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A Matrix Iteration for Finding Drazin Inverse with Ninth-Order Convergence

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A Matrix Iteration for Finding Drazin Inverse with Ninth-Order Convergence Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 137486, 7 pages http://dx.doi.org/10.1155/2014/137486 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. Soleymani3 1 Department of Mathematics, Faculty of Sciences, King AbdulazizUniversity,P.O.Box80203,Jeddah21589,SaudiArabia 2 Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa 3 Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran Correspondence should be addressed to S. Shateyi; [email protected] Received 31 January 2014; Accepted 11 March 2014; Published 14 April 2014 Academic Editor: Sofiya Ostrovska Copyright © 2014 A. S. Al-Fhaid 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. 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. 1. Preliminary Notes Generally speaking, applying Schroder’s¨ general method (often called Schroder-Traub’s¨ sequence [2]) to the nonlinear C× C× × Let and denote the set of all complex matrix equation =, one obtains the following scheme matrices and the set of all complex ×matrices of rank , ∗ [3]: respectively. By , R(),rank(),andN(), we denote the conjugate transpose, the range, the rank, and the null space of 2 −1 × +1 = (+ + +⋅⋅⋅+ ) ∈C ,respectively. () Important matrix-valued functions are, for exam- = ( + ( + (⋅ ⋅ ⋅ + )⋅⋅⋅)), =0,1,2,..., −1 √ ple, the inverse , the (principal) square root ,andthe (1) matrix sign function. Their evaluation for large matrices aris- ing from partial differential equations or integral equations of order ,requiring Horner’s matrix multiplications, where (e.g., resulting from wavelet-like methods) is not an easy task =−. and needs techniques exploiting appropriate structures of the The application of such (fixed-point type) matrix iterative () matrices and . methods is not limited to the matrix inversion for square In this paper, we focus on the matrix function of inverse nonsingular matrices [4, 5]. In fact and under some fair for square matrices. To this goal, we construct a matrix iter- conditions, one may construct a sequence of iterates converg- ative method for finding approximate inverses quickly. It is ing to the Moore-Penrose inverse [6], the weighted Moore- proven that the new method possesses the high convergence Penrose inverse [7], the Drazin inverse [8], or the outer order nine using only seven matrix-matrix multiplications. inverse in the field of generalized inverses. Such extensions We will then discuss how to apply the new method for Drazin alongside the asymptotical stability of matrix iterations in the inverse. The Drazin inverse is investigated in the matrix form (1) encouraged many authors to present new schemes or theory (particularly in the topic of generalized inverses) and work on the application of such methods in different fields of also in the ring theory; see, for example, [1]. sciences and engineering; see, for example, [9–12]. 2 Abstract and Applied Analysis −1 Choosing =2and =3in (1)reducestothe on the matrix equation () = −. Next, we obtain well-known methods of Schulz [13]andChebyshevin matrix inversion. Note that any method extracted from Sen- 1 = () Prabhu scheme (1)requires matrix-matrix multiplications +1 729 to achieve the convergence order . In this work, we are 1 interested in proposing a new scheme, at which a convergence 2 = (7047 − 30726 + 79712 order can be attained by fewer matrix-matrix multiplica- 729 tions than . − 1366383 + 1624504 It is of great importance to arrive at the convergence phase (5) by a valid initial value 0 in matrix iterative methods. An − 1367525 + 816846 interesting initial matrix was developed and introduced by Ben-IsraelandGrevillein[14] as follows: 7 8 9 − 34137 + 9663 − 1746 +18010 −811), =0,1,2,..., ∗ 0 = , (2) where =.Notethatabackgroundontheconstruction 2 of iterative methods for solving nonlinear equations and their where 0 < < 2/‖‖2,oncetheuserwantstofindtheMoore- Penrose inverse. applications might be found in [17–20]. The rest of the paper has been organized as follows. Using proper factorizing, we attain the following iteration Section 2 describes a contribution of the paper alongside for matrix inversion, at which 0 is an initial approximation −1 a convergence analysis, while, in Section 3, we will extend to : the new method for finding the Drazin inverse as well. Section 4 is devoted to the computational examples. Section 5 =, concludes the paper. = −29 + (33 + (−15 + 2)) , =, 1 2. A New Method =− (243 + (27 + )), =0,1,2,.... +1 729 Let us consider the inverse-finder informational efficiency (6) index [15], which states that if and stand for the rate of convergence