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Research Article A Matrix Iteration for Finding Drazin...
8
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, ohoyandou 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. is 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. e 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 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. eir evaluation for large matrices aris- ing from partial differential 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 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. e 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 (oſten 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 = − . e application of such (fixed-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 field 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 different fields 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
Transcript of Research Article A Matrix Iteration for Finding Drazin...
Research ArticleA Matrix Iteration for Finding Drazin Inverse with Ninth-OrderConvergence
A S Al-Fhaid1 S Shateyi2 M Zaka Ullah1 and F Soleymani3
1 Department of Mathematics Faculty of Sciences King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics and Applied Mathematics School of Mathematical and Natural Sciences University of VendaPrivate Bag X5050 Thohoyandou 0950 South Africa
3 Department of Mathematics Islamic Azad University Zahedan Branch Zahedan Iran
Correspondence should be addressed to S Shateyi stanfordshateyiunivenacza
Received 31 January 2014 Accepted 11 March 2014 Published 14 April 2014
Academic Editor Sofiya Ostrovska
Copyright copy 2014 A S Al-Fhaid et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 isconstructed Second how the newmethod could be applied for computing the Drazin inverse is discussed It is theoretically proventhat the contributed method possesses the convergence rate nine Numerical studies are brought forward to support the analyticalparts
1 Preliminary Notes
Let C119899times119899 and C119899times119899119903
denote the set of all complex 119899 times 119899
matrices and the set of all complex 119899 times 119899matrices of rank 119903respectively By119860lowastR(119860) rank(119860) andN(119860) we denote theconjugate transpose the range the rank and the null space of119860 isin C119899times119899 respectively
Important matrix-valued functions 119891(119860) are for exam-ple the inverse 119860minus1 the (principal) square root radic119860 and thematrix sign functionTheir evaluation for large matrices aris-ing from partial differential equations or integral equations(eg resulting from wavelet-like methods) is not an easy taskand needs techniques exploiting appropriate structures of thematrices 119860 and 119891(119860)
In this paper we focus on the matrix function of inversefor square matrices To this goal we construct a matrix iter-ative method for finding approximate inverses quickly It isproven that the new method possesses the high convergenceorder nine using only seven matrix-matrix multiplicationsWewill then discuss how to apply the newmethod for Drazininverse The Drazin inverse is investigated in the matrixtheory (particularly in the topic of generalized inverses) andalso in the ring theory see for example [1]
Generally speaking applying Schroderrsquos general method(often called Schroder-Traubrsquos sequence [2]) to the nonlinearmatrix equation 119860119883 = 119868 one obtains the following scheme[3]
119883119896+1
= 119883119896(119868 + 119884
119896+ 1198842
119896+ sdot sdot sdot + 119884
119899minus1
119896)
= 119883119896(119868 + 119884
119896(119868 + 119884
119896(sdot sdot sdot + 119884
119896) sdot sdot sdot )) 119896 = 0 1 2
(1)
of order 119899 requiring 119899Hornerrsquosmatrixmultiplications where119884119896= 119868 minus 119860119883
119896
The application of such (fixed-point type) matrix iterativemethods is not limited to the matrix inversion for squarenonsingular matrices [4 5] In fact and under some fairconditions onemay 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 outerinverse in the field of generalized inverses Such extensionsalongside the asymptotical stability of matrix iterations in theform (1) encouragedmany authors to present new schemes orwork on the application of such methods in different fields ofsciences and engineering see for example [9ndash12]
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 137486 7 pageshttpdxdoiorg1011552014137486
2 Abstract and Applied Analysis
Choosing 119899 = 2 and 119899 = 3 in (1) reduces to thewell-known methods of Schulz [13] and Chebyshev inmatrix inversion Note that any method extracted from Sen-Prabhu scheme (1) requires 119899 matrix-matrix multiplicationsto achieve the convergence order 119899 In this work we areinterested in proposing a new scheme at which a convergenceorder 119901 can be attained by fewer matrix-matrix multiplica-tions than 119901
It is of great importance to arrive at the convergence phaseby a valid initial value 119883
0in matrix iterative methods An
interesting initial matrix was developed and introduced byBen-Israel and Greville in [14] as follows
1198830= 120572119860lowast
(2)
where 0 lt 120572 lt 211986022 once the userwants to find theMoore-
Penrose inverseThe rest of the paper has been organized as follows
Section 2 describes a contribution of the paper alongsidea convergence analysis while in Section 3 we will extendthe new method for finding the Drazin inverse as wellSection 4 is devoted to the computational examples Section 5concludes the paper
2 A New Method
Let us consider the inverse-finder informational efficiencyindex [15] which states that if 120585 and 120603 stand for therate of convergence and the number of matrix-by-matrixmultiplications in floating point arithmetics for a matrixmethod in matrix inversion then the index is
IIEI = 120585
120603
(3)
Based on (3) we must design a method at which thenumber of matrix-matrix products is fewer than the localconvergence order The first of such an attempt dated back tothe earlier works of Ostrowski in [16] wherein he suggestedthat a robust way for achieving such a goal is in propermatrixfactorizing
Now let us first apply the following new nonlinear equa-tion solver
119910119896= 119909119896minus 1198911015840
(119909119896)minus1
119891 (119909119896)
119911119896= 119909119896minus 119891(119910
119896)minus1
[119891 (119909119896) minus (
2
3
)119891 (119910119896)]
times (1 minus (
1
3
)1198911015840
(119909119896)minus1
119891 (119910119896))
119909119896+1
= 119911119896minus 1198911015840
(119911119896)minus1
119891 (119911119896)
minus (
1
2
) [1198911015840
(119911119896)minus1
11989110158401015840
(119911119896)] [1198911015840
(119911119896)minus1
119891 (119911119896)]
2
119896 = 0 1 2
(4)
on the matrix equation 119891(119909) = 119909minus1 minus 119886 Next we obtain
119883119896+1
=
1
729
119883119896120593 (119896)
=
1
729
119883119896(7047119868 minus 30726120595
119896+ 79712120595
2
119896
minus 1366381205953
119896+ 162450120595
4
119896
minus 1367521205955
119896+ 81684120595
6
119896
minus 341371205957
119896+ 9663120595
8
119896minus 1746120595
9
119896
+18012059510
119896minus 812059511
119896) 119896 = 0 1 2
(5)
where120595119896= 119860119883
119896 Note that a background on the construction
