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A new iterative scheme has been constructed for finding minimal solution of a rational matrix equation of the form

In this paper, we will discuss the following nonlinear matrix equation:

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 [

The most common iterative method for finding the maximal solution of (

The maximal solution of (

In 2010, Monsalve and Raydan in [

We remark that there are several other well-known iterative methods for solving (

The rest of this paper is organized as follows. In Section

An equivalent formulation of (

In order to obtain an iterative method for finding the minimal HPD solution of (

We remark that there is a tight relationship between iterative methods for nonlinear systems and the construction of higher-order methods for matrix equations ([

The matrix iteration (

In the meantime, it is easy to show that the zeros of the map

Following Remark

The proposed method (

The initial matrix

By considering that

Let us consider

On the other hand, Chebyshev’s method for matrix inversion problem is convergent if the initial approximation reads

Note that since

The only problem that happens in this process is the fact that the convergence order is

The reason is that the matrix

The sequence of matrices produced by (

First since

Consequently, one has the error inequality (

In this section, we mainly investigate the performance of the new method (

Note that recently Zhang in [

(1) Choose

(2) for

(3) end for

(4) set

(5) for

(6) end for

We compare Algorithm

In this experiment, we compare the results of different methods for finding the minimal solution of (

The results are given in Figure

Number of iterations against accuracies for experiment 1 (a) and experiment 2 (b).

Note that PM and M2 converge to

Furthermore, we have chosen this number empirically. In fact, varying

In Example

Applying the stopping criterion

We have studied the fact that the minimal HPD solution of (

The developed method requires the computation of one matrix inverse at the beginning of the process and it is hence an inversion-free method. The convergence and the rate of convergence have been studied for this scheme. Furthermore, using a proper acceleration technique from the literature, we have further speeded up the process of finding the HPD solution of (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very helpful suggestions that improved the manuscript substantially. The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having project number GP-IBT/2013/9420100.