Iterative Methods of Higher Order for Nonlinear Equations

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Vietnam J Math (2016)44:387-398 DOI 10.I007/S10013-015.0I35-1

Iterative Methods of Higher Order for Nonlinear Equations

Sukhjit Singh • Dharmendra Kumar Gupta

Received: 22 Apnl 2014 / Accepted: 24 November 2014 /Published online 5 March 2015

Abstract Two new three-step higher-order iterative methods are developed by modifying two third-order two-step methods for solving nonlinear equations. This is done by introducing Newton's method as a third step in both methods. We approximate the denvative in Ihe third step by the technique of linear interpolation and divided differences leading to new sixth-order methods with efficiency indices equal to 1.565. The number of iterations and the total number of function evaluations required to get a root correct up to 15 decimal places are taken as performance measures of the methods. A number of numerical examples are worked out to demonstrate the efficacy of the proposed methods. The results obtained are compared with some existing methods. It is observed that the methods give improved results.

Keywords Newton's method Nonlinear equations Iterative method • Taylor series • Convergence analysis • Efficiency index

Mathematics Subject Classification (2010) 65H04 • 65H05

1 Introduction

One of the most promising problems in numerical analysis and apphed mathematics is to solve nonlinear equations

f(x) = 0. il) where f : D c K - > M i s a continuously differentiable real function on an open interval

D. A number of applications can be found fliat give rise to Uiese equations depending on one or more parameters. For example, die problems of kinetic theory of gases, elasticity and

S Singh (S) D K Gupta

Department of Mathematics, Indian Insiiluie of Technology Kharagpur. Kharagpur 721302, India e-mail' [email protected]

D. K Gupia

e-mail: dkg@maths iitkgp.ernet in

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S, Singh. D. K, Gupia optimization are reduced to solving nonlinear equations. Many researchers [4, 11, 14, 15, 17,18,21] have extensively studied iterative methods along with their convergence analysis for them. These methods require one or more initial guesses for the desired root. Bisection and regula falsi methods are globally convergent iterative methods used to solve (1). The quadratically convergent Newton's method given for fc = 0, 1, 2 , . . . by

fiXk)

with suitably chosen initial approximation near to the root is generally used for solving (I).

But it may fail to converge in case the initial point is far from the root or the derivative van- ishes in the vicinity of the root. If the efficiency index [3] of an iterative mefliod is defined as q " ' ' , where q is the order of the method and d is the number of function evaluations per iteration, then the efficiency index of Newton's method is 1.414. Newton's method is modified in a number of ways at the additional cost of evaluation of a function, denvative and changes in the points of iteration in order to increase its efficiency index and order of convergence.

Recendy, a number of higher-order iterative methods are also developed for solving (1).

Though these methods require more computational costs, fliey are advantageous in applica- uons such as stiff system of equations where quick convergence is required. A third-order method requiring evaluations of one function and two first derivatives per iteration is proposed in [21]. A family of third-order methods has been developed in [4] which requires one function, one first derivative and one second derivative evaluations per iteration It is given for t = 0 , 1 , 2 , . . . by

A , I Efixi,) \ f(xk) Xk+i =Xk-[l-\--- ' - • T T T " ; ' V 2l-^Lf{xk)J f'(Xk)

starting with a suitably chosen initial approximation to the root x" and Lfix/i) = ,f'(/}\2' where ^ is a real parameter. One can easily see that for particular values of fi, the Chebyshev's (fi ^ 0), die Halley's (fi ^ \) and the Super-Halley's (fi = I) methods are special cases of this family. But the disadvantage of these methods are that diey involve the evaluation second-order derivative which is either computationally difficult to compute or remain unbounded. Using forward, backward and central difference approximation of the first derivative, Khattri and Log [7] developed nine derivative-free families of iterative methods from the three well-known classical mediods namely Cheby- shev, Halley and Euler iterative methods for solving nonlinear equations. Convergence analysis is established to show that the methods of these famihes are cubically convergent.

