Spectral Analysis of Some Iterations in the Chandrasekhar's H-Functions

Abstract

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 where is a nonlinear map from Rⁿ to Rⁿ . Here , One such method is essentially a nonlinear Gauss-Seidel iteration with respect to [Ftilde]. The other ingenious approach is to normalize each iterate after a nonlinear Gauss-Jacobi iteration with respect to [Ftilde] 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 . 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.

Keywords:

• H-function
• Radiative transfer
• Nonnegative matrices
• Convergence
• Perron-Frobenius theorem
• Gauss-Jacobi
• Gauss-Seidel
• Eigenvalues
• AMS Subject Classifications: 65H10
• 65R20
• 85A25

Acknowledgment

We wish to thank Professor C. T. Kelley for helpful suggestions concerning this work.