TY - JOUR

T1 - Spectral analysis of some iterations in the Chandrasekhar's H-functions

AU - Jonq, Juang

AU - Lin, Kun Yi

AU - Lin, Wen-Wei

PY - 2003/1/1

Y1 - 2003/1/1

N2 - 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 = (f̃1, f̃ 2,...,f̃n)T is a nonlinear map from R n to Rn. Here f̃i = 1/(√1-c + ∑k=1n (ckμkh k/μi + μk)), 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.

AB - 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 = (f̃1, f̃ 2,...,f̃n)T is a nonlinear map from R n to Rn. Here f̃i = 1/(√1-c + ∑k=1n (ckμkh k/μi + μk)), 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.

KW - Convergence

KW - Eigenvalues

KW - Gauss-Jacobi

KW - Gauss-Seidel

KW - H-function

KW - Nonnegative matrices

KW - Perron-Frobenius theorem

KW - Radiative transfer

UR - http://www.scopus.com/inward/record.url?scp=0041419271&partnerID=8YFLogxK

U2 - 10.1081/NFA-120023909

DO - 10.1081/NFA-120023909

M3 - Article

AN - SCOPUS:0041419271

VL - 24

SP - 575

EP - 586

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 5-6

ER -