Apart from this, computational work is also carried out to demonstrate that the developed methods are better than the three classical methods.

Ostrowski [15] proposed a fourth-order iterative method which requires two evaluations of a function and one evaluation of a derivative. King [8] developed a one-parameter family of two-step fourth-order mediods to solve {!). A number of sixth-order methods [2, 5, 9, 12, 13, 16, 17, 19, 20] are also developed to solve (1). Khattri and Argyros [6] developed a four-parameter family of sixth-order convergent iterative methods for solving nonlinear scaler equations. These methods are totally free of derivatives and require evaluation of four functions per iteration. Convergence analysis is carried out to show that the family is sixth- order convergent, which is verified through the numerical work. Based on a fourth-order

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Iterative Methods of Higher Order for Nonlinear Equations

^ ^ ^ ^ ^ method, Sharma and Guha [17] developed a one-parameter family of three-step sixlh-order methods defined for A = 0, 1, 2 by

/ ' ( « ) f(.xt)Ha-2)f{zt)

with a suitable chosen XQ near to the root x'. Another family of three-step sixdi-order mediods in [12] forsolving(l)isgivenforjt = 0 , 1 , 2 by

-jr. -IS^

~'^* r(xk)'

"" '^* /'(«) f(j:j)-l-(^-2)/(«n)- i^>

, - ,, _ /(:*) f(.XL)-f<.wO+y.f{zt)

" ' * /'tr*) f{xi)-3/{wi)+yf{zi) •

where ^, y e M. Starting with a suitably chosen XQ near to the root x', Kou and Li

|9] proposed a new variant of Jarratt method with sixth order of convergence defined for it = 0,1,2 by

Jrtx,) = 3 / ' ( y t ) + / > )

Z,^Xk-Jf(x,)jl^^, (4)

Xk+i - Z k - -; ^ ^ . ^ ,

Wherey, = xk - \ j ^ ,

The aim of this paper is to develop two new three-step higher-order iterative methods by modifying two third-order two-step methods for solving nonlinear equations. This is done by introducing a third step in first of them and approximating its denvative by linear interpolation making it a sixth-order method In a similar manner, a third step is added in a second third-order method but its derivative is approximated by divided dilTerences up to second order leading it also to a sixth-order method. Convergence analysis of both the metiiods is established. This enhances the efficiency indices of the new methods from 1,442 to 1 565. A number of numerical examples are worked out to demonstrate the efficacy of the proposed methods. The number of iterations and flie total number of function evaluations requu-ed to get a root correct up to 15 decimal places are taken as performance measures of the mediods The results obtained are compared widi some existing mediods. it is observed that the methods give improved results.

This paper is organized as follows. Section 1 is the introduction. In Section 2, two new three-step sixth-order iterative methods for solving nonlinear equations are described. The convergence analysis of the methods to show their sixfli order of convergence is established in Section 3. In Section 4, several numerical examples are worked out to demonstrate the etficiency of die proposed methods. Finally, conclusions are included in Section 5.

2 Two New Higher-Order Iterative Methods

In tills section, two new three-step sixdi-order iterative methods for solving nonlinear

f

fions are described. A diird-order metiiod to solve (I) developed in [1] is given for 0 . 1 . 2 . . . ,by

^ S p r i i

(

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S Singh, D. K, Gupia yk=Xk- fi'k)'

x.+.=y>.-^{imfm)i \f(xL)+rixk)jTw'

⁽⁵⁾

with an initial approximation JCQ m the neighborhood of the root x*. The first sixth-order iierative mediod is developed from (5) as follows. Starting with a suitably chosen initial approximation JTQ near to the root, defined for A = 0 . 1 . 2 , . . . , by

yk=xt- f'iXi) i (f\xt)-f' -JML

f'bO

The efficiency index of this method is 6 ' = 1.430. In order to reduce the number of function evaluations, f'(zk) is approximated by linear interpolation on two points: {J:^, f'ixt)) and (yk. /'(y*)). This gives