of iterativemethods for solving nonlinear equations and theirapplications might be found in [17ndash20]
Using proper factorizing we attain the following iterationfor matrix inversion at which 119883
0is an initial approximation
to 119860minus1
120595119896= 119860119883
119896
120577119896= minus29119868 + 120595
119896(33119868 + 120595
119896(minus15119868 + 2120595
119896))
120581119896= 120595119896120577119896
119883119896+1
= minus
1
729
119883119896120577119896(243119868 + 120581
119896(27119868 + 120581
119896)) 119896 = 0 1 2
(6)
The scheme (6) falls in the category of Schulz-typemethods which possesses matrix-by-matrix multiplicationsto provide approximate inverses Let us prove the rate ofconvergence for (6) using the theory of matrix analysis [21]in what follows
Theorem 1 Let119860 = [119886119894119895]119899times119899
be a nonsingular complexmatrixIf the initial value119883
0satisfies
10038171003817100381710038171198640
1003817100381710038171003817=1003817100381710038171003817119868 minus 119860119883
0
1003817100381710038171003817lt 1 (7)
then thematrix iterativemethod (6) converges with ninth orderto 119860minus1
Proof Let (7) hold and further assume that 119864119896= 119868 minus 119860119883
119896
119896 ge 0 It is straightforward to have
119864119896+1
=
1
729
[3431198649
119896+ 294119864
10
119896+ 84119864
11
119896+ 811986412
119896] (8)
The rest of the proof for this theorem is similar to Theorem21 of [22] It is hence omitted
3 Extension to the Drazin Inverse
TheDrazin inverse named after Drazin [23] is a generalizedinverse which has spectral properties similar to the ordinary
Abstract and Applied Analysis 3
inverse of a given square matrix In some cases it alsoprovides a solution of a given system of linear equations Notethat the Drazin inverse of a matrix 119860 mostly resembles thetrue inverse of 119860
Definition 2 The smallest nonnegative integer 119870 such thatrank(119860119870+1) = rank(119860119870) is called the index of119860 and denotedby ind(119860)
Definition 3 Let 119860 isin C119899times119899 be a complex matrix the Drazininverse of119860 denoted by119860119863 is the uniquematrix119883 satisfyingthe following
(1) 119860119870+1119883 = 119860119870
(2) 119883119860119883 = 119883(3) 119860119883 = 119883119860
where119870 = ind(119860) is the index of 119860
The Drazin inverse has applications in the theory of finiteMarkov chains as well as in the study of differential equationsand singular linear difference equations and so forth [24] Toillustrate further the solution of singular systems has beenstudied by several authors For instance in [25] an analyticalsolution for continuous systems using the Drazin inverse waspresented
Note that a projection matrix 119875 defined as a matrix suchthat 1198752 = 119875 has 119870 = 1 (or 0) and has the Drazin inverse119875119863
= 119875 Also if 119860 is a nilpotent matrix (eg a shift matrix)then 119860119863 = 0 See for more [26]
In 2004 Li andWei in [27] proved that thematrixmethodof Schulz (the case for 119899 = 2 in (1)) can be used for findingthe Drazin inverse of square matrices They proposed thefollowing initial matrix
1198830= 120572119860119897
119897 ge ind (119860) = 119870 (9)
where the parameter 120572 must be chosen such that the condi-tion 119865
0 = 119860119860
119863
minus 1198601198830 lt 1 is satisfied Using this initial
matrix yields a numerically (asymptotical) stable methodfor finding the famous Drazin inverse with quadraticalconvergence
Using the above descriptions it is easy to apply theefficient method (6) for finding the Drazin inverse in whatfollows
Theorem 4 Let119860 = [119886119894119895]119899times119899
be a square matrix and ind(119860) =119870 ge 1 Choosing the initial approximation 119883
0as
1198830=
2
tr (119860119870+1)119860119870
(10)
or
1198830=
1
2119860119896+1
lowast
119860119870
(11)
wherein tr(sdot) stands for the trace of an arbitrary square matrixand sdot
lowastis a matrix norm then the iterative method (6)
converges with ninth order to 119860119863
Proof Consider the notation 1198650= 119860119860
119863
minus 1198601198830and sub-
sequently the residual matrix as 119865119896= 119860119860
119863
minus119860119883119896for finding
the Drazin inverse Then similar to (8) we get
119865119896+1
= 119860119860119863
minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119868 minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119864119896+1
= 119860119860119863
minus 119868 +
1
729
[3431198649
119896+ 294119864
10
119896+ 84119864
11
119896+ 811986412
119896]
= 119860119860119863
minus 119868 +
1
729
[343(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
9
+ 294(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
10
+ 84(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
11
+ 8(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
12
]
= 119860119860119863
minus 119868 +
1
729
[343 (119868 minus 119860119860119863
+ 1198659
119896)
+ 294 (119868 minus 119860119860119863
+ 11986510
119896)
+ 84 (119868 minus 119860119860119863
+ 11986511
119896)
+ 8 (119868 minus 119860119860119863
+ 11986512
119896)]
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(12)
By taking an arbitrary matrix norm to both sides of (12) weattain
1003817100381710038171003817119865119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
(13)
In addition since 1198650 lt 1 (the result of choosing the
appropriate initial matrices (10) and (11)) by relation (13) weobtain that 119865
1 le 119865
09 Similarly 119865
119896+1 le 119865
1198969 Using
mathematical induction we obtain1003817100381710038171003817119865119896
1003817100381710038171003817le10038171003817100381710038171198650
1003817100381710038171003817
9119896
119896 ge 0 (14)
So the sequence 1198651198962 is strictly monotonically decreasing
Now by considering 120575119896= 119860119863
minus 119883119896 as the error matrix for
finding the Drazin inverse we have
119860120575119896+1
= 119860119860119863
minus 119860119883119896+1
= 119865119896+1
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(15)
4 Abstract and Applied Analysis
Taking into account (12) and using elementary algebraictransformations we further derive
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
le1003817100381710038171003817119865119896
1003817100381710038171003817
9
=1003817100381710038171003817119860120575119896
1003817100381710038171003817
9
le 11986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(16)
It is now easy to find the error inequality of the new scheme(6) using (16) as follows
1003817100381710038171003817120575119896+1
1003817100381710038171003817=
10038171003817100381710038171003817119883119896+1
minus 11986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
119860119883119896+1
minus 119860119863
11986011986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
(119860119883119896+1
minus 119860119860119863
)
10038171003817100381710038171003817
le
1003817100381710038171003817100381711986011986310038171003817100381710038171003817
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817
le
100381710038171003817100381710038171198601198631003817100381710038171003817100381711986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(17)
Therefore the inequalities in (17) immediately lead to theconclusion that119883
119896rarr 119860119863 as 119896 rarr +infin with the ninth order
of convergence
