求解非对称代数Riccati 方程几个新的预估-校正法

doi:10.12286/jssx.2013.4.401

计算数学 ›› 2013, Vol. 35 ›› Issue (4): 401-418.DOI: 10.12286/jssx.2013.4.401

求解非对称代数Riccati 方程几个新的预估-校正法

黄娜¹, 马昌凤¹, 谢亚君²

1. 福建师范大学数学与计算机科学学院, 福州 350007;
2. 福建江夏学院信息系, 福州 350108

收稿日期:2013-04-11 出版日期:2013-11-15 发布日期:2013-12-03
基金资助:
国家自然科学基金（11071041，11201074）资助项目；福建省自然科学基金（2013J01006）资助项目。

PDF ( 1170 ) 引用

黄娜, 马昌凤, 谢亚君. 求解非对称代数Riccati 方程几个新的预估-校正法[J]. 计算数学, 2013, 35(4): 401-418.

Huang Na, Ma Changfeng, Xie Yajun. SOME PREDICTOR-CORRECTOR-TYPE ITERATIVE SCHEMES FOR SOLVING NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS ARISING IN TRANSPORT THEORY[J]. Mathematica Numerica Sinica, 2013, 35(4): 401-418.

SOME PREDICTOR-CORRECTOR-TYPE ITERATIVE SCHEMES FOR SOLVING NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS ARISING IN TRANSPORT THEORY

Huang Na¹, Ma Changfeng¹, Xie Yajun²

1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China;
2. Department of Information, Fujian jiangxia University, Fuzhou 350108, China

Received:2013-04-11 Online:2013-11-15 Published:2013-12-03

来源于输运理论的非对称代数Riccati 方程可等价地转化成向量方程组来求解. 本文提出了求解该向量方程组的几个预估—校正迭代格式，证明了这些迭代格式所产生的序列是严格单调递增且有上界，并收敛于向量方程组的最小正解. 最后，给出了一些数值实验，实验结果表明，本文所提出的算法是有效的.

关键词： 非对称代数Riccati方程, 迭代格式, 最小正解, 收敛性分析, 数值实验

It is as well known that nonsymmetric algebraic Riccati equations arising in transport theory can be translated to vector equations. In this paper, we propose some predictorcorrector-type iterative schemes to solve the vector equations. And we prove that all the sequence generated by the iterative schemes, which converges to the minimal positive solution of the vector equations, are strictly and monotonically increasing and bounded above. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.

Key words: Nonsymmetric algebraic Riccati equation, iterative scheme, minimal positive solution, convergence analysis, numerical experiment

MR(2010)主题分类:

分享此文:

[1] Juang J, Lin W W. Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices[J]. SIAM J. Matrix Anal. Appl., 1999, 20: 228-243.

[2] Juang J. Existence of algebraic matrix Riccati equations arising in transport theory[J]. Linear Algebra Appl., 1995, 230: 89-100.

[3] Lu L Z. Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory[J]. SIAM J. Matrix Anal. Appl., 2005, 26: 679-685.

[4] Guo C H, Laub A J. On the iterative solution of a class of nonsymmetric algebraic Riccati equations[J]. SIAM J. Matrix Anal. Appl., 2000, 22: 376-391.

[5] Bao L, Lin Y, Wei Y. A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory[J]. Appl. Math. Comput., 2006, 181: 1499-1504.

[6] Bai Z Z, Gao Y H, Lu L Z. Fast iterative schemes for nonsymmetric algebraic Riccati equations arising from transport theory[J]. SIAM J. Sci. Comput., 2008, 30: 804-818.

[7] C.H. Guo, Lin W W. Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory[J]. Linear Algebra Appl., 2010, 432: 283-291.

[8] Wu S, Huang C. Two-step relaxation Newton method for nonsymmetric algebraic Riccati equations arising from transport theory[J]. Math. Probl. Eng., 2009, 12: 1-17.

[9] Lin Y. A class of iterative methods for solving nonsymmetric algebraic Riccati equations arising in transport theory[J]. Comput. Math. Appl., 2008, 56: 3046-3051.

[10] Lin Y, Bao L, Wu Q. On the convergence rate of an iterative method for solving nonsymmetric algebraic Riccati equations[J]. Comput. Math. Appl., 2011, 62: 4178-4184.

[11] Berman A, Plemmons R J. Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.

[12] Fiedler M, Ptak V. On matrices with non-positive off-diagonal elements and positive principal minors[J]. Czechoslovak Math. J., 1962, 12: 382-400.

[13] Bini D A, Meini B and Poloni F. From algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms, Proceeding of Numerical Methods for Structured Markov Chains, Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany 2008.

[14] Bini D A, Meini B and Poloni F. Fast solution of a certain Riccati equation through Cauchy-like matrices[J]. Electron. Trans. Numer. Anal., 2009, 33: 84-104.

[15] Gao Y H and Bai Z Z. On inexact Newton methods based on doubling iteration scheme for nonsymmetric algebraic Riccati equations[J]. Numer. Linear Algebra Appl., 2011, 18: 325-341.

[16] Guo C H. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for Mmatrices[J]. SIAM J. Matrix Anal. Appl., 2001, 23: 225-242.

[17] Guo X X, Lin W W and Xu S F. A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation[J]. Numer. Math., 2006, 103: 393-412.

