The Chandrasekhar equation describes the particles emerging from the atmospheric radiation and its solution of physical significance is the minimal positive solution. This paper analyses the efficiency index of Newton’s iteration in detail, which then helps to design a structured Shamanskii method for calculating the minimal positive solution. The monotone convergence of the presented algorithm is subsequently established as well as the elementary monotonicity of the solution. Preliminary numerical experiments are listed to indicate that the newly developed two-step structured Shamanskii method outperforms the Newton’s method in terms of CPU time and iterative number with almost no loss in accuracy.
1. Bini, D. A., Iannazzo, B., and Poloni, F. A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation. SIAM J. Matrix Anal. Appl., 2008, 30(1), 276-290.
https://doi.org/10.1137/070681478
2. Benner, P., Li, R. C., and Truhar, N. On the ADI method for Sylvester equations. J. Comp. Appl. Math., 2009, 233, 1035-1045.
https://doi.org/10.1016/j.cam.2009.08.108
3. Chandrasekhar, S. Radiative transfer. Dover, New York, 1960.
4. Chandrasekhar, S. Radiative transfer. Courier Corporation, 2013.
5. Deeba, E. Y. and Khuri, S. A. The decomposition method applied to Chandrasekhar H-equation. Appl. Math. Comput., 1996, 1, 67-78.
https://doi.org/10.1016/0096-3003(95)00188-3
6. Dong, N. and Yu, B. On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure. J. Comput. Appl. Math., 2015, 288, 18-32.
https://doi.org/10.1016/j.cam.2015.03.036
7. Golub, G. H. and Van Loan C. F. Matrix computations, 3rd Edition. Johns Hopkins University Press, Baltimore, MD, 1996
8. Guo, C.-H. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl., 2001, 23, 225-242.
https://doi.org/10.1137/S0895479800375680
9. Guo, C.-H. Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations. Numer. Algor., 2013, 64, 295-309.
https://doi.org/10.1007/s11075-012-9666-7
10. Guo, C.-H. and Laub, A. J. On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations. SIAM J. Matrix Anal. Appl., 2000, 22(2), 376-391.
https://doi.org/10.1137/S089547989834980X
11. Juang, J. On Coupled Integral H-Like Equations of Chandrasekhar. SIAM J. Math. Anal., 1995, 4, 869-879.
https://doi.org/10.1137/S0036141093250475
12. Juang, J. and Lin, W.-W. Nonsymmetric Algebraic Riccati Equations and Hamiltonian-like Matrices. SIAM J. Matrix Anal. Appl., 1998, 20(1), 228-243.
https://doi.org/10.1137/S0895479897318253
13. Kelley, C. T. Solving nonlinear equations with Newton's method. Society for lndustrial and Applied Mathematics, Philadelphia, 2003.
https://doi.org/10.1137/1.9780898718898
14. Kelley, C. T., Kevrekidis, I. G., and Qiao, L. Newton-Krylov solvers for time-steppers. Tech. Rep. CRSC-TR04-10, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 2004.
https://doi.org/10.21236/ADA446724
15. Khuri, S. A. On the solution of coupled H-like equations of Chandrasekhar. Appl. Math. Comput., 2002, 2-3, 479-485.
https://doi.org/10.1016/S0096-3003(01)00251-X
16. Li, J.-R. and White, J. Low-rank Solution of Lyapunov Equations. SIAM J. Matrix Anal. Appl., 2002, 24(1), 260-280.
https://doi.org/10.1137/S0895479801384937
17. Li, R.-C. Solving secular equations stably and efficiently. Technical Report, Department of Mathematics, University of California, Berkeley, CA, LAPACK Working Note 1993, 89.
https://doi.org/10.21236/ADA608792
http://www2.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-851.pdf
18. Mehrmann, V. and Xu, H.-G. Explicit Solutions for a Riccati Equation from Transport Theory. SIAM J. Matrix Anal. Appl., 2008, 30(4), 1339-1357.
https://doi.org/10.1137/070708743
19. Traub, J. F. Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, 1964.
20. Benner, P., Kurschner, P., and Saak, J. Low-rank Newton-ADI methods for large nonsymmtric algebraic Riccati euqaitons. J. Franklin Instit., 2016, 353(5), 1147-1167.
https://doi.org/10.1016/j.jfranklin.2015.04.016
21. Varga, R. Matrix iterative analysis, 2nd Edition. Springer-Verlag, Berlin, Heidelberg, 2000.
https://doi.org/10.1007/978-3-642-05156-2
22. Wachspress, E. L. The ADI model problem, Windsor, CA, 1995.
23. Wachspress, E. L. ADI iteration parameters for solving Lyapunov and Sylvester equations. Technical Report, March, 2009.
24. Yu, B., Li, D.-H., and Dong, N. Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory. J. Comput. Appl. Math., 2013, 250, 175-189.
https://doi.org/10.1016/j.cam.2013.03.017