Theorem 5 Considering the same assumptions as inTheorem 4 the iterative method (6) has asymptotical stabilityfor finding the Drazin inverse
Proof The steps of proving the asymptotical stability of (6)are similar to those that have recently been taken for a generalfamily of methods in [28] Hence the proof is omitted
Remark 6 It should be remarked that the generalization ofour proposed scheme for generalized outer inverses that is119860(2)
119879119878 is straightforward according to the recent work [29]
Remark 7 The new iteration (6) is free frommatrix power inits implementation and this allows one to apply it for findinggeneralized inverses easily
4 Numerical Experiments
We herein present several numerical tests to illustrate theefficiency of the new iterativemethod to compute the approx-
2 4 6 8 100
5
10
15
20
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 1 Comparison of the number of iterations for solvingExample 8
imate inverses mathematica 8 [30] has been employed inour calculations We have carried out the numerical testswith machine precision on a computer with characteris-tics Microsoft Windows XP Intel(R) Pentium(R) 4CPU320GHz with 4GB of RAM
For comparisons we have used the methods ldquoSchulzrdquo(119899 = 2) ldquoChebyshevrdquo (119899 = 3) ldquoKMSrdquo (119899 = 9) in (1) andthe proposed method (6) As the programs were running inthis paper we found the running time using the commandAbsoluteTiming[] to report the elapsed CPU time (inseconds) for the examples
Example 8 The computations of approximate inverses for 10dense random complex matrices of the dimension 100 areconsidered and compared as follows
n = 100 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Note that 119868 = radicminus1 For this example the stopping criterion is119883119896+1
minus 1198831198962le 10minus5 and the maximum number of iterations
allowed is set to 100 The initial choice has been constructedusing (2) with 120572 = 1120590
2
max The result of comparisons hasbeen presented in Figures 1 and 2 As could be observed in allthe 10 test problems our iterative method (6) beats the otherexisting schemes
Example 9 The computations of approximate inverses for 10dense random complex matrices of the dimension 200 areconsidered and compared as follows
n = 200 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Abstract and Applied Analysis 5
2 4 6 8 1000
01
02
03
04
Matrices
PMKMS
ChebyshevSchulz
Com
puta
tiona
l tim
e (s)
Figure 2 Comparison of the elapsed time for solving Example 8
2 4 6 8 100
5
10
15
20
25
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 3 Comparison of the number of iterations for solvingExample 9
Note again that 119868 = radicminus1 For this test the stoppingcriterion for finding the approximate inverse for the gen-erated random matrices is 119883
119896+1minus 1198831198962le 10
minus5 and themaximum number of iterations allowed is set to 100 by theinitial choice as in Example 8 The numerics are provided inFigures 3 and 4 Results are in harmony with the theoreticalaspects and show that the new method is efficient in matrixinversion
Example 10 The computations of approximate inverses for10 dense random matrices of the size 200 are investigated inwhat follows
n = 200 number = 10 SeedRandom[1]
Table[A[l] = RandomReal[0 1
n n] l number]
here the stopping criterion is 119883119896+1
minus 119883119896119865le 10minus8 Results
are provided in Figures 5 and 6 which indicate that the
2 4 6 8 1000
05
10
15
20
25
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 4 Comparison of the elapsed time for solving Example 9
2 4 6 8 100
10
20
30
40
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 5 Comparison of the number of iterations for solvingExample 10
2 4 6 8 1000
02
04
06
08
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 6 Comparison of the elapsed time for solving Example 10
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Choosing 119899 = 2 and 119899 = 3 in (1) reduces to thewell-known methods of Schulz [13] and Chebyshev inmatrix inversion Note that any method extracted from Sen-Prabhu scheme (1) requires 119899 matrix-matrix multiplicationsto achieve the convergence order 119899 In this work we areinterested in proposing a new scheme at which a convergenceorder 119901 can be attained by fewer matrix-matrix multiplica-tions than 119901
It is of great importance to arrive at the convergence phaseby a valid initial value 119883
0in matrix iterative methods An
interesting initial matrix was developed and introduced byBen-Israel and Greville in [14] as follows
1198830= 120572119860lowast
(2)
where 0 lt 120572 lt 211986022 once the userwants to find theMoore-
Penrose inverseThe rest of the paper has been organized as follows
Section 2 describes a contribution of the paper alongsidea convergence analysis while in Section 3 we will extendthe new method for finding the Drazin inverse as wellSection 4 is devoted to the computational examples Section 5concludes the paper
2 A New Method
Let us consider the inverse-finder informational efficiencyindex [15] which states that if 120585 and 120603 stand for therate of convergence and the number of matrix-by-matrixmultiplications in floating point arithmetics for a matrixmethod in matrix inversion then the index is
IIEI = 120585
120603
(3)
Based on (3) we must design a method at which thenumber of matrix-matrix products is fewer than the localconvergence order The first of such an attempt dated back tothe earlier works of Ostrowski in [16] wherein he suggestedthat a robust way for achieving such a goal is in propermatrixfactorizing
Now let us first apply the following new nonlinear equa-tion solver
119910119896= 119909119896minus 1198911015840
(119909119896)minus1
119891 (119909119896)
119911119896= 119909119896minus 119891(119910
119896)minus1
[119891 (119909119896) minus (
2
3
)119891 (119910119896)]
times (1 minus (
1
3
)1198911015840
(119909119896)minus1
119891 (119910119896))
119909119896+1
= 119911119896minus 1198911015840
(119911119896)minus1
119891 (119911119896)
minus (
1
2
) [1198911015840
(119911119896)minus1
11989110158401015840
(119911119896)] [1198911015840
(119911119896)minus1
119891 (119911119896)]
2
119896 = 0 1 2
(4)
on the matrix equation 119891(119909) = 119909minus1 minus 119886 Next we obtain
119883119896+1
=
1
729
119883119896120593 (119896)
=
1
729
119883119896(7047119868 minus 30726120595
119896+ 79712120595
2
119896
minus 1366381205953
119896+ 162450120595
4
119896
minus 1367521205955
119896+ 81684120595
6
119896
minus 341371205957
119896+ 9663120595
8
119896minus 1746120595
9
119896
+18012059510
119896minus 812059511
119896) 119896 = 0 1 2
(5)
where120595119896= 119860119883
119896 Note that a background on the construction
of iterativemethods for solving nonlinear equations and theirapplications might be found in [17ndash20]
Using proper factorizing we attain the following iterationfor matrix inversion at which 119883
0is an initial approximation
to 119860minus1
120595119896= 119860119883
119896
120577119896= minus29119868 + 120595
119896(33119868 + 120595
119896(minus15119868 + 2120595
119896))
120581119896= 120595119896120577119896
119883119896+1
= minus
1
729
119883119896120577119896(243119868 + 120581
119896(27119868 + 120581
119896)) 119896 = 0 1 2
(6)
The scheme (6) falls in the category of Schulz-typemethods which possesses matrix-by-matrix multiplicationsto provide approximate inverses Let us prove the rate ofconvergence for (6) using the theory of matrix analysis [21]in what follows