[18] Juang J and Chen I D. Iterative solution for a certain class of algebraic matrix Riccati equations arising in transport theory[J]. Tansport Theory Statist. Phys., 1993, 22: 65-80.

[19] Juang J and Lin Z T. Convergence of an iterative technique for algebraic matrix Riccati euqations and applications to transport theory[J]. Tansport Theory Statist. Phys., 1992, 21: 87-100.

[20] Bai Z Z, Guo X X and Xu S F. Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations[J]. Numer. Linear Algebra Appl., 2006, 13: 655-674.

[21] Benner P, Mena H and Saak J. On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations[J]. Electr. Trans. Num. Anal., 2008, 29: 136-149.

[22] Benner P and Saak J. A Galerkin-Newton-ADI method for solving largescale algebraic Riccati equations, Preprint SPP1253-090, DFG Priority Programme Optimization with Partial Differential Equations, (2010) . URL http:././www.am.unierlangen.de./ home./spp1253./wiki./index.php./Preprints

[23] Bini D A, Iannazzo B and Poloni F. A fast Newton's method for a nonsymmetric algebraic Riccati equation[J]. SIAM J. Matrix Anal. Appl., 2008, 30: 276-290.

[24] Guo C H, Iannazzo B and Meini B. On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equations[J]. SIAM J. Matrix Anal. Appl., 2007, 29: 1083-1100.

[25] Lu L Z. Newton iterations for a nonsymmetric algebraic Riccati equation[J]. Numer. Linear Algebra Appl., 2005, 12: 191-200.

[26] Li J R and White J. Low-rank solution of Lyapunov equations[J]. SIAM J. Matrix Anal. Appl., 2002, 24: 260-280.

[27] Martinsson P G, Rokhlin V and Tygert M. A fast algorithm for the inversion of general Toeplitz matrices[J]. Comput. Math. Appl., 2005, 50: 741-752.

[28] Penzl T. A cyclic low-rank smith method for large sparse Lyapunov equations[J]. SIAM J. Sci. Comput., 2000, 21: 1401-1418.

[29] Wachspress E L. Optimum alternating-direction-implicit iteration parameters for a model problem[J]. J. Soc. Indust. Appl. Math., 1962, 10: 339-350.

[30] Wachspress E L. ADI iteration parameters for solving Lyapunov and Sylvester equations, Note of private communication, 2009.

[31] Bai Z Z. On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations[J]. J. Comp. Math., 2011, 29: 185-198.

[32] Bai Z Z, Golub G H and Ng M K. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems[J]. SIAM J. Matrix Anal. Appl., 2003, 24: 603-626.

[1]	杨学敏, 牛晶, 姚春华. 椭圆型界面问题的破裂再生核方法[J]. 计算数学, 2022, 44(2): 217-232.
[2]	古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.
[3]	甘小艇. 状态转换下欧式Merton跳扩散期权定价的拟合有限体积方法[J]. 计算数学, 2021, 43(3): 337-353.
[4]	李旭, 李明翔. 连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速[J]. 计算数学, 2021, 43(3): 354-366.
[5]	古振东, 孙丽英. 非线性第二类Volterra积分方程的Chebyshev谱配置法[J]. 计算数学, 2020, 42(4): 445-456.
[6]	闫熙, 马昌凤. 求解矩阵方程AXB+CXD=F参数迭代法的最优参数分析[J]. 计算数学, 2019, 41(1): 37-51.
[7]	王志强, 文立平, 朱珍民. 时间延迟扩散-波动分数阶微分方程有限差分方法[J]. 计算数学, 2019, 41(1): 82-90.
[8]	陈圣杰, 戴彧虹, 徐凤敏. 稀疏线性规划研究[J]. 计算数学, 2018, 40(4): 339-353.
[9]	古振东, 孙丽英. 一类弱奇性Volterra积分微分方程的级数展开数值解法[J]. 计算数学, 2017, 39(4): 351-362.
[10]	刘丽华, 马昌凤, 唐嘉. 求解广义鞍点问题的一个新的类SOR算法[J]. 计算数学, 2016, 38(1): 83-95.
[11]	陈绍春, 梁冠男, 陈红如. Zienkiewicz元插值的非各向异性估计[J]. 计算数学, 2013, 35(3): 271-274.
[12]	任志茹. 三阶线性常微分方程Sinc方程组的结构预处理方法[J]. 计算数学, 2013, 35(3): 305-322.
[13]	范斌, 马昌凤, 谢亚君. 求解非线性互补问题的一类光滑Broyden-like方法[J]. 计算数学, 2013, 35(2): 181-194.
[14]	张亚东, 石东洋. 各向异性网格下抛物方程一个新的非协调混合元收敛性分析[J]. 计算数学, 2013, 35(2): 171-180.
[15]	陈争, 马昌凤. 求解非线性互补问题一个新的 Jacobian 光滑化方法[J]. 计算数学, 2010, 32(4): 361-372.

阅读次数

全文

摘要

求解非对称代数Riccati 方程几个新的预估-校正法

SOME PREDICTOR-CORRECTOR-TYPE ITERATIVE SCHEMES FOR SOLVING NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS ARISING IN TRANSPORT THEORY

扫码分享