such systems, that is usually the simplest and most efficient approach to use. For larger linear systems, however, iteration is usually more efficient, and it is often the only practical means of solution. There is a large literature on general iteration methods for solving linear systems, but many of these general methods are often not efficient (or possibly, not even convergent) when used to solve the linear systems we have seen in the preceding chapters. In this chapter we define and analyze several iteration methods that seem especially suitable for solving the linear systems associated with the numerical solution of integral equations.

In §6.1 we give an iteration method for solving degenerate kernel integral equations and the associated linear systems. In §6.2 we define and analyze two-grid iteration methods for solving the systems associated with the Nyström method. And in §6.3 we consider related two-grid methods for projection methods. In our experience these are the most efficient numerical methods for solving the linear systems obtained when solving integral equations of the second kind. In §6.4 we define multigrid iteration methods, which are closely related to two-grid methods. Multigrid methods are among the most efficient methods for solving the linear systems associated with the numerical solution of elliptic partial differential equations, and they are also very efficient when solving Fredholm integral equations.