Theorem 1 Let119860 = [119886119894119895]119899times119899
be a nonsingular complexmatrixIf the initial value119883
0satisfies
10038171003817100381710038171198640
1003817100381710038171003817=1003817100381710038171003817119868 minus 119860119883
0
1003817100381710038171003817lt 1 (7)
then thematrix iterativemethod (6) converges with ninth orderto 119860minus1
Proof Let (7) hold and further assume that 119864119896= 119868 minus 119860119883
119896
119896 ge 0 It is straightforward to have
119864119896+1
=
1
729
[3431198649
119896+ 294119864
10
119896+ 84119864
11
119896+ 811986412
119896] (8)
The rest of the proof for this theorem is similar to Theorem21 of [22] It is hence omitted
3 Extension to the Drazin Inverse
TheDrazin inverse named after Drazin [23] is a generalizedinverse which has spectral properties similar to the ordinary
Abstract and Applied Analysis 3
inverse of a given square matrix In some cases it alsoprovides a solution of a given system of linear equations Notethat the Drazin inverse of a matrix 119860 mostly resembles thetrue inverse of 119860
Definition 2 The smallest nonnegative integer 119870 such thatrank(119860119870+1) = rank(119860119870) is called the index of119860 and denotedby ind(119860)
Definition 3 Let 119860 isin C119899times119899 be a complex matrix the Drazininverse of119860 denoted by119860119863 is the uniquematrix119883 satisfyingthe following
(1) 119860119870+1119883 = 119860119870
(2) 119883119860119883 = 119883(3) 119860119883 = 119883119860
where119870 = ind(119860) is the index of 119860
The Drazin inverse has applications in the theory of finiteMarkov chains as well as in the study of differential equationsand singular linear difference equations and so forth [24] Toillustrate further the solution of singular systems has beenstudied by several authors For instance in [25] an analyticalsolution for continuous systems using the Drazin inverse waspresented
Note that a projection matrix 119875 defined as a matrix suchthat 1198752 = 119875 has 119870 = 1 (or 0) and has the Drazin inverse119875119863
= 119875 Also if 119860 is a nilpotent matrix (eg a shift matrix)then 119860119863 = 0 See for more [26]
In 2004 Li andWei in [27] proved that thematrixmethodof Schulz (the case for 119899 = 2 in (1)) can be used for findingthe Drazin inverse of square matrices They proposed thefollowing initial matrix
1198830= 120572119860119897
119897 ge ind (119860) = 119870 (9)
where the parameter 120572 must be chosen such that the condi-tion 119865
0 = 119860119860
119863
minus 1198601198830 lt 1 is satisfied Using this initial
matrix yields a numerically (asymptotical) stable methodfor finding the famous Drazin inverse with quadraticalconvergence
Using the above descriptions it is easy to apply theefficient method (6) for finding the Drazin inverse in whatfollows
Theorem 4 Let119860 = [119886119894119895]119899times119899
be a square matrix and ind(119860) =119870 ge 1 Choosing the initial approximation 119883
0as
1198830=
2
tr (119860119870+1)119860119870
(10)
or
1198830=
1
2119860119896+1
lowast
119860119870
(11)
wherein tr(sdot) stands for the trace of an arbitrary square matrixand sdot
lowastis a matrix norm then the iterative method (6)
converges with ninth order to 119860119863
Proof Consider the notation 1198650= 119860119860
119863
minus 1198601198830and sub-
sequently the residual matrix as 119865119896= 119860119860
119863
minus119860119883119896for finding
the Drazin inverse Then similar to (8) we get
119865119896+1
= 119860119860119863
minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119868 minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119864119896+1
= 119860119860119863
minus 119868 +
1
729
[3431198649
119896+ 294119864
10
119896+ 84119864
11
119896+ 811986412
119896]
= 119860119860119863
minus 119868 +
1
729
[343(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
9
+ 294(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
10
+ 84(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
11
+ 8(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
12
]
= 119860119860119863
minus 119868 +
1
729
[343 (119868 minus 119860119860119863
+ 1198659
119896)
+ 294 (119868 minus 119860119860119863
+ 11986510
119896)
+ 84 (119868 minus 119860119860119863
+ 11986511
119896)
+ 8 (119868 minus 119860119860119863
+ 11986512
119896)]
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(12)
By taking an arbitrary matrix norm to both sides of (12) weattain
1003817100381710038171003817119865119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
(13)
In addition since 1198650 lt 1 (the result of choosing the
appropriate initial matrices (10) and (11)) by relation (13) weobtain that 119865
1 le 119865
09 Similarly 119865
119896+1 le 119865
1198969 Using
mathematical induction we obtain1003817100381710038171003817119865119896
1003817100381710038171003817le10038171003817100381710038171198650
1003817100381710038171003817
9119896
119896 ge 0 (14)
So the sequence 1198651198962 is strictly monotonically decreasing
Now by considering 120575119896= 119860119863
minus 119883119896 as the error matrix for
finding the Drazin inverse we have
119860120575119896+1
= 119860119860119863
minus 119860119883119896+1
= 119865119896+1
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(15)
4 Abstract and Applied Analysis
Taking into account (12) and using elementary algebraictransformations we further derive
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
le1003817100381710038171003817119865119896
1003817100381710038171003817
9
=1003817100381710038171003817119860120575119896
1003817100381710038171003817
9
le 11986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(16)
It is now easy to find the error inequality of the new scheme(6) using (16) as follows
1003817100381710038171003817120575119896+1
1003817100381710038171003817=
10038171003817100381710038171003817119883119896+1
minus 11986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
119860119883119896+1
minus 119860119863
11986011986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
(119860119883119896+1
minus 119860119860119863
)
10038171003817100381710038171003817
le
1003817100381710038171003817100381711986011986310038171003817100381710038171003817
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817
le
100381710038171003817100381710038171198601198631003817100381710038171003817100381711986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(17)
Therefore the inequalities in (17) immediately lead to theconclusion that119883
119896rarr 119860119863 as 119896 rarr +infin with the ninth order
of convergence
Theorem 5 Considering the same assumptions as inTheorem 4 the iterative method (6) has asymptotical stabilityfor finding the Drazin inverse
Proof The steps of proving the asymptotical stability of (6)are similar to those that have recently been taken for a generalfamily of methods in [28] Hence the proof is omitted
Remark 6 It should be remarked that the generalization ofour proposed scheme for generalized outer inverses that is119860(2)
119879119878 is straightforward according to the recent work [29]
Remark 7 The new iteration (6) is free frommatrix power inits implementation and this allows one to apply it for findinggeneralized inverses easily
