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Numerical Study on Nonsymmetric Algebraic Riccati Equations

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Abstract

In this paper, we consider the nonsymmetric algebraic Riccati equation whose four coefficient matrices form an M-matrix K. When K is a regular M-matrix, the Riccati equation is known to have a minimal nonnegative solution. We present two new numerical methods which can be applied directly in the case where K is a regular M-matrix. Furthermore, we find that the alternately linearized implicit iteration method is also feasible. In addition, we study the monotone convergence property of the proposed methods. Numerical experiments show that the above three numerical iteration methods are feasible and effective for solving the nonsymmetric algebraic Riccati equation.

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References

  1. Guo, C.H.: On algebraic Riccati Equations Associated with M-Matrices. Linear Algebra Appl. 439(10), 2800–2814 (2013)

  2. Juang J.: Existence of algebraic matrix Riccati equations arising in transport theory. Linear Algebra Appl. 230, 89–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Rogers L.C.G.: Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Probab. 4, 390–413 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bini D.A., Iannazzo B., Latouche G., Meini B.: On the solution of Riccati equations arising in fluid queues. Linear Algebra Appl. 413, 474–494 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bini D.A., Meini B., Poloni F.: Transforming algebraic Riccati equations into unilateral quadratic matrix equations. Numer. Math. 116, 553–578 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chiang C.-Y., Chu E.K.-W., Guo C.-H., Huang T.-M., Lin W.-W., Xu S.-F.: Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case. SIAM J. Matrix Anal. Appl. 31, 227–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao Y.-H., Bai Z.-Z.: On inexact Newton methods based on doubling iteration scheme for non-symmetric algebraic Riccati equations. Numer. Linear Algebra Appl. 18, 325–341 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Laub A.: A Schur method for solving algebraic Riccati equations. Autom. Control IEEE Trans. 24(6), 913–921 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Iannazzo, B., Poloni, F.: A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M-matrix. Numer. Linear Algebra Appl. 20, 440–452 (2013)

  11. Juang J., Lin W.-W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20, 228–243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xue J., Xu S., Li R.-C.: Accurate solutions of M-matrix algebraic Riccati equations. Numer. Math. 120, 671–700 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo C.H.: Nonsymmetric algebraic Riccati equations and Wiener–Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl. 23(1), 225–242 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo C.H., Laub A.J.: On a Newton-like method for solving algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 21(2), 694–698 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo X.X., Lin W.W., Xu S.F.: A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer. Math. 103(3), 393–412 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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(8), 655–674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang W., Wang W., Li R.C.: Alternating-directional doubling algorithm for M-matrix algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 33(1), 170–194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo C.H.: A note on the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation. Linear Algebra Appl. 357, 299–302 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo C.H.: Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models. J. Comput. Appl. Math. 192, 353–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo C.H., Higham N.J.: Iterative solution of a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 29, 396–412 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo C.H., Iannazzo B., Meini B.: On the doubling algorithm for a (shifted) nonsym-metric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 29, 1083–1100 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo C.H., Bai Z.Z.: On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation. J. Comput. Math. 23, 305–320 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Gao Y.H., Bai Z.Z.: On inexact Newton methods based on doubling iteration scheme for non-symmetric algebraic Riccati equations. Numer. Linear Algebra Appl. 18, 325–341 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bini, D.A.; Iannazzo, B.; Meini, B.: Numerical Solution of Algebraic Riccati Equations. SIAM, Philadelphia, PA (2012)

  25. Lin W.W., Xu S.F.: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl. 28, 26–39 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Guo, C.H.: Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations. Numer. Algorithms 64(2), 295–309 (2013)

  27. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences, re-vised reprint of the 1979 Academic Press original. SIAM, Philadelphia (1994)

  28. Guo, C.H.: On Algebraic Riccati equations associated with M-matrices. Linear Algebra Appl. 439, 2800–2814 (2013)

  29. Guo C.H., Laub A.J.: On the iterative solution of a class of nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 22(2), 376–391 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computer Science and FJKLMAA, Fujian Normal University, Fuzhou, 350117, China

    Changfeng Ma & Huaize Lu

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  1. Changfeng Ma
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  2. Huaize Lu
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Correspondence to Changfeng Ma.

Additional information

The project is supported by the National Natural Science Foundation of China (Grant No. 11071041) and Fujian Natural Science Foundation (Grant No. 2016J01005).

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Ma, C., Lu, H. Numerical Study on Nonsymmetric Algebraic Riccati Equations. Mediterr. J. Math. 13, 4961–4973 (2016). https://doi.org/10.1007/s00009-016-0786-5

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  • DOI: https://doi.org/10.1007/s00009-016-0786-5

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