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Convergence Rates of a Class of Predictor-Corrector Iterations for the Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory

Part of: Numerical linear algebra Basic linear algebra

Published online by Cambridge University Press:  18 January 2017

Ning Dong ,
Jicheng Jin  and
Bo Yu
Ning Dong*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, China School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
Jicheng Jin*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, China
Bo Yu*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, China
*
*Corresponding author. Email:dongning_158@sina.com (N. Dong), jcjin2008@sina.com (J. C. Jin), boyu_hut@126.com (B. Yu)
*Corresponding author. Email:dongning_158@sina.com (N. Dong), jcjin2008@sina.com (J. C. Jin), boyu_hut@126.com (B. Yu)
*Corresponding author. Email:dongning_158@sina.com (N. Dong), jcjin2008@sina.com (J. C. Jin), boyu_hut@126.com (B. Yu)
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Abstract

In this paper, we analyse the convergence rates of several different predictor-corrector iterations for computing the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. We have shown theoretically that the new predictor-corrector iteration given in [Numer. Linear Algebra Appl., 21 (2014), pp. 761–780] will converge no faster than the simple predictor-corrector iteration and the nonlinear block Jacobi predictor-corrector iteration. Moreover the last two have the same asymptotic convergence rate with the nonlinear block Gauss-Seidel iteration given in [SIAM J. Sci. Comput., 30 (2008), pp. 804–818]. Preliminary numerical experiments have been reported for the validation of the developed comparison theory.

Keywords

Convergence ratepredictor-corrector iterationsnonsymmetric algebraic Riccati equationregular splitting

MSC classification

Secondary: 65F50: Sparse matrices 15A24: Matrix equations and identities
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 944 - 963
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bai, Z. Z., Gao, Y. H. and Lu, L. Z., Fast iterative schemes for nonsymmetric algebraic Riccati equations arising from transport theory, SIAM J. Sci. Comput., 30 (2008), pp. 804–818.CrossRefGoogle Scholar
[2] Bao, L., Lin, Y. Q. and Wei, Y. M., A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory, Appl. Math. Comput., 181 (2006), pp. 1499–1504.Google Scholar
[3] Bini, D. A., Iannazzo, B. and Poloni, F., A fast Newton's method for a nonsymmetric algebraic Riccati equation, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 276–290.CrossRefGoogle Scholar
[4] Gao, Y. H., Theories and Algorithms for Several Quadratic Matrix Equations, Ph.D. thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, 2007.Google Scholar
[5] Guo, C. H., A new class of nonsymmetric algebraic Riccati equations, Linear Algebra Appl., 426 (2007), pp. 636–649.CrossRefGoogle Scholar
[6] Guo, C. H. and Laub, A. J., On a Newton-like method for solving algebraic Riccati equations, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 694–698.CrossRefGoogle Scholar
[7] Guo, C. H. and Lin, W. W., Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory, Linear Algebra Appl., 432 (2010), pp. 283–291.CrossRefGoogle Scholar
[8] Gu, B., Sheng, V. S., Tay, K. Y., Romano, W. and Li, S., Incremental support vector learning for ordinal regression, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), pp. 1403–1416.CrossRefGoogle ScholarPubMed
[9] Huang, N. and Ma, C. F., Some predictor-corrector-type iterative schemes for solving nonsymmetric algebraic Riccati equations arising in transport theory, Numer. Linear Algebra Appl., 21 (2014), pp. 761–780.CrossRefGoogle Scholar
[10] Juang, J., Existence of algebraic matrix Riccati equations arising in transport theory, Linear Algebra Appl., 230 (1995), pp. 89–100.CrossRefGoogle Scholar
[11] Juang, J. and Chen, I. D., Iterative solution for a certain class of algebraic matrix Riccati equations arising in transport theory, Tansport Theory Statist. Phys., 22 (1993), pp. 65–80.CrossRefGoogle Scholar
[12] Juang, J. and Lin, W. W., Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 228–243.CrossRefGoogle Scholar
[13] Krasnoselskii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, YA. B. and Stetsenko, V. YA., Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972.CrossRefGoogle Scholar
[14] Li, J., Li, X., Yang, B. and Sun, X., Segmentation-based image copy-move forgery detection scheme, IEEE Trans. Inf. Forensics Security, 10 (2015), pp. 507–518.Google Scholar
[15] Li, T. X., Chu, E. K. W., Lin, W. W. and Weng, P. C. Y., Solving large-scale continuous-time algebraic Riccati equations by doubling, J. Comput. Appl. Math., 237 (2013), pp. 373–383.CrossRefGoogle Scholar
[16] Li, T. X., Chu, E. K. W., Kuo, Y. C. and Lin, W. W., Solving large-scale nonsymmetric algebraic Riccati equations by doubling, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1129–1147.CrossRefGoogle Scholar
[17] Liu, C. and Xue, J., Complex nonsymmetric algebraic Riccati equations arising in Markov modulated fluid flows, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 569–596.CrossRefGoogle Scholar
[18] Lu, L. Z., Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 679–685.CrossRefGoogle Scholar
[19] Mehrmann, V. and Xu, H., Explicit solutions for a Riccati equation from transport theory, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1339–1357.CrossRefGoogle Scholar
[20] Varga, R., Matrix Iterative Analysis, 2nd Edn. Springer-Verlag: Berlin Heidelberg, 2000.CrossRefGoogle Scholar
[21] Wen, F. H., He, Z. F., Dai, Z. F. and Yang, X. G., Characteristics of investors risk preference forstock markets, Econ. Comput. Econ. Cyb., 3(48) (2014), pp. 235–254.Google Scholar
[22] Weng, P. C. Y., Fan, H. Y. and Chu, E. K. W., Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory, Appl.Math. Comput., 219 (2012), pp. 729–740.Google Scholar
[23] 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., 250 (2013), pp. 175–189.CrossRefGoogle Scholar