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Solving Symmetric Algebraic Riccati Equations with High Order Iterative Schemes

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Abstract

We solve symmetric algebraic Riccati equation using efficient high-order iterative schemes which improve the speed of convergence of others widely used in the literature. These high-order iterative schemes involve the Riccati operator and the first Fréchet derivative. Applying these iterative schemes is equivalent to solving a fixed number of Lyapunov equations with the same matrix. This key fact makes efficient these methods. Moreover, a local convergence result of one of these high-order iterative schemes is analyzed. Finally, numerical experiments confirm the advantageous performance of these schemes.

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References

  1. Altman, G.: Iterative methods of higher order. Bull. Acad. Pollon. Sci. Sér. des Sci. Math. Astron. Phys. IX, 62–68 (1961)

  2. Amat, S., Busquier, S.: Geometry and convergence of some third-order methods. Southwest J. Pure Appl. Math. 2, 61–72 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1), 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Argyros, I.K.: Newton-like methods under mild differentiability conditions with error analysis. Bull. Aust. Math. Soc. 37, 131–147 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Argyros, I.K., Chen, D., Qian, Q.S.: A local convergence theorem for the super-Halley method in a Banach space. Appl. Math. Lett. 7(5), 49–52 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, Z.Z., Guo, X.X., Yin, J.F.: On two iteration methods for the quadratic matrix equations. Int. J. Numer. Anal. Model 2, 114–122 (2005)

    MathSciNet  Google Scholar 

  8. Bartels, R.H., Stewart, G.W.: Algorithm 432: solution of the matrix equation \(AX + XB = C\). Commun. Assoc. Comput. Mach. 15(9), 820–826 (1972)

    MATH  Google Scholar 

  9. Benner, P., Byers, R., Quintana-Ortí, E.S., Quintana-Ortí, G.: Solving algebraic Riccati equations on parallel computers using Newton’s method with exact line search. Parallel Comput. 26, 1345–1368 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Benner, P., Laub, A.J., Mehrmann, V.: A collection of benchmark examples for the numerical solution of algebraic Riccati equations. IEEE Control Syst. 17(5), 18–28 (1997)

    Article  Google Scholar 

  11. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  12. Golub, G.H., Van Loan, C.F.: Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  13. Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: Chebyshev-like methods and quadratic equations. Rev. Anal. Numér. Théor. Approx. 28(1), 23–35 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Ezquerro, J.A., Hernández, M.A., Romero, N.: A modification of Cauchy’s method for quadratic equations. J. Math. Anal. Appl. 339(2), 954–969 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ezquerro, J.A., Hernández, M.A., Romero, N.: An extension of Gander’s result for quadratic equations. J. Comput. Appl. Math. 234, 960–971 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, C.H., Lancaster, P.: Analysis and modification of Newton’s method for algebraic Riccati equations. Math. Comput. 67, 1089–1105 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55(1), 113–130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hernández, M.A.: A note on Halley’s method. Numer. Math. 59(3), 273–276 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41, 433–445 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hernández, M.A., Romero, N.: On a characterization of some Newton-like methods of \(R\)-order at least three. J. Comput. Appl. Math. 183(1), 53–66 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Higham, N.J., Kim, H.M.: Solving a quadratic matrix equation by Newton’s method with exact line searches. SIAM J. Matrix Anal. Appl. 23, 303–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hilton, P., Pedersen, J.: Catalan numbers, their generalization and their uses. Math. Intell. 13, 64–75 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  25. Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  26. Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford Science Publications, Oxford (1995)

    MATH  Google Scholar 

  27. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  28. Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1966)

    MATH  Google Scholar 

  29. Patnaik, L.M., Viswanadham, N., Sarma, I.G.: Computer control algorithms for a tubular ammonia reactor. IEEE Trans. Autom. Control 25(4), 642–651 (1980)

    Article  MATH  Google Scholar 

  30. Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computation, University of La Rioja, Calle Madre de Dios, 53, 26006, Logroño, Spain

    M. A. Hernández-Verón & N. Romero

Authors
  1. M. A. Hernández-Verón
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  2. N. Romero
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Correspondence to N. Romero.

Additional information

The research has been partially supported by the project MTM2014-52016-C2-1-P of the Spanish Ministry of Economy and Competitiveness.

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Hernández-Verón, M.A., Romero, N. Solving Symmetric Algebraic Riccati Equations with High Order Iterative Schemes. Mediterr. J. Math. 15, 51 (2018). https://doi.org/10.1007/s00009-018-1092-1

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  • DOI: https://doi.org/10.1007/s00009-018-1092-1

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