4 Numerical Experiments
We herein present several numerical tests to illustrate theefficiency of the new iterativemethod to compute the approx-
2 4 6 8 100
5
10
15
20
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 1 Comparison of the number of iterations for solvingExample 8
imate inverses mathematica 8 [30] has been employed inour calculations We have carried out the numerical testswith machine precision on a computer with characteris-tics Microsoft Windows XP Intel(R) Pentium(R) 4CPU320GHz with 4GB of RAM
For comparisons we have used the methods ldquoSchulzrdquo(119899 = 2) ldquoChebyshevrdquo (119899 = 3) ldquoKMSrdquo (119899 = 9) in (1) andthe proposed method (6) As the programs were running inthis paper we found the running time using the commandAbsoluteTiming[] to report the elapsed CPU time (inseconds) for the examples
Example 8 The computations of approximate inverses for 10dense random complex matrices of the dimension 100 areconsidered and compared as follows
n = 100 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Note that 119868 = radicminus1 For this example the stopping criterion is119883119896+1
minus 1198831198962le 10minus5 and the maximum number of iterations
allowed is set to 100 The initial choice has been constructedusing (2) with 120572 = 1120590
2
max The result of comparisons hasbeen presented in Figures 1 and 2 As could be observed in allthe 10 test problems our iterative method (6) beats the otherexisting schemes
Example 9 The computations of approximate inverses for 10dense random complex matrices of the dimension 200 areconsidered and compared as follows
n = 200 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Abstract and Applied Analysis 5
2 4 6 8 1000
01
02
03
04
Matrices
PMKMS
ChebyshevSchulz
Com
puta
tiona
l tim
e (s)
Figure 2 Comparison of the elapsed time for solving Example 8
2 4 6 8 100
5
10
15
20
25
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 3 Comparison of the number of iterations for solvingExample 9
Note again that 119868 = radicminus1 For this test the stoppingcriterion for finding the approximate inverse for the gen-erated random matrices is 119883
119896+1minus 1198831198962le 10
minus5 and themaximum number of iterations allowed is set to 100 by theinitial choice as in Example 8 The numerics are provided inFigures 3 and 4 Results are in harmony with the theoreticalaspects and show that the new method is efficient in matrixinversion
Example 10 The computations of approximate inverses for10 dense random matrices of the size 200 are investigated inwhat follows
n = 200 number = 10 SeedRandom[1]
Table[A[l] = RandomReal[0 1
n n] l number]
here the stopping criterion is 119883119896+1
minus 119883119896119865le 10minus8 Results
are provided in Figures 5 and 6 which indicate that the
2 4 6 8 1000
05
10
15
20
25
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 4 Comparison of the elapsed time for solving Example 9
2 4 6 8 100
10
20
30
40
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 5 Comparison of the number of iterations for solvingExample 10
2 4 6 8 1000
02
04
06
08
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 6 Comparison of the elapsed time for solving Example 10
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
inverse of a given square matrix In some cases it alsoprovides a solution of a given system of linear equations Notethat the Drazin inverse of a matrix 119860 mostly resembles thetrue inverse of 119860
Definition 2 The smallest nonnegative integer 119870 such thatrank(119860119870+1) = rank(119860119870) is called the index of119860 and denotedby ind(119860)
Definition 3 Let 119860 isin C119899times119899 be a complex matrix the Drazininverse of119860 denoted by119860119863 is the uniquematrix119883 satisfyingthe following
(1) 119860119870+1119883 = 119860119870
(2) 119883119860119883 = 119883(3) 119860119883 = 119883119860
where119870 = ind(119860) is the index of 119860
The Drazin inverse has applications in the theory of finiteMarkov chains as well as in the study of differential equationsand singular linear difference equations and so forth [24] Toillustrate further the solution of singular systems has beenstudied by several authors For instance in [25] an analyticalsolution for continuous systems using the Drazin inverse waspresented
Note that a projection matrix 119875 defined as a matrix suchthat 1198752 = 119875 has 119870 = 1 (or 0) and has the Drazin inverse119875119863
= 119875 Also if 119860 is a nilpotent matrix (eg a shift matrix)then 119860119863 = 0 See for more [26]
In 2004 Li andWei in [27] proved that thematrixmethodof Schulz (the case for 119899 = 2 in (1)) can be used for findingthe Drazin inverse of square matrices They proposed thefollowing initial matrix
1198830= 120572119860119897
119897 ge ind (119860) = 119870 (9)
where the parameter 120572 must be chosen such that the condi-tion 119865
0 = 119860119860
119863
minus 1198601198830 lt 1 is satisfied Using this initial
matrix yields a numerically (asymptotical) stable methodfor finding the famous Drazin inverse with quadraticalconvergence
Using the above descriptions it is easy to apply theefficient method (6) for finding the Drazin inverse in whatfollows
Theorem 4 Let119860 = [119886119894119895]119899times119899
be a square matrix and ind(119860) =119870 ge 1 Choosing the initial approximation 119883
0as
1198830=
2
tr (119860119870+1)119860119870
(10)
or
1198830=
1
2119860119896+1
lowast
119860119870
(11)
wherein tr(sdot) stands for the trace of an arbitrary square matrixand sdot
lowastis a matrix norm then the iterative method (6)
converges with ninth order to 119860119863
Proof Consider the notation 1198650= 119860119860
119863
minus 1198601198830and sub-
sequently the residual matrix as 119865119896= 119860119860
119863
minus119860119883119896for finding
the Drazin inverse Then similar to (8) we get
119865119896+1
= 119860119860119863
minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119868 minus 119860119883119896+1
= 119860119860119863
minus 119868 + 119864119896+1
= 119860119860119863
minus 119868 +
1
729
[3431198649
119896+ 294119864
10
119896+ 84119864
11
119896+ 811986412
119896]
= 119860119860119863
minus 119868 +
1
729
[343(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
9
+ 294(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
10
+ 84(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
11
+ 8(119868 minus 119860119860119863
+ 119860119860119863
minus 119860119883119896)
12
]
= 119860119860119863
minus 119868 +
1
729
[343 (119868 minus 119860119860119863
+ 1198659
119896)
+ 294 (119868 minus 119860119860119863
+ 11986510
119896)
+ 84 (119868 minus 119860119860119863
+ 11986511
119896)
+ 8 (119868 minus 119860119860119863
+ 11986512
119896)]
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(12)
By taking an arbitrary matrix norm to both sides of (12) weattain
1003817100381710038171003817119865119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
(13)
In addition since 1198650 lt 1 (the result of choosing the
appropriate initial matrices (10) and (11)) by relation (13) weobtain that 119865
1 le 119865
09 Similarly 119865
119896+1 le 119865
1198969 Using
mathematical induction we obtain1003817100381710038171003817119865119896
1003817100381710038171003817le10038171003817100381710038171198650
1003817100381710038171003817
9119896
119896 ge 0 (14)
