### 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 h = F̃(h), where F̃ = (f̃_{1}, f̃ _{2},...,f̃_{n})^{T} is a nonlinear map from R ^{n} to R^{n}. Here f̃_{i} = 1/(√1-c + ∑_{k=1}^{n} (c_{k}μ_{k}h _{k}/μ_{i} + μ_{k})), 0 < c ≤ 1, i = 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 (√3 - 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.

Original language | English |
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Pages (from-to) | 575-586 |

Number of pages | 12 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 24 |

Issue number | 5-6 |

DOIs | |

State | Published - 1 Jan 2003 |

### Keywords

- Convergence
- Eigenvalues
- Gauss-Jacobi
- Gauss-Seidel
- H-function
- Nonnegative matrices
- Perron-Frobenius theorem
- Radiative transfer