Research Article
An Inversion-Free Method for Finding Positive Definite
Solution of a Rational Matrix Equation
Fazlollah Soleymani,
1
Mahdi Sharifi,
1
Solat Karimi Vanani,
1
Farhad Khaksar Haghani,
1
and Adem KJlJçman
2
1
Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran
2
Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Malaysia
Correspondence should be addressed to Adem Kılıc ¸man; akilic@upm.edu.my
Received 15 July 2014; Accepted 26 July 2014; Published 19 August 2014
Academic Editor: Hassan Saberi Nik
Copyright © 2014 Fazlollah Soleymani et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
A new iterative scheme has been constructed for fnding minimal solution of a rational matrix equation of the form +
∗
−1
=.
Te new method is inversion-free per computing step. Te convergence of the method has been studied and tested via numerical
experiments.
1. Introduction
In this paper, we will discuss the following nonlinear matrix
equation:
+
∗
−1
= , (1)
where is an × nonsingular complex matrix, is the unit
matrix of the appropriate size, and ∈ C
×
is an unknown
Hermitian positive defnite (HPD) matrix that should be
found. It was proved in [1] that if (1) has an HPD solution,
then all its Hermitian solutions are positive defnite and,
moreover, it has the maximal solution
and the minimal
solution
in the sense that
≤≤
for any HPD
solution .
A lot of papers have been published regarding the iterative
HPD solutions of such nonlinear rational matrix equations
in the literature due to their importance in some practical
problems arising in control theory, dynamical problems, and
so forth (see [2, 3]).
Te most common iterative method for fnding the
maximal solution of (1) is the following fxed-point iteration
[4]:
0
= ,
+1
=−
∗
−1
.
(2)
Te maximal solution of (1) can be obtained through
=−
, where
is the minimal solution of the dual
equation +
−1
∗
=.
In 2010, Monsalve and Raydan in [5] proposed the
following iteration method (also known as Newton’s method)
for fnding the minimal solution:
0
=
∗
,
+1
=
(2 −
−∗
( −
)
−1
),
(3)
which is an inversion-free scheme. Since,
−1
should be
computed only once in contrast to the matrix iteration (2).
Note that
−∗
=
−1
*
, and similar notations are used
throughout.
Remark 1. We remark that there are several other well-known
iterative methods for solving (1) rather than Newton’s method
(3). To the best of our knowledge, the procedure of extending
higher-order iterative methods for fnding the solution of(1)
has not been exploited up to now. Hence, we hope that this
interlink among the felds of root-fnding and solving (1) may
lead to discovering novel and innovative techniques.
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 560931, 5 pages
http://dx.doi.org/10.1155/2014/560931