Iterative Solution and Stability
The approximation methods for integral equations described in Chapters 11-13 lead to full linear systems. Only if the number of unknowns is reasonably small may these equations be solved by direct methods like Gaussian elimination. But, in general, a satisfying accuracy of the approximate solution to the integral equation will require a comparatively large number of unknowns, in particular for integral equations in more than one dimension. Therefore iterative methods for the resulting linear systems will be preferable. For this, in principle, in the case of positive definite symmetric matrices the classical conjugate gradient method (see Problem 13.2) can be used. In the general case, when the matrix is not symmetric more general Krylov subspace iterations may be used among which a method called generalized minimum residual method (GMRES)due to Saad and Schultz  is widely used. Since there is a large literature on these and other general iteration methods for large linear systems (see Freud, Golub, and Nachtigal , Golub and van Loan , Greenbaum , Saad , and Trefethen and Bau , among others), we do not intend to present them in this book. At the end of this chapter we will only briefly describe the main idea of the panel clustering methodsand the fast multipole methodsbased on iterative methods and on a speed-up of matrix-vector multiplications for the matrices arising from the discretization of integral equations.
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