and the number of matrix-by-matrix The scheme (6)fallsinthecategoryofSchulz-type multiplications in floating point arithmetics for a matrix methods, which possesses matrix-by-matrix multiplications method in matrix inversion, then the index is to provide approximate inverses. Let us prove the rate of convergence for (6), using the theory of matrix analysis [21] in what follows. = . (3) IIEI Theorem 1. Let =[ ]× be a nonsingular complex matrix. If the initial value 0 satisfies Based on (3), we must design a method, at which the number of matrix-matrix products is fewer than the local 0 = −0 <1, (7) convergence order. The first of such an attempt dated back to the earlier works of Ostrowski in [16], wherein he suggested then the matrix iterative method (6) converges with ninth order that a robust way for achieving such a goal is in proper matrix −1 factorizing. to . Now,letusfirstapplythefollowingnew nonlinear equa- =− tion solver: Proof . Let (7) hold, and further assume that , ≥0. It is straightforward to have = −( )−1( ), 1 9 10 11 12 2 +1 = [343 + 294 + 84 +8 ] . (8) = −( )−1 [ ( )−( )( )] 729 3 1 ×(1−( )( )−1( )) , The rest of the proof for this theorem is similar to Theorem 3 (4) 2.1 of [22]. It is hence omitted. −1 +1 = −() () 1 2 −( )[( )−1 ( )] [( )−1( )] , 3. Extension to the Drazin Inverse 2 The Drazin inverse, named after Drazin [23], is a generalized =0,1,2,..., inverse which has spectral properties similar to the ordinary Abstract and Applied Analysis 3 inverse of a given square matrix. In some cases, it also Proof. Consider the notation 0 = −0 and sub- provides a solution of a given system of linear equations. Note sequently the residual matrix as = − for finding that the Drazin inverse of a matrix mostly resembles the theDrazininverse.Thensimilarto(8), we get true inverse of . +1 = −+1 = −+−+1 Definition 2. The smallest nonnegative integer ,suchthat +1 rank( )=rank( ), is called the index of and denoted = −++1 by ind(). 1 9 10 11 12 × = −+ [343 + 294 + 84 +8 ] Definition 3. Let ∈C be a complex matrix; the Drazin 729 inverse of , denoted by , is the unique matrix satisfying 1 9 the following: = −+ [343( − + − ) 729 +1 (1) = , 10 + 294( − + − ) (2) =, = (3) , 11 +84(− + −) where =ind() is the index of . 12 + 8( − + − ) ] The Drazin inverse has applications in the theory of finite Markovchains,aswellasinthestudyofdifferentialequations and singular linear difference equations and so forth24 [ ]. To 1 9 = −+ [343 ( − + ) illustrate further, the solution of singular systems has been 729 studied by several authors. For instance, in [25], an analytical 10 solution for continuous systems using the Drazin inverse was + 294 ( − + ) presented. Note that a projection matrix , defined as a matrix such +84(− +11) 2 that =,has=1(or 0) and has the Drazin inverse =.Also,if is a nilpotent matrix (e.g., a shift matrix), +8(− +12)] then =0.Seeformore[26]. In2004,LiandWeiin[27]provedthatthematrixmethod 1 9 10 11 12 =2 = [343 + 294 + 84 +8 ]. of Schulz (the case for in (1))canbeusedforfinding 729 the Drazin inverse of square matrices. They proposed the (12) following initial matrix: 0 =,≥ind () =, (9) By taking an arbitrary matrix norm to both sides of (12), we attain where the parameter must be chosen such that the condi- ‖ ‖ = ‖ −‖<1 tion 0 0 is satisfied. Using this initial 1 9 10 11 12 ≤ [343 + 294 +84 +8 ]. matrix yields a numerically (asymptotical) stable method +1 729 for finding the famous Drazin inverse with quadratical (13) convergence. Using the above descriptions, it is easy to apply the ‖ ‖<1 efficient method (6) for finding the Drazin inverse in what In addition, since 0 (the result of choosing the appropriate initial matrices (10)and(11)) by relation (13), we follows. 9 9 obtain that ‖1‖≤‖0‖ .Similarly,‖+1‖≤‖‖ .Using Theorem 4. Let =[ ]× be a square matrix and ind() = mathematical induction, we obtain ≥1 . Choosing the initial approximation 0 as 9 ≤ 0 ,≥0. (14) 2 0 = , (10) tr (+1) So, the sequence {‖‖2} is strictly monotonically decreasing. or Now, by considering = −,astheerrormatrixfor 1 = , finding the Drazin inverse, we have 0 +1 (11) 2‖‖∗ +1 = −+1 =+1 wherein tr(⋅) stands for the trace of an arbitrary square matrix (15) and ‖⋅‖∗ is a matrix norm, then the iterative method (6) 1 9 10 11 12 = [343 + 294 + 84 +8 ]. converges with ninth order to . 729 4 Abstract and Applied Analysis Taking into account (12) and using elementary algebraic transformations, we further derive 20 15 1 9 10 ≤ [343 + 294 +1 729 10 11 12 +84 +8 ] (16) Number of iterations 5 9 9 9 9 ≤ = ≤ ‖‖ . 0 2 4 6 8 10 Matrices Schulz KMS It is now easy to find the error inequality of the new scheme Chebyshev PM (6)using(16) as follows: Figure 1: Comparison of the number of iterations for solving Example 8.
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