So the sequence 1198651198962 is strictly monotonically decreasing
Now by considering 120575119896= 119860119863
minus 119883119896 as the error matrix for
finding the Drazin inverse we have
119860120575119896+1
= 119860119860119863
minus 119860119883119896+1
= 119865119896+1
=
1
729
[3431198659
119896+ 294119865
10
119896+ 84119865
11
119896+ 811986512
119896]
(15)
4 Abstract and Applied Analysis
Taking into account (12) and using elementary algebraictransformations we further derive
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
le1003817100381710038171003817119865119896
1003817100381710038171003817
9
=1003817100381710038171003817119860120575119896
1003817100381710038171003817
9
le 11986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(16)
It is now easy to find the error inequality of the new scheme(6) using (16) as follows
1003817100381710038171003817120575119896+1
1003817100381710038171003817=
10038171003817100381710038171003817119883119896+1
minus 11986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
119860119883119896+1
minus 119860119863
11986011986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
(119860119883119896+1
minus 119860119860119863
)
10038171003817100381710038171003817
le
1003817100381710038171003817100381711986011986310038171003817100381710038171003817
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817
le
100381710038171003817100381710038171198601198631003817100381710038171003817100381711986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(17)
Therefore the inequalities in (17) immediately lead to theconclusion that119883
119896rarr 119860119863 as 119896 rarr +infin with the ninth order
of convergence
Theorem 5 Considering the same assumptions as inTheorem 4 the iterative method (6) has asymptotical stabilityfor finding the Drazin inverse
Proof The steps of proving the asymptotical stability of (6)are similar to those that have recently been taken for a generalfamily of methods in [28] Hence the proof is omitted
Remark 6 It should be remarked that the generalization ofour proposed scheme for generalized outer inverses that is119860(2)
119879119878 is straightforward according to the recent work [29]
Remark 7 The new iteration (6) is free frommatrix power inits implementation and this allows one to apply it for findinggeneralized inverses easily
4 Numerical Experiments
We herein present several numerical tests to illustrate theefficiency of the new iterativemethod to compute the approx-
2 4 6 8 100
5
10
15
20
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 1 Comparison of the number of iterations for solvingExample 8
imate inverses mathematica 8 [30] has been employed inour calculations We have carried out the numerical testswith machine precision on a computer with characteris-tics Microsoft Windows XP Intel(R) Pentium(R) 4CPU320GHz with 4GB of RAM
For comparisons we have used the methods ldquoSchulzrdquo(119899 = 2) ldquoChebyshevrdquo (119899 = 3) ldquoKMSrdquo (119899 = 9) in (1) andthe proposed method (6) As the programs were running inthis paper we found the running time using the commandAbsoluteTiming[] to report the elapsed CPU time (inseconds) for the examples
Example 8 The computations of approximate inverses for 10dense random complex matrices of the dimension 100 areconsidered and compared as follows
n = 100 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Note that 119868 = radicminus1 For this example the stopping criterion is119883119896+1
minus 1198831198962le 10minus5 and the maximum number of iterations
allowed is set to 100 The initial choice has been constructedusing (2) with 120572 = 1120590
2
max The result of comparisons hasbeen presented in Figures 1 and 2 As could be observed in allthe 10 test problems our iterative method (6) beats the otherexisting schemes
Example 9 The computations of approximate inverses for 10dense random complex matrices of the dimension 200 areconsidered and compared as follows
n = 200 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Abstract and Applied Analysis 5
2 4 6 8 1000
01
02
03
04
Matrices
PMKMS
ChebyshevSchulz
Com
puta
tiona
l tim
e (s)
Figure 2 Comparison of the elapsed time for solving Example 8
2 4 6 8 100
5
10
15
20
25
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 3 Comparison of the number of iterations for solvingExample 9
Note again that 119868 = radicminus1 For this test the stoppingcriterion for finding the approximate inverse for the gen-erated random matrices is 119883
119896+1minus 1198831198962le 10
minus5 and themaximum number of iterations allowed is set to 100 by theinitial choice as in Example 8 The numerics are provided inFigures 3 and 4 Results are in harmony with the theoreticalaspects and show that the new method is efficient in matrixinversion
Example 10 The computations of approximate inverses for10 dense random matrices of the size 200 are investigated inwhat follows
n = 200 number = 10 SeedRandom[1]
Table[A[l] = RandomReal[0 1
n n] l number]
here the stopping criterion is 119883119896+1
minus 119883119896119865le 10minus8 Results
are provided in Figures 5 and 6 which indicate that the
2 4 6 8 1000
05
10
15
20
25
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 4 Comparison of the elapsed time for solving Example 9
2 4 6 8 100
10
20
30
40
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 5 Comparison of the number of iterations for solvingExample 10
2 4 6 8 1000
02
04
06
08
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 6 Comparison of the elapsed time for solving Example 10
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Taking into account (12) and using elementary algebraictransformations we further derive
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817le
1
729
[3431003817100381710038171003817119865119896
1003817100381710038171003817
9
+ 2941003817100381710038171003817119865119896
1003817100381710038171003817
10
+ 841003817100381710038171003817119865119896
1003817100381710038171003817
11
+ 81003817100381710038171003817119865119896
1003817100381710038171003817
12
]
le1003817100381710038171003817119865119896
1003817100381710038171003817
9
=1003817100381710038171003817119860120575119896
1003817100381710038171003817
9
le 11986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(16)
It is now easy to find the error inequality of the new scheme(6) using (16) as follows
1003817100381710038171003817120575119896+1
1003817100381710038171003817=
10038171003817100381710038171003817119883119896+1
minus 11986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
119860119883119896+1
minus 119860119863
11986011986011986310038171003817100381710038171003817
=
10038171003817100381710038171003817119860119863
(119860119883119896+1
minus 119860119860119863
)
10038171003817100381710038171003817
le
1003817100381710038171003817100381711986011986310038171003817100381710038171003817
1003817100381710038171003817119860120575119896+1
1003817100381710038171003817
le
100381710038171003817100381710038171198601198631003817100381710038171003817100381711986091003817100381710038171003817120575119896
1003817100381710038171003817
9
(17)
Therefore the inequalities in (17) immediately lead to theconclusion that119883
119896rarr 119860119863 as 119896 rarr +infin with the ninth order
of convergence
Theorem 5 Considering the same assumptions as inTheorem 4 the iterative method (6) has asymptotical stabilityfor finding the Drazin inverse
