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On an iterative algorithm converging to the solution of \(XCX=D\)

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Abstract

Based on a modified Newton method in the Banach algebra of square matrices, we construct here a simple iterative algorithm that converges to the unique positive solution of the algebraic Riccati equation \(XCX=D\). Numerical examples illustrating the theoretical study and showing the speed of convergence of our approach are discussed as well. At the end, we put some open questions as purposes for future research.

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References

  1. Ando, T., Li, C.K., Mathias, R.: Geometric means. Linear Algebra Appl. 38, 305–334 (2004)

    Article  MathSciNet  Google Scholar 

  2. Atteia, M., Raïssouli, M.: Self dual operators on convex functionals, geometric mean and square root of convex functionals. J. Convex Anal. 8, 223–240 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Bhatia, R.: Matrix Analysis. Verlag, New York (1997)

    Book  Google Scholar 

  4. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1984)

    MATH  Google Scholar 

  5. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    Book  Google Scholar 

  6. Higham, N.J.: Newton’s method for the matrix square root 46(174), 537–549 (1986)

  7. Jung, C., Lee, H., Lim, Y., Yamazaki, T.: Weighted geometric mean of \(n\)-operators and \(n\)-parameters. Linear Algebra Appl. 432, 1515–1530 (2010)

    Article  MathSciNet  Google Scholar 

  8. Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)

    Article  MathSciNet  Google Scholar 

  9. Nakamura, N.: Geometric operator mean induced from the Riccati Equation, Sc. Math. Jap. Online, e-2007, 387–391

  10. Zeidler, E.: Nonlinear Functional Analysis and its Applications III. Springer, Berlin (1986)

    Book  Google Scholar 

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Acknowledgements

We would like to express our sincere thanks to the anonymous referee for his/her useful comments and suggestions which substantially helped improving the quality of the present manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Science Faculty, Taibah University, P.O.Box 30097, Medina, 41477, Saudi Arabia

    Nawal Al-Harbi & Mustapha Raïssouli

  2. Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia

    Nawal Al-Harbi

  3. Department of Mathematics, Science Faculty, Moulay Ismail University, Meknes, Morocco

    Mustapha Raïssouli

Authors
  1. Nawal Al-Harbi
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  2. Mustapha Raïssouli
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Correspondence to Mustapha Raïssouli.

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Al-Harbi, N., Raïssouli, M. On an iterative algorithm converging to the solution of \(XCX=D\). Afr. Mat. 31, 997–1007 (2020). https://doi.org/10.1007/s13370-020-00776-3

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  • DOI: https://doi.org/10.1007/s13370-020-00776-3

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