Research Article
A Matrix Iteration for Finding Drazin Inverse with Ninth-Order
Convergence
A. S. Al-Fhaid,
1
S. Shateyi,
2
M. Zaka Ullah,
1
and F. Soleymani
3
1
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2
Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda,
Private Bag X5050, Tohoyandou 0950, South Africa
3
Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
Correspondence should be addressed to S. Shateyi; stanford.shateyi@univen.ac.za
Received 31 January 2014; Accepted 11 March 2014; Published 14 April 2014
Academic Editor: Sofya Ostrovska
Copyright © 2014 A. S. Al-Fhaid 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.
Te aim of this paper is twofold. First, a matrix iteration for fnding 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.
1. Preliminary Notes
Let C
×
and C
×
denote the set of all complex ×
matrices and the set of all complex × matrices of rank ,
respectively. By
∗
, R(), rank(), and N(), we denote the
conjugate transpose, the range, the rank, and the null space of
∈ C
×
, respectively.
Important matrix-valued functions () are, for exam-
ple, the inverse
−1
, the (principal) square root
√
, and the
matrix sign function. Teir evaluation for large matrices aris-
ing from partial diferential equations or integral equations
(e.g., resulting from wavelet-like methods) is not an easy task
and needs techniques exploiting appropriate structures of the
matrices and ().
In this paper, we focus on the matrix function of inverse
for square matrices. To this goal, we construct a matrix iter-
ative method for fnding 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. Te Drazin inverse is investigated in the matrix
theory (particularly in the topic of generalized inverses) and
also in the ring theory; see, for example, [1].
Generally speaking, applying Schr¨ oder’s general method
(ofen called Schr¨ oder-Traub’s sequence [2]) to the nonlinear
matrix equation = , one obtains the following scheme
[3]:
+1
=
(+
+
2
+⋅⋅⋅+
−1
)
=
(+
(+
(⋅⋅⋅+
)⋅⋅⋅)), =0,1,2,...,
(1)
of order , requiring Horner’s matrix multiplications, where
=−
.
Te application of such (fxed-point type) matrix iterative
methods is not limited to the matrix inversion for square
nonsingular matrices [4, 5]. In fact and under some fair
conditions, one may construct a sequence of iterates converg-
ing to the Moore-Penrose inverse [6], the weighted Moore-
Penrose inverse [7], the Drazin inverse [8], or the outer
inverse in the feld of generalized inverses. Such extensions
alongside the asymptotical stability of matrix iterations in the
form (1) encouraged many authors to present new schemes or
work on the application of such methods in diferent felds of
sciences and engineering; see, for example, [9–12].
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 137486, 7 pages
http://dx.doi.org/10.1155/2014/137486