Thus, an approximation to f'(zk) is given by

- / ' U i ) . On simplification, this gives

) + 4f'ixk)f'iyk) - (fixk))^ - if'(yk))^

f(Xk) + f'iXk) Substituting (7) in (6), the three-step sixth-order method {PMl) is given for k -- by

Xk+l

m

2f(xt)f(yk)+'if'(xi)f'(yi)-(.f'(xO)^-(f'(yk))^

The efficiency index of this method is 1,5651, The third-order method of [10] given for it = 0 , 1 , 2 , . .,by

I

V, — j-i -I- /t-^*' Xk+i -Xk 7^;-) ••.

is also extended in a similar manner to get another three-s Starting with a suitably chosen initial approximation XQ r A = 0 , 1 , 2 by

(9) IS sixth-order method as follows, r to the root, it is given for yk=Xk-^-

Zk = Xk- fiXk)'

iSi' •

⁽¹⁰⁾

The efficiency index of diis three-step method is 65 = 1.430. In order to reduce the number of function evaluations, f'(zk) is approximated by divided differences up to the second order. Expanding f(zk) at yt by Taylor's expansion up to second order, we get

f(Zk) ~ fiyk) + f'iyOizk - (11)

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Iterative Methods of Higher Order for Nonlinear Equations This gives

f'(yk) ^ f[zk. yk] - i^f"iyk)izk - yk),

where f[zk. yk] = ^''^Zyj^— is die divided difference of fu-st order. In a similar manner, f'iyk) can be approximated by

Now, differentiating (11), we get

f'izk) ^ f'iyk) + f"(yk)izk - yk). (12) Substituting Ihe expressions for f'iyk) and f"(yk) in (12), we get

f'(zk) ^ f[Zk, yk] + fiZk. xk,xk]izk - yk). (13) Substituting (13) in (10), another three-steps sixth-order method CPM2) is given for

k = 0 , 1 , 2 , . , . , by

fiXk) ' 7 ^ * } '

f'ixi,) ' i^^>

fUk)

^' ~ •'* fUt.yk]+f[zk.'^k.xd(zk-yky The efficiency index of this method is enhanced to 1.5651.

3 Convergence Analysis

Theorem 1 Let f : D c R -> R be a sufficiently differentiable function in an open interval D and XQ is close to its simple root x* 6 D. The iterative method (PMl) defined by (8) satisfies the error equation

ek+i = ^ {4cl + I6ct + I6cl - 8c|c3 - 16c^c3 - 5c2c|) ef + Oiel), where cj = /f'rc-) / " ' ' <: = 2, 3 , . ..

Proof Letet = Xk — jc* be the error in Adi iterate. Using Taylor expansion o f / ( . t i ) and f'ixk) about j ; * , we get

f(xk) = fix") [ek + C2el + cêl -i- CACI + cêl + cêl 4- 0 ( e j ) ] , f'(xk) = fix*) [l 4- 2c2«i + 3cêl + 4c4el + Scêj -I- 6c6e| -I- 0(ef)'j . Substimting f(xk) and fixk) in (8), we get

yt = [x* 4- C2el -I- ( - 2 c ^ 4- 2ci) el -I- (4c^ - 7c2C3 -H 3C4) e^

4- ( - 8 c | -I- 20c^C3 - 6cl - 10C2C4 -H 4C5) e |

-1- (l6c^ - 52clci -F 33c2c| -I- 28c^c4 - 17oc4 - l3c2C5 -t- Scs) ef -\- O(el)'^ .

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S Singh. D. K Gupta Again, using tile Taylor expansion of fiyk) about x*, we get

/ ( » ) = fix') [ ( » - X') + C2(» - X'f + C3(n - X'f + C4(V1 - X'f +C5(» -x')'+ C6(» - * • ) « + 0 ( e ] ) ] .

