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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.

The purpose of this paper is two fold. First, we shed some light on the slow convergence behaviour of a certain iterative procedure for solving the reflection matrix in a nonmultiplying half-space. In particular, we show that the asymptotic convergence rate of the iterative procedure of Shimiju and Aoki is 1 – □1–c. Here c, 0 ≤ c ≤ 1. the expected number of particles emerging from a collision. Second, a convergence accerelation procedure is proposed. The asymptotic convergence rate of the accerelation procedure is given. Some very satisfactory numerical results are also presented.

A monotone iterative technique is applied to a class of algebraic matrix Riccati equations that is satisfied by the reflection matrix for linear particle transport in a half-space. The convergence of the iteration as well as the rate of convergence are addressed. Moreover, we show theoretically and numerically that our iterative procedure, in general, is faster than that of Shimizu and Aoki.

Of concern is the degenerate Riccati equation of the form TR+RT=TA+TBR+RTC+RTDR (∗). This models a certain transport equation in the half-space. Our first result will concern a unique, positive solution of the operator equation SR+RT=K in a anach space equipped with a cone structure. We then proceed to prove existence of positive solutions for (∗) on an L1 space of σ-finite measure and a Banach space X, respectively, under different assumptions. Some estimates of the solution with respect to certain norms are obtained.