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The aim of this paper is twofold. First, a matrix iteration for finding approximate inverses of nonsingular square matrices is constructed. Second, how the new method could be applied for computing the Drazin inverse is discussed. It is theoretically proven that the contributed method possesses the convergence rate nine. Numerical studies are brought forward to support the analytical parts.

Let

Important matrix-valued functions

In this paper, we focus on the matrix function of inverse for square matrices. To this goal, we construct a matrix iterative method for finding approximate inverses quickly. It is proven that the new method possesses the high convergence order nine using only seven matrix-matrix multiplications. We will then discuss how to apply the new method for Drazin inverse. The Drazin inverse is investigated in the matrix theory (particularly in the topic of generalized inverses) and also in the ring theory; see, for example, [

Generally speaking, applying Schröder's general method (often called Schröder-Traub's sequence [

The application of such (fixed-point type) matrix iterative methods is not limited to the matrix inversion for square nonsingular matrices [

Choosing

It is of great importance to arrive at the convergence phase by a valid initial value

The rest of the paper has been organized as follows. Section

Let us consider the

Based on (

Now, let us first apply the following

Using proper factorizing, we attain the following iteration for matrix inversion, at which

The scheme (

Let

Let (

The Drazin inverse, named after Drazin [

The smallest nonnegative integer

Let

The Drazin inverse has applications in the theory of finite Markov chains, as well as in the study of differential equations and singular linear difference equations and so forth [

Note that a projection matrix

In 2004, Li and Wei in [

Using the above descriptions, it is easy to apply the efficient method (

Let

Consider the notation

Therefore, the inequalities in (

Considering the same assumptions as in Theorem

The steps of proving the asymptotical stability of (

It should be remarked that the generalization of our proposed scheme for generalized outer inverses, that is,

The new iteration (

We herein present several numerical tests to illustrate the efficiency of the new iterative method to compute the approximate inverses.

For comparisons, we have used the methods “Schulz” (

The computations of approximate inverses for 10 dense random complex matrices of the dimension 100 are considered and compared as follows:

Note that

Comparison of the number of iterations for solving Example

Comparison of the elapsed time for solving Example

The computations of approximate inverses for 10 dense random complex matrices of the dimension 200 are considered and compared as follows:

Note again that

Comparison of the number of iterations for solving Example

Comparison of the elapsed time for solving Example

The computations of approximate inverses for 10 dense random matrices of the size 200 are investigated in what follows:

here, the stopping criterion is

Comparison of the number of iterations for solving Example

Comparison of the elapsed time for solving Example

The order of convergence and the number of matrix-matrix products are not the only factors to govern the efficiency of an algorithm per computing step in matrix iterations. Generally speaking, the stopping criterion could be reported as one of the important factors, which could indirectly affect the computational time of an algorithm in implementations, especially when trying to find the generalized inverses.

Although in the above implementations we considered the stop termination on two successive iterates, this is not a reliable termination when dealing with some large ill-conditioned matrices. For example, the reliable stopping criterion in the above examples is

The aim of this example is to apply the discussions of Section

with

Due to the efficiency of the new method, we did not compare different methods for this test and just bring forward the results of (

In this paper, we have developed a high order matrix method for finding approximate inverses for nonsingular square matrices. It has been proven that the contributed method reaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational index as

We have also discussed the importance of the well-known Drazin inverse and how to find it numerically by the proposed method. Numerical examples were also employed to support the underlying theory of the paper.

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

This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 159/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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