On the positive definite solutions of a nonlinear matrix equation.

*(English)*Zbl 1268.15013Summary: The positive definite solutions of the nonlinear matrix equation \(X^s + A^\ast f(X)A = Q\) are discussed. A necessary and sufficient condition for the existence of positive definite solutions for this equation is derived. Then, the uniqueness of the Hermitian positive definite solution is studied based on an iterative method proposed in this paper. Lastly, the perturbation analysis for this equation is discussed.

##### MSC:

15A24 | Matrix equations and identities |

65F30 | Other matrix algorithms (MSC2010) |

65H10 | Numerical computation of solutions to systems of equations |

15B48 | Positive matrices and their generalizations; cones of matrices |

##### Keywords:

positive definite solutions; nonlinear matrix equation; iterative method; perturbation analysis
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\textit{P. Liu} et al., J. Appl. Math. 2013, Article ID 676978, 6 p. (2013; Zbl 1268.15013)

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##### References:

[1] | A. C. M. Ran and M. C. B. Reurings, “On the nonlinear matrix equation X+A* \Cal F(X)A=Q: solutions and perturbation theory,” Linear Algebra and Its Applications, vol. 346, pp. 15-26, 2002. · Zbl 1086.15013 |

[2] | I. G. Ivanov, “On positive definite solutions of the family of matrix equations X+A\ast X - nA=Q,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 277-301, 2006. · Zbl 1096.15003 |

[3] | J. F. Wang, Y. H. Zhang, and B. R. Zhu, “The Hermitian positive definite solutions of the matrix equation X+A\ast X - qA=I(q>0),” Mathematica Numerica Sinica, vol. 26, no. 1, pp. 61-72, 2004 (Chinese). |

[4] | S. M. El-Sayed and A. M. Al-Dbiban, “On positive definite solutions of the nonlinear matrix equation X+A\ast X - nA=I,” Applied Mathematics and Computation, vol. 151, no. 2, pp. 533-541, 2004. · Zbl 1055.15022 |

[5] | S. M. El-Sayed and A. C. M. Ran, “On an iteration method for solving a class of nonlinear matrix equations,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 3, pp. 632-645, 2001/02. · Zbl 1002.65061 |

[6] | S. M. El-Sayed and M. G. Petkov, “Iterative methods for nonlinear matrix equations X+A\ast X - \alpha A=I(\alpha >0),” Linear Algebra and Its Applications, vol. 403, pp. 45-52, 2005. · Zbl 1074.65057 |

[7] | V. I. Hasanov, “Positive definite solutions of the matrix equations X\pm A\ast X - qA=Q(0<q\leq 1),” Linear Algebra and Its Applications, vol. 404, pp. 166-182, 2005. · Zbl 1078.15012 |

[8] | X. F. Duan and A. P. Liao, “The Hermitian positive definite solution of the matrix equation X+A\ast X - qA=Q(q\geq 1) and its perturbation analysis,” Numerical Mathematics, vol. 30, no. 3, pp. 280-288, 2008 (Chinese). · Zbl 1174.15338 |

[9] | X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations Xs\pm ATX - tA=In,” Linear Algebra and Its Applications, vol. 368, pp. 83-97, 2003. · Zbl 1025.15018 |

[10] | Y. Yueting, “The iterative method for solving nonlinear matrix equation Xs+A\ast X - tA=Q,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 46-53, 2007. · Zbl 1131.65039 |

[11] | G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press, Boston, Mass, USA, 1990. · Zbl 0706.65013 |

[12] | R. Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1997. · Zbl 0863.15001 |

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