On the convergence of iterative solutions of the integral magnetic field equation

The solution of the integral magnetic field equation by numerical iteration is discussed. Using a simple linear example, it is shown rigorously that relaxation techniques are required to obtain convergence. The range of permissible relaxation parameters is examined and that particular value which yields most rapid convergence is determined. An iterative solution to a simple nonlinear problem is shown to converge rapidly if the relaxation parameter is adjusted appropriately at each step in the iteration. For the general case of a saturable media of complex geometric shape, a relaxation matrix method is proposed in order to achieve rapid convergence.