Multiple Wielandt deflations in the numerical solution of singular integral equations of potential theory

A technique for obtaining numerical approximations to Equation (1) will be briefly discussed. The convergence (under appropriate assumptions) of these approximations to a solution of Equation (1) has been shown elsewhere (see references ^1,4). The emphasis here will be on the efficient solution of the linear algebraic systems of equations which arise from the approximating scheme. The solution of the algebraic equations will be based on Jacobi iteration, whose rate of convergence is enhanced by the use of multiple deflations of the Wielandt type ^5,6,7,8,9 the key factor being that certain eigenvalues and eigenvectors are known a priori. These eigenvalues and eigenvectors may be used to obtain an equivalent system of equations. “Equivalent” is used to mean that the solution to the original algebraic equations may be obtained from the new system in relatively few operations. The new system has the computational advantage that the spectral radius of the appropriate iteration matrix has been considerably reduced.