Proof The steps of proving the asymptotical stability of (6)are similar to those that have recently been taken for a generalfamily of methods in [28] Hence the proof is omitted
Remark 6 It should be remarked that the generalization ofour proposed scheme for generalized outer inverses that is119860(2)
119879119878 is straightforward according to the recent work [29]
Remark 7 The new iteration (6) is free frommatrix power inits implementation and this allows one to apply it for findinggeneralized inverses easily
4 Numerical Experiments
We herein present several numerical tests to illustrate theefficiency of the new iterativemethod to compute the approx-
2 4 6 8 100
5
10
15
20
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 1 Comparison of the number of iterations for solvingExample 8
imate inverses mathematica 8 [30] has been employed inour calculations We have carried out the numerical testswith machine precision on a computer with characteris-tics Microsoft Windows XP Intel(R) Pentium(R) 4CPU320GHz with 4GB of RAM
For comparisons we have used the methods ldquoSchulzrdquo(119899 = 2) ldquoChebyshevrdquo (119899 = 3) ldquoKMSrdquo (119899 = 9) in (1) andthe proposed method (6) As the programs were running inthis paper we found the running time using the commandAbsoluteTiming[] to report the elapsed CPU time (inseconds) for the examples
Example 8 The computations of approximate inverses for 10dense random complex matrices of the dimension 100 areconsidered and compared as follows
n = 100 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Note that 119868 = radicminus1 For this example the stopping criterion is119883119896+1
minus 1198831198962le 10minus5 and the maximum number of iterations
allowed is set to 100 The initial choice has been constructedusing (2) with 120572 = 1120590
2
max The result of comparisons hasbeen presented in Figures 1 and 2 As could be observed in allthe 10 test problems our iterative method (6) beats the otherexisting schemes
Example 9 The computations of approximate inverses for 10dense random complex matrices of the dimension 200 areconsidered and compared as follows
n = 200 number = 10 SeedRandom[1234]
Table[A[l] = RandomComplex[minus2 + I
2 - I n n]l number]
Abstract and Applied Analysis 5
2 4 6 8 1000
01
02
03
04
Matrices
PMKMS
ChebyshevSchulz
Com
puta
tiona
l tim
e (s)
Figure 2 Comparison of the elapsed time for solving Example 8
2 4 6 8 100
5
10
15
20
25
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 3 Comparison of the number of iterations for solvingExample 9
Note again that 119868 = radicminus1 For this test the stoppingcriterion for finding the approximate inverse for the gen-erated random matrices is 119883
119896+1minus 1198831198962le 10
minus5 and themaximum number of iterations allowed is set to 100 by theinitial choice as in Example 8 The numerics are provided inFigures 3 and 4 Results are in harmony with the theoreticalaspects and show that the new method is efficient in matrixinversion
Example 10 The computations of approximate inverses for10 dense random matrices of the size 200 are investigated inwhat follows
n = 200 number = 10 SeedRandom[1]
Table[A[l] = RandomReal[0 1
n n] l number]
here the stopping criterion is 119883119896+1
minus 119883119896119865le 10minus8 Results
are provided in Figures 5 and 6 which indicate that the
2 4 6 8 1000
05
10
15
20
25
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 4 Comparison of the elapsed time for solving Example 9
2 4 6 8 100
10
20
30
40
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 5 Comparison of the number of iterations for solvingExample 10
2 4 6 8 1000
02
04
06
08
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 6 Comparison of the elapsed time for solving Example 10
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
2 4 6 8 1000
01
02
03
04
Matrices
PMKMS
ChebyshevSchulz
Com
puta
tiona
l tim
e (s)
Figure 2 Comparison of the elapsed time for solving Example 8
2 4 6 8 100
5
10
15
20
25
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 3 Comparison of the number of iterations for solvingExample 9
Note again that 119868 = radicminus1 For this test the stoppingcriterion for finding the approximate inverse for the gen-erated random matrices is 119883
119896+1minus 1198831198962le 10
minus5 and themaximum number of iterations allowed is set to 100 by theinitial choice as in Example 8 The numerics are provided inFigures 3 and 4 Results are in harmony with the theoreticalaspects and show that the new method is efficient in matrixinversion
Example 10 The computations of approximate inverses for10 dense random matrices of the size 200 are investigated inwhat follows
n = 200 number = 10 SeedRandom[1]
Table[A[l] = RandomReal[0 1
n n] l number]
here the stopping criterion is 119883119896+1
minus 119883119896119865le 10minus8 Results
are provided in Figures 5 and 6 which indicate that the
2 4 6 8 1000
05
10
15
20
25
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 4 Comparison of the elapsed time for solving Example 9
2 4 6 8 100
10
20
30
40
Matrices
Num
ber o
f ite
ratio
ns
PMKMS
ChebyshevSchulz
Figure 5 Comparison of the number of iterations for solvingExample 10
2 4 6 8 1000
02
04
06
08
Matrices
Com
puta
tiona
l tim
e (s)
PMKMS
ChebyshevSchulz
Figure 6 Comparison of the elapsed time for solving Example 10
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
new method is better in terms of the number of iterationsand behaves similarly to the other well-known schemesof Schulz and Chebyshev in terms of the computationaltime
The order of convergence and the number of matrix-matrix products are not the only factors to govern the effi-ciency of an algorithm per computing step in matrix iter-ations Generally speaking the stopping criterion couldbe reported as one of the important factors which couldindirectly affect the computational time of an algorithm inimplementations especially when trying to find the general-ized inverses
Although in the above implementations we consideredthe stop termination on two successive iterates this is nota reliable termination when dealing with some large ill-conditioned matrices For example the reliable stoppingcriterion in the above examples is 119868 minus 119883
119896119860119865le 120598
Example 11 The aim of this example is to apply the dis-cussions of Section 4 for finding the Drazin inverse of thefollowing square matrix (taken from [27])
119860
=
(
(
(
(
(
(
(
(
(
(
(
(
(
2 04 0 0 0 0 0 0 0 0 0 0
minus2 04 0 0 0 0 0 0 0 0 0 0
minus1 minus1 1 minus1 0 0 0 0 minus1 0 0 0
minus1 minus1 minus1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 minus1 minus1 0 0 minus1 0
0 0 0 0 1 1 minus1 minus1 0 0 0 0
0 0 0 minus1 minus2 04 0 0 0 0 0 0
0 0 0 0 2 04 0 0 0 0 0 0
0 minus1 0 0 0 0 0 0 1 minus1 minus1 minus1
0 0 0 0 0 0 0 0 minus1 1 minus1 minus1
0 0 0 0 0 0 0 0 0 0 04 minus2
0 0 0 0 0 0 0 0 0 0 04 2
)
)
)
)
)
)
)
)
)
)
)
)
)
(18)
with119870 = ind(119860) = 3 Using (6) and the stopping termination119883119896+1
minus 119883119896infin
le 10minus8 and (10) we could obtain the Drazin
inverse as follows
119860119863
=
(
(
(
(
(
(
(
(
(
(
(
(
(
025 minus025 0 0 0 0 0 0 0 0 0 0
125 125 0 0 0 0 0 0 0 0 0 0
minus166406 minus0992187 025 minus025 0 0 0 0 minus00625 minus00625 0 015625
minus119531 minus0679687 minus025 025 0 0 0 0 minus00625 01875 06875 134375
