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Research Article

On an inversion-free algorithm for the nonlinear matrix problem Xα+A∗X−βA+B∗X−γB=I,

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Pages 2555-2567 | Received 19 Dec 2021, Accepted 02 May 2022, Published online: 18 May 2022

References

  • R. Bhatia, Matrix Analysis, Springer-Velag, New York, 1997.
  • Ph. Bougerol, Kalman filtering with random coefficients and contractions, SIAM J. Control Optim.31 (1993), pp. 942–959.
  • J. Cai and G. Chen, Some investigation on Hermitian positive definite solutions the matrix equation Xs+A∗X−tA=Q, Linear Algebra Appl. 430 (2009), pp. 2448–2456.
  • J. Cai and G. Chen, On the Hermitian positive definite solutions of nonlinear matrix equation Xs+A∗X−tA=Q, Appl. Math. Comput. 217 (2010), pp. 117–123.
  • E. Catinaş, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput. 343 (2019), pp. 1–20.
  • S. El-Sayed, A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation, ANZIAM J. 45 (2003), pp. 145–152.
  • S.M. El-Sayed and A.C.M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2002), pp. 632–645.
  • J.C. Engwerda, C.M.A. Ran and A.L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A∗X−1A=Q, Linear Algebra Appl. 186 (1993), pp. 255–275.
  • T. Furuta, Operator inequalities associated with Holder–McCarthy and Kantorovich inequalities, J. Inequal. Appl. 6 (1998), pp. 137–148.
  • V.I. Hasanov and A.A. Ali, On convergence of three iterative methods for solving of the matrix equation X+A∗X−1A+B∗X−1B=Q, Comput. App. Math. 36 (2017), pp. 79–87.
  • V.I. Hasanov and S. Hakkaev, Newton's method for nonlinear matrix equation, C. R. Acad. Bulg. Sci.68 (2015), pp. 973–982.
  • N. Huang and C.F. Ma, The inversion-free iterative methods for solving the nonlinear matrix equation X=I−AHX−1A−BHX−1B, Abstr. Appl. Anal. 2013 (2013), Article ID 843785, 7 pages.
  • B. Huang and C. Ma, Some iterative algorithms for positive definite solution to nonlinear matrix equations, J. Appl. Anal. Comput. 9 (2019), pp. 526–546.
  • N. Huang, C. Ma and J. Tang, The inversion-free iterative methods for a system of nonlinear matrix equations, Int. J. Comput. Math. 93 (2016), pp. 1470–1483.
  • I.G. Ivanov, V.I. Hasanov and B.V. Minchev, On matrix equations X±A∗X−2A=I, Linear Algebra Appl. 326 (2001), pp. 27–44.
  • A. Liu and G. Chen, On the Hermitian positive definite solutions of nonlinear matrix equation Xs+A∗X−t1A+B∗X−t2B=Q, Math. Probl. Eng. 2011 (2011), Article ID 163585, 18 pages.
  • X.G. Liu and H. Gao, The positive definite solutions of the matrix equations Xs±ATX−tA=I, Linear Algebra Appl.368 (2003), pp. 83–97.
  • J.H. Long, X.Y. Hu and L. Yhang, On the Hermitian positive definite solution of the nonlinear matrix equation X=I−A∗X−1A−B∗X−1B, Bull. Braz. Math. Soc. 39 (2008), pp. 371–386.
  • F. Mannel, On the order of convergence of Broyden's method: faster convergence on mixed linear-nonlinear systems of equations and a conjecture on the q-order, Calcolo 58 (2021), Article ID 47.
  • J. Meng, H. Chen, Y.J. Kim and H.M. Kim, A further study on a nonlinear matrix equation, Jpn. J. Ind. Appl. Math. 37 (2020), pp. 831–849.
  • S. Pakhira, S. Bose and Sk.M. Hossein, Solution of a class of nonlinear matrix equations, Bull. Iran. Math. Soc. 47 (2021), pp. 415–434.
  • Y.Y. Peng, S.M. El-sayed and X.L. Yhang, Iterative methods for the extremal positive definite solution of the matrix equation X+A∗X−αA=Q, J. Comput. Appl. Math. 200 (2007), pp. 520–527.
  • S. Shil and H. Kumar Nashine, Latest inversion-free iterative scheme for solving a pair of nonlinear matrix equations, J. Math. 2021 (2021), Article ID 2667885, 22 pages.
  • A. Taherian and F. Toutounian, Preconditioned global GPBiCG method for solving saddle point problems with multiple right-hand sides and its convergence analysis, Iran. J. Numer. Anal. Optim. 12 (2022), pp. 111–129.
  • S. Vaezzadeh, S.M. Vaezpour and R. Saadati, On nonlinear matrix equations, Appl. Math. Lett. 26 (2013), pp. 919–923.
  • S. Vaezzadeh, S.M. Vaezpour, R. Saadati and C. Park, The iterative methods for solving nonlinear matrix equation, X+AHX−1A+BHX−1B=Q, Adv. Differ. Equ. 2013 (2013), Article ID 229.
  • D.M. Yhou, G.I. Chen, G.X. Wu and X.Y. Yhang, Some properties of the nonlinear matrix equation Xs+A∗X−tA=Q, J. Math. Anal. Appl. 392 (2012), pp. 75–82.
  • T. Zhanlav, C. Chun, Kh. Otgondorj and V. Ulziibayar, High-order iterations for systems of nonlinear equations, Int. J. Comput. Math. 97 (2020), pp. 1704–1724.

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