The Scientific World Journal

Volume 2015 (2015), Article ID 964257, 6 pages

http://dx.doi.org/10.1155/2015/964257

## On a Cubically Convergent Iterative Method for Matrix Sign

^{1}Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran^{2}Department of Mathematics and Applied Mathematics, University of Venda, Thohoyandou 0950, South Africa

Received 20 July 2014; Revised 29 August 2014; Accepted 30 August 2014

Academic Editor: Predrag S. Stanimirovic

Copyright © 2015 M. Sharifi et al. This 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.

#### Abstract

We propose an iterative method for finding matrix sign function. It is shown that the scheme has global behavior with cubical rate of convergence. Examples are included to show the applicability and efficiency of the proposed scheme and its reciprocal.

#### 1. Introduction

It is known that the function of sign in the scalar case is defined for any not on the imaginary axis byAn extension of (1) for the matrix case was given firstly by Roberts in [1]. This extended matrix function is of clear importance in several applications (see, e.g., [2] and the references therein).

Assume that is a matrix with no eigenvalues on the imaginary axis. To define this matrix function formally, letbe a Jordan canonical form arranged so that , where the eigenvalues of lie in the open left half-plane and those of lie in the open right half-plane; thenwhere . A simplified definition of the matrix sign function for Hermitian case (eigenvalues are all real) iswhereis a diagonalization of .

The importance of computing is also due to the fact that the sign function plays a fundamental role in iterative methods for matrix roots and the polar decomposition [3].

Note that although is a square root of the identity matrix, it is not equal to or unless the spectrum of lies entirely in the open right half-plane or open left half-plane, respectively. Hence, in general, is a nonprimary square root of .

In this paper, we focus on iterative methods for finding . In fact, such methods are Newton-type schemes which are in essence fixed-point-type methods by producing a convergent sequence of matrices via applying a suitable initial matrix.

The most famous method of this class is the quadratic Newton method defined by

It should be remarked that iterative methods, such as (6), and the Newton-Schultz iterationor the cubically convergent Halley methodare all special cases of the Padé family proposed originally in [4]. The Padé approximation belongs to a broader category of rational approximations. Coincidentally, the best uniform approximation of the sign function on a pair of symmetric but disjoint intervals can be expressed as a rational function.

Note that although (7) does not possess a global convergence behavior, on state-of-the-art parallel computer architectures, matrix inversions scale less satisfactorily than matrix multiplications do, and subsequently (7) is useful in some problems. However, due to local convergence behavior, it is excluded from our numerical examples in this work.

The rest of this paper is organized as follows. In Section 2, we discuss how to construct a new iterative method for finding (3). It is also shown that the constructed method is convergent with cubical rate. It is noted that its reciprocal iteration obtained from our main method is also convergent. Numerical examples are furnished to show the higher numerical accuracy for the constructed solvers in Section 3. The paper ends in Section 4 with some concluding comments.

#### 2. A New Method

The connection of matrix iteration methods with the sign function is not immediately obvious, but in fact such methods can be derived by applying a suitable root-finding method to the nonlinear matrix equationand when of course is one solution of this equation (see for more [5]).

Here, we consider the following root-solver:with . In what follows, we observe that (10) possesses third order of convergence.

Theorem 1. *Let be a simple zero of a sufficiently differentiable function , which contains as an initial approximation. Then the iterative expression (10) satisfies**where , .*

*Proof. *The proof would be similar to the proofs given in [6].

Applying (10) on the matrix equation (9) will result in the following new matrix fixed-point-type iteration for finding (3):where . This is named PM1 from now on.

The proposed scheme (12) is not a member of Padé family [4]. Furthermore, applying (10) on the scalar equation provides a global convergence in the complex plane (except the points lying on the imaginary axis). This global behavior, which is kept for matrix case, has been illustrated in Figure 1 by drawing the basins of attraction for (6) and (8). The attraction basins for (7) (local convergence) and (12) (global convergence) are also portrayed in Figure 2.