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Remarks on Invariant Method of the Second-Order Linear Differential Equations with Variable Coefficients Read chapter Read first chapter

2013 | OriginalPaper | Chapter

On the Existence of Extremal Positive Definite Solutions of a Class of Nonlinear Matrix Equation

Authors : Bo Wang, Qingchun Li

Published in: Proceedings of The Eighth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013

Publisher: Springer Berlin Heidelberg

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Abstract

The nonlinear matrix equation \( X^{r} + \sum\nolimits_{i = 1}^{m} {A_{i}^{*} } X^{{\delta_{i} }} A_{i} = Q \) is studied. A necessary condition for the existence of positive definite solutions of this equation is derived. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed.

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previous chapter Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equation
next chapter An Iterative Algorithm for the Generalized Center Symmetric Solutions of a Class of Linear Matrix Equation and Its Optimal Approximation
Literature
1.
Duan X, Liao A, Tang B (2008) On the nonlinear matrix equation \( X - \sum\nolimits_{i = 1}^{m} {A_{i}^{*} } X^{{\delta_{i} }} A_{i} = Q \). Linear Algebra Appl 429:110–121
2.
El-Sayed SM, Ramadan MA (2001) On the existence of a positive definite solution of the matrix equation \( X - A^{*} \root{{2^{m} }} \of {{X^{ - 1} }}A = I \). Int J Comput Math 76:331–338
3.
Peng ZY, El-sayed SM, Zhang XL (2007) Iterative method for the extremal positive definite solution of the matrixequation \( X + A^{*} X^{ - \alpha } A = Q \). Appl Math Comput 200:520–527
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Mohamed A (2005) Ramadan: on the existence of extremal positive definite solutions of a kind of matrix. Int J Nonlinear Sci Numer Simul 6:115–126
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Ramadan MA (2005) Necessary and sufficient conditions to the existence of positive definite solution of the matrix equation \( X + A^{T} X^{ - 2} A = I \). Int J Comput Math 82:865–870
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7.
Liu XG, Gao H (2003) On the positive definite solutions of a matrix equations \( X^{s} \pm A^{T} X^{ - t} A = I_{n} \). Linear Algebra Appl 368:83–97
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Duan X, Liao A (2008) On the existence of hermitian positive definite solutions of the matrix equation \( X^{s} + A^{*} X^{ - t} A = Q \). Linear Algebra Appl 429:673–687
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Yueting Y (2007) The iterative method for solving nonlinear matrix equation \( X^{s} + A^{*} X^{ - t} A = Q \). Appl Math Comput 188:46–53
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Metadata
Title
On the Existence of Extremal Positive Definite Solutions of a Class of Nonlinear Matrix Equation
Authors
Bo Wang
Qingchun Li
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Proceedings of The Eighth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013

Print ISBN: 978-3-642-37501-9

Electronic ISBN: 978-3-642-37502-6

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-37502-6_18

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