Spectral analysis of some iterations in the Chandrasekhar's H-functions

Two very general, fast and simple iterative methods were proposed by Bosma and de Rooij (Bosma, P. B., de Rooij, W. A. (1983). Efficient methods to calculate Chandrasekhar's H functions. Astron. Astrophys. 126:283--292.) to determine Chandrasekhar's H-functions. The methods are based on the use of the equation 'j h = (F) over tilde (h), where (F) over tilde = ((f1) over tilde, (f2) over tilde, . . . (f(n)) over tilde")(T) is a nonlinear map from R-n to R-n. Here (f(i)) over tilde = 1 /(root1 - c + Sigma(k=1)(n) (c(k)mu(k)h(k)/mu(i) + mu(k)), 0 < c less than or equal to 1, i = 1,2, . . . ,n. One such method is essentially a nonlinear Gauss-Seidel iteration with respect to (F) over tilde. The other ingenious approach is to normalize each iterate after a nonlinear Gauss-Jacobi iteration with respect to (F) over tilde is taken. The purpose of this article is two-fold. First, we prove that both methods converge locally. Moreover, the convergence rate of the second iterative method is shown to be strictly less than (root3 - 1)/2. Second, we show that both the Gauss-Jacobi method and Gauss-Seidel method with respect to some other known alternative forms of the Chandrasekhar's H-functions either do not converge or essentially stall for c = 1.