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A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

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Abstract

A large scale nonsymmetric algebraic Riccati equation X C X − X E − A X + B = 0 arising in transport theory is considered, where the n × n coefficient matrices B,C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop count of original SDA_ls by half. Numerical experiments illustrate the effectiveness of our method.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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, 655–574 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bini, D.A., Meini, B., Poloni, F.: Fast solution of a certain Riccati equation through Cauchy-like matrices. Electron. Trans. Numer. Anal. 33(1), 84–104 (Unknown Month 2008)

  4. Bini, D.A., Iannazzo, B., Poloni, F.: A fast Newton’s method for a nonsymmetric algebraic Riccati equation. SIAM. J. Matrix Anal. Appl. 30, 276–290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Guo, C.-H., Lin, W.-W.: Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory. Linear Algebra Appl. 432(1), 283–291 (2010)

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

    MathSciNet  MATH  Google Scholar 

  11. Guo, P.-C., Guo, X.-X.: A modified structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equations from transport theory. J. Comput. Appl. Math. 261, 213–220 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lu, L.-Z.: Newton iterations for a non-symmetric algebraic Riccati equation. Numer. Linear Algebra Appl. 12, 191–200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, T.-X., Chu, E. K., Kuo, Y.-C., Lin, W.-W.: Solving large-scale nonsymmetric algebraic Riccati equations by doubling. SIAM J. Matrix Anal. Appl. 34, 1129–1147 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mehrmann, V., Xu, H.: Explicit solutions for a Riccati equation from transport theory. SIAM J. Matrix Anal. Appl. 30, 1339–1357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu, B., Li, D.-H., Dong, N.: Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory. J. Comput. Appl. Math. 250, 175–189 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Pei-Chang Guo

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  1. Pei-Chang Guo
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Guo, PC. A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory. Numer Algor 71, 541–552 (2016). https://doi.org/10.1007/s11075-015-0008-4

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  • DOI: https://doi.org/10.1007/s11075-015-0008-4

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