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 h 1⁄4 ~ FðhÞ, where ~ F 1⁄4 ð ~ f1, ~ f2, . . . , ~ fnÞ is a nonlinear map from R to R. Here ~ fi 1⁄4 1=ð ffiffiffiffiffiffiffiffiffiffi 1 c p þ Pn k1⁄41 ðck khk= i þ kÞÞ, 0 < c 1, i 1⁄4 1, 2, . . . , n: One such method is essentially a nonlinear Gauss-Seidel iteration with respect to F̃. The other ingenious approach is to normalize each iterate after a nonlinear Gauss-Jacobi iteration with respect to F̃ 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 ð ffiffiffi 3 p 1Þ=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 c1⁄4 1. *Correspondence: Jonq Juang, Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu, Taiwan 30050, R.O.C.; E-mail: [email protected].