On positive definite solutions of a class of nonlinear matrix equation
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This paper is concerned with the positive definite solutions of the nonlinear matrix equation $X-A^{*}\bar{X}^{-1}A=Q$, where $A, Q$ are given complex matrices with $Q$ positive definite. We show that such a matrix equation always has a unique positive definite solution and if $A$ is nonsingular, it also has a unique negative definite solution. Moreover, based on Sherman-Morrison-Woodbury formula, we derive elegant relationships between solutions of $X-A^{*}\bar{X}^{-1}A=I$ and the well-studied standard nonlinear matrix equation $Y+B^{*}Y^{-1}B=Q$, where $B, Q$ are uniquely determined by $A$. Then several effective numerical algorithms for the unique positive definite solution of $X-A^{*}\bar{X}^{-1}A=Q$ with linear or quadratic convergence rate such as inverse-free fixed-point iteration, structure-preserving doubling algorithm, Newton algorithm are proposed. Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.
Abstract
This paper is concerned with the positive definite solutions of the nonlinear matrix equation $X-A^{*}\bar{X}^{-1}A=Q$, where $A, Q$ are given complex matrices with $Q$ positive definite. We show that such a matrix equation always has a unique positive definite solution and if $A$ is nonsingular, it also has a unique negative definite solution. Moreover, based on Sherman-Morrison-Woodbury formula, we derive elegant relationships between solutions of $X-A^{*}\bar{X}^{-1}A=I$ and the well-studied standard nonlinear matrix equation $Y+B^{*}Y^{-1}B=Q$, where $B, Q$ are uniquely determined by $A$. Then several effective numerical algorithms for the unique positive definite solution of $X-A^{*}\bar{X}^{-1}A=Q$ with linear or quadratic convergence rate such as inverse-free fixed-point iteration, structure-preserving doubling algorithm, Newton algorithm are proposed. Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.
참고문헌 (28)
- M. Berzig, "Positive definite solution of the matrix equation <TEX>$X\;=\;Q\;-\;A*X^{-1}\;A+B*X^{-1}\;B$<", TEX> via Bhaskar-Lakshmikantham fixed point theorem, 2012, pp.6, 55~62
- TEX>, in Current trends in matrix theory, "Consimilarity and the matrix equation <TEX>$A{\overline{X}}\;-\;XB\;=\;C$<", pp.1986, J. H. Bevis
- J. H. Bevis, "The matrix equation <TEX>$A{\overline{X}}\;-\;XB\;=\;C$<", TEX> and its special cases, 1988, pp.9(3), 348~359
- R. Bhatia, "Matrix Analysis, Graduate Texts in Mathematics, 169" (1997), Springer-Verlag
- M. T. Chu, "On the first degree Fejer-Riesz factorization and its applications to <TEX>$X\;+\;A*X^{-1}\;A\;=\;Q$<", TEX>, 2016, pp.489, 123~143
- XP.-F. Duan, "On the nonlinear matrix equation <TEX>$X\;-\;{{\sum}_{i=1}^{m}}A_{i}^{*}X^{{\delta}_{i}}A_{i}\;=\;Q$<", TEX>, 2008, pp.429(1), 110~121
- X.-F. Duan, "Positive definite solution of a class of nonlinear matrix equation", Linear Multilinear Algebra, 62(6), pp.839~852 (2014)[DOI]
- J. C. Engwerda, "Necessary and suffcient conditions for the existence of a positive definite solution of the matrix equation <TEX>$X\;+\;A*X^{-1}\;A\;=\;Q$<", TEX>, 1993, pp.186, 255~275
- A. Ferrante, "Hermitian solutions of the equation <TEX>$X\;=\;Q\;+\;NX^{-1}\;N*$<", TEX>, 1996, pp.247, 359~373
- C.-H. Guo, "Iterative solution of two matrix equations", Math. Comp., 68(228), pp.1589~1603 (1999)[DOI]
- V. I. Hasanov, "Positive definite solutions of the matrix equations <TEX>$X\;{\pm}\;A*X^{-q}A\;=\;Q$<", TEX>, 2005, pp.404, 166~182
- N. Huang, "Two structure-preserving-doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation", Comput. Math. Appl., 69(6), pp.494~502 (2015)[DOI]
- I. G. Ivanov, "Improved methods and starting values to solve the matrix equations <TEX>$X\;{\pm}\;A*X^{-1}\;A\;=\;I$<", TEX> iteratively, 2005, pp.74(249), 263~278
- T. Jiang, "An algebraic relation between consimilarity and similarity of complex matrices and its applications", J. Phys. A, 39(29), pp.9215~9222 (2006)[DOI]
- T. Jiang, "On solutions of the matrix equations X - AXB = C and <TEX>$X\;-\;A{\overline{X}}B\;=\;C$<", TEX>, 2003, pp.367, 225~233
- J. Li, "Perturbation analysis of the matrix equation <TEX>$X\;{\pm}\;A*X^{-p}A\;=\;Q$<", TEX>, 2009, pp.431(9), 1489~1501
- Z.-Y. Li, "Towards positive definite solutions of a class of nonlinear matrix equations", Appl. Math. Comput., 237, pp.546~559 (2014)
- W.-W. Lin, "Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations", SIAM J. Matrix Anal. Appl., 28(1), pp.26~39 (2006)[DOI]
- X.-G. Liu, "On the positive definite solutions of the matrix equations <TEX>$X^{s}\;{\pm}\;A*X^{-t}A\;=\;I_{n}$<", TEX>, 2003, pp.368, 83~97
- B. Meini, "Effcient computation of the extreme solutions of <TEX>$X\;+\;A*X^{-1}A\;=\;Q\;and\;X\;-\;A*X^{-1}A\;=\;Q$<", TEX>, 2002, pp.71(239), 1189~1204
- Z. Peng, "Iterative methods for the extremal positive definite solution of the matrix equation <TEX>$X\;+\;A*X^{-{\alpha}}A\;=\;Q$<", TEX>, 2007, pp.200(2), 520~527
- A. C. M. Ran, "On the nonlinear matrix equation X+AF(X)A = Q : solutions and perturbation theory", Linear Algebra Appl., 346, pp.15~26 (2002)[DOI]
- S. Vaezzadeh, "On nonlinear matrix equations", Appl. Math. Lett., 26(9), pp.919~923 (2013)[DOI]
- Y. Yao, "Numerical methods to solve the complex symmetric stabilizing solution of the complex matrix equation <TEX>$X\;+\;A^{T}X^{-1}A\;=\;Q$<", TEX>, 2015, pp.48(1), 53~65
- X.-X. Yin, "Solutions and perturbation estimates for the matrix equation <TEX>$X^{S}\;+\;A*X^{-t}A\;=\;Q$<", TEX>, 2009, pp.431(9), 1409~1421
- J. Yong, "Stochastic Controls, Applications of Mathematics (New York), 43" (1999), Springer-Verlag
- L. Zhang, "An improved inversion-free method for solving the matrix equation <TEX>$X\;+\;A*X^{-{\alpha}}A\;=\;Q$<", TEX>, 2013, pp.253, 200~203
- B. Zhou, "Positive definite solutions of the nonlinear matrix equation <TEX>$X\;+\;A^{H}\overline{X}^{-1}\;A\;=\;I$<", TEX>, 2013, pp.219(14), 7377~7391