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 332031 664063
minus276367 minus104492 minus1875 minus125 minus125 125 125 125 148438 257813 457031 851563
141094 630078 6625 3375 5 minus3 minus5 minus5 minus41875 minus85 minus105078 minus224609
minus193242 minus850781 minus975 minus525 minus75 45 75 75 6375 125625 159766 337891
minus0625 minus03125 0 0 0 0 0 0 025 minus025 minus0875 minus1625
minus125 minus09375 0 0 0 0 0 0 minus025 025 minus0875 minus1625
0 0 0 0 0 0 0 0 0 0 125 125
0 0 0 0 0 0 0 0 0 0 minus025 025
)
)
)
)
)
)
)
)
)
)
)
)
)
(19)
Due to the efficiency of the new method we did notcompare different methods for this test and just bringforward the results of (6) Checking the conditions ofDefinition 42 yields 119860119896+1119860119863 minus 119860119896
infin= 369482 times 10
minus13119860119863
119860119860119863
minus 119860119863
infin
= 100933 times 10minus10 and 119860119860
119863
minus
119860119863
119860infin
= 231148 times 10minus11 which supports the theoretical
discussions
5 Summary
In this paper we have developed a high order matrix methodfor finding approximate inverses for nonsingular squarematrices It has been proven that the contributed methodreaches the convergence order nine by using seven matrix-matrix multiplications which makes its informational indexas 97 asymp 1288 gt 1
We have also discussed the importance of the well-knownDrazin inverse andhow to find it numerically by the proposedmethod Numerical examples were also employed to supportthe underlying theory of the paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1591301434 The authors therefore acknowledgewith thanks DSR technical and financial support
References
[1] B Canto C Coll and E Sanchez ldquoIdentifiability for a classof discretized linear partial differential algebraic equationsrdquoMathematical Problems in Engineering vol 2011 Article ID510519 12 pages 2011
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
[3] S K Sen and S S Prabhu ldquoOptimal iterative schemes forcomputing the Moore-Penrose matrix inverserdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 8 pp 748ndash753 1976
[4] F Soleymani ldquoA rapid numerical algorithm to compute matrixinversionrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2012 Article ID 134653 11 pages 2012
[5] F Soleymani ldquoA new method for solving ill-conditioned linearsystemsrdquo Opuscula Mathematica vol 33 no 2 pp 337ndash3442013
[6] F Toutounian and F Soleymani ldquoAn iterative method forcomputing the approximate inverse of a square matrix andthe Moore-Penrose inverse of a non-square matrixrdquo AppliedMathematics and Computation vol 224 pp 671ndash680 2013
[7] F Soleymani P S Stanimirovic and M Z Ullah ldquoAn acceler-ated iterative method for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 222 pp365ndash371 2013
[8] F Soleymani and P S Stanimirovic ldquoA higher order iterativemethod for computing the Drazin inverserdquoThe Scientific WorldJournal vol 2013 Article ID 708647 11 pages 2013
[9] X Liu H Jin and Y Yu ldquoHigher-order convergent iterativemethod for computing the generalized inverse and its applica-tion to Toeplitz matricesrdquo Linear Algebra and Its Applicationsvol 439 no 6 pp 1635ndash1650 2013
[10] G Montero L Gonzalez E Florez M D Garcıa and ASuarez ldquoApproximate inverse computation using Frobeniusinner productrdquoNumerical Linear Algebra with Applications vol9 no 3 pp 239ndash247 2002
[11] F Soleymani ldquoOn a fast iterative method for approximateinverse of matricesrdquo Korean Mathematical Society Communi-cations vol 28 no 2 pp 407ndash418 2013
[12] X Sheng ldquoExecute elementary row and column operations onthe partitioned matrix to compute M-P inverse Adaggerrdquo AbstractandApplied Analysis vol 2014 Article ID 596049 6 pages 2014
[13] G Schulz ldquoIterative Berechnung der Reziproken matrixrdquoZeitschrift fur Angewandte Mathematik und Mechanik vol 13pp 57ndash59 1933
[14] A Ben-Israel and T N E Greville Generalized InversesSpringer New York NY USA 2nd edition 2003
[15] F Soleymani ldquoA fast convergent iterative solver for approximateinverse of matricesrdquo Numerical Linear Algebra with Applica-tions 2013
[16] A M Ostrowski ldquoSur quelques transformations de la serie deLiouvilleNewmanrdquo Comptes Rendus de lrsquoAcademie des Sciencesvol 206 pp 1345ndash1347 1938
[17] F Soleimani F Soleymani and S Shateyi ldquoSome iterativemethods free from derivatives and their basins of attraction fornonlinear equationsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 301718 10 pages 2013
[18] F Soleymani and D K R Babajee ldquoComputing multiple rootsusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 pp 531ndash541 2013
[19] F Soleymani ldquoEfficient optimal eighth-order derivative-freemethods for nonlinear equationsrdquo Japan Journal of Industrialand Applied Mathematics vol 30 no 2 pp 287ndash306 2013
[20] J R Torregrosa I K Argyros C Chun A Cordero andF Soleymani ldquoIterative methods for nonlinear equations orsystems and their applications [Editorial]rdquo Journal of AppliedMathematics vol 2013 Article ID 656953 2 pages 2013
[21] G W Stewart and J G Sun Matrix Perturbation TheoryAcademic Press Boston Mass USA 1990
[22] M Zaka Ullah F Soleymani and A S Al-Fhaid ldquoAn effi-cient matrix iteration for computing weighted Moore-Penroseinverserdquo Applied Mathematics and Computation vol 226 pp441ndash454 2014
[23] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958
[24] I Kyrchei ldquoExplicit formulas for determinantal representationsof the Drazin inverse solutions of some matrix and differentialmatrix equationsrdquo Applied Mathematics and Computation vol219 no 14 pp 7632ndash7644 2013
[25] S L Campbell C D Meyer Jr and N J Rose ldquoApplicationsof the Drazin inverse to linear systems of differential equationswith singular constant coefficientsrdquo SIAM Journal on AppliedMathematics vol 31 no 3 pp 411ndash425 1976
[26] L Zhao ldquoThe expression of the Drazin inverse with rankconstraintsrdquo Journal of Applied Mathematics vol 2012 ArticleID 390592 10 pages 2012
[27] X Li and Y Wei ldquoIterative methods for the Drazin inverse ofa matrix with a complex spectrumrdquo Applied Mathematics andComputation vol 147 no 3 pp 855ndash862 2004
[28] F Soleymani and P S Stanimirovic ldquoA note on the stability ofa 119901th order iteration for finding generalized inversesrdquo AppliedMathematics Letters vol 28 pp 77ndash81 2014
[29] P S Stanimirovic and F Soleymani ldquoA class of numerical algo-rithms for computing outer inversesrdquo Journal of Computationaland Applied Mathematics vol 263 pp 236ndash245 2014
[30] M TrottTheMathematica Guide-Book For Numerics SpringerNew York NY USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![ON THE GENERALIZED DRAZIN INVERSE AND GENERALIZED …nasport.pmf.ni.ac.rs/publikacije/7/ROSE1.pdf · The Drazin inverse is investigated in the matrix theory [2, 3, 17, 26, 27], in](https://static.fdocuments.us/doc/165x107/5f6460212813764a924bb37c/on-the-generalized-drazin-inverse-and-generalized-the-drazin-inverse-is-investigated.jpg)
![arXiv:math/0003224v1 [math.RA] 30 Mar 2000 · rank equalities for matrix expressions involving Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose](https://static.fdocuments.us/doc/165x107/5f961236b05e766be7258ab6/arxivmath0003224v1-mathra-30-mar-2000-rank-equalities-for-matrix-expressions.jpg){kind=icon}
