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The King-Werner method for solving nonsymmetric algebraic Riccati equation. (English) Zbl 1202.65055
The authors consider a nonsymmetric algebraic Riccati equation that is transformed into a vector form. The King-Werner method is used to solve this equation for the minimal positive solution of the vector equation. The authors give convergence and error analysis of the method and use numerical tests to show that this method is able to obtain the minimal positive solution of the vector equation.
MSC:
65F30 Other matrix algorithms (MSC2010)
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[1] Lu, L.-Z., Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory, SIAM J. matrix anal. appl., 26, 679-685, (2005) · Zbl 1072.15016
[2] Juang, J.; Chen, I., Iterative solution for a certain class of algebraic matrix Riccati equations aring in transport theory, Trans. theory stat. phys., 22, 65-80, (1993) · Zbl 0774.65020
[3] Juang, J.; Lin, Z.-T., Convergence of an iterative technique for algebraic matrix Riccati equations and applications in to transport theory, Trans. theory stat. phys., 21, 87-100, (1992) · Zbl 0759.65022
[4] Juang, J., Existence of algebraic matrix Riccati equations arising in transport theory, Linear algebra appl., 230, 89-100, (1995) · Zbl 0839.15006
[5] Juang, J.; Lin, W.-W., Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices, SIAM J. matrix anal. appl., 20, 228-243, (1999) · Zbl 0922.15003
[6] Guo, C.-H., Nonsymmetric algebraic Riccati equations and wiener – hopf factorization for mmatrices, SIAM J. matrix anal. appl., 23, 225-242, (2001) · Zbl 0996.65047
[7] Bao, L.; Lin, Y.-Q.; Wei, Y.-M., A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory, Appl. math. comput., 181, 1499-1504, (2006) · Zbl 1115.65040
[8] Lu, L.-Z., Newton iterations for a nonsymmetric algebraic Riccati equation, Numer. linear algebra appl., 12, 191-200, (2005) · Zbl 1164.65386
[9] Lin, Y.-Q.; Bao, L.; Wei, Y.-M., A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory, IMA J. numer. anal., 28, 215-224, (2008) · Zbl 1144.65030
[10] Bai, Z.-Z.; Gao, Y.-H.; Lu, L.-Z., Fast iterative schemes for nonsymmetric algebraic Riccati equations arising from transport theory, SIAM J. sci. comput., 30, 804-818, (2008) · Zbl 1166.65065
[11] Lin, Y.-Q., A class of iterative methods for solving nonsymmetric algebraic Riccati equations arising in transport theory, Comput. math. appl., 56, 3046-3051, (2008) · Zbl 1165.65348
[12] Guo, C.-H.; Laub, A.J., On the iterative solution of a class of nonsymmetric algebraic Riccati equations, SIAM J. matrix anal. appl., 22, 376-391, (2000) · Zbl 0973.65025
[13] Bai, Z.-Z.; Guo, X.-X.; Xu, S.-F., Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations, Numer. linear algebra appl., 13, 655-674, (2006) · Zbl 1174.65381
[14] Guo, X.-X.; Lin, W.-W.; Xu, S.-F., A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation, Numer. math., 103, 393-412, (2006) · Zbl 1097.65055
[15] Bini, D.A.; Iannazzo, B.; Poloni, F., A fast Newton method for a nonsymmetric algebraic Riccati equation, SIAM J. matrix anal. appl., 30, 276-290, (2008) · Zbl 1166.15300
[16] Werner, W., Über ein verfahren der ordnung \(1 + \sqrt{2}\) zur nullstellenbestimmung, Numer. math., 32, 333-342, (1979) · Zbl 0431.65040
[17] King, R.F., Tangent method for nonlinear equations, Numer. math., 18, 298-304, (1972) · Zbl 0215.27403
[18] Werner, W., Some supplementary results on the \(1 + \sqrt{2}\) order method for the solution of nonlinear equations, Numer. math., 38, 383-392, (1982) · Zbl 0478.65029
[19] Wang, X.-H.; Zheng, S.-M., On the convergence of king – werner’s iteration procedure for solving nonlinear equations, Math. numer. sinica, 2, 70-79, (1982) · Zbl 0538.65025
[20] Berman, A.; Plemmons, R., Nonnegative matrices in the mathematical sciences, (1979), Academic Press NY · Zbl 0484.15016
[21] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press NY · Zbl 0576.15001
[22] Chen, J.-L.; Chen, X.-H., Special matrices (in Chinese), (2000), Tsinghua University Press Beijing
[23] Decker, D.W.; Keller, H.B.; Kelley, C.T., Convergence rates for newton’s method at singular points, SIAM J. numer. anal., 20, 296-314, (1983) · Zbl 0571.65046
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