Tlus gives

/ ( » ) = / ' ( * • ) [nel + ( - 2 1 5 + 2C3) el + (54 - 7 Q C 3 + 3 Q ) e*

+ ( - 1 2 4 + 24cjc3 - 6c^ - 10C2C4 + 4n) el

+ (2^4 - 73c|c3 + 37c2c| + 14cjc, - nc^c^ - 13c2« + Scj) ej + 0{e\-^ . Similarly

/ ' ( » ) = f(x•)[l+24el + (~44+^C2C3)el+(sci-u4cl + 6c2Ct)e'^

+ ( - 1 6 c | + 28c|c3 - 20clc, + S Q C J ) C ' + (32cf - 68c^C3 +12c^ + 60cjc4 - 16C2C3C4 - 26c|c5 + 10C2C6) «i + "(<?!)] • Therefore,

1 /'(g) - /'(») /to)

2 / ( « ) + / ' ( i » ) / ' ( « )

+ 156c|c4 -41C3C4 - 5c, - 41C2C5 + 6<r6) ef + 0(<fj)l. (15) Substituting (15) in (8), we get

Again expanding the Taylor expansion of fizO about x*, we get

/(Zt) = fix') [(zt - x') + C2(Zi - x'f + C3(Zi - ! * ) ' + C4(zt - x'f +C5(z» -x')'+ aUk - x')'' + OCe^)] .

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Iterative Methods of Higher Order for Nonlinear Equa This gives

/ ( z . ) = f\x') [ ( c , + 14+'^) 4 + (-C2 - 5 t | - 9 4 + ^ + ? f 2 + C4) 4 +- (2C2 + 12c| + 38c| + 60c5 - 3c3 - 3OC2C3 - W4ci +icl + 4c4 + 12C2C4 + 3i;5) ef + 7 ( - 4 e 2 - 28cl - 100c| - 236c5 -3i64 + 6C3 + 74<;2C3 + 340c|c3 + 634r|c3 - 42c| - 147ir2c|

-8C4 - 84C2C4 - 200c|c4 + 14C3C4 + IOC5 + 3OC2C5 + 8C6) 4

+0(el)].

From (7), we get

/'(Zl) - / ' ( i * ) I 1 + 2 (c5 + 2c5 - C2C3) « | + J ( - 4 « 1 - 20c] - 36c; + 12c2C3 +42f;< 3 - 9c] - 4C2C4) ej + - Oicj + 24c| + 76cJ + 120c^ - 12c2C3 -84c^C3 - 204c|c3 + 9C3 + 78C2C3 + 16C2C4 + 48C5C4 - 24c3C4 - 4C2C5) el 4-- ( - 4 c | - 28c| - 104c3 - 252c| - 352c| + 12c2C3 + 104c|c3 + 408cjc3 +776t;lc3 - 9c5 - 11 lC2c| - 430c;c| + 5 l c | - I6C2C4 - 116c5c4 - 256cjc4

+24C3C4 + i80c2C3C4 - 16c] + 20C2C5 + 60c|c5 - 3OC3C5 - 4C2C6) cf + 0 ( c , ' ) ] .

Substituting fUk) and / ' ( z t ) in (8), we get

ci+l = « + i - ;r' = i (4cl + \6ci+ I6c] - Sc^ct - IScia - 5C2cfj ef + 0 ( c j ) .

Thus,thePMl isof sixthorrJer. ItsefFiciencyindexis65 = 1.5651. D Theorem 2 Z.er f . D c M - * H be a sufficiently differentiable function in an open

interval D and jro be close 10 its simple root x* e D.The iterative method (PM2i defined by{\4) satisfies the error equation

C14.1 = 4 (cS - 3c|c3 + 2C2c] + c;c4 - C3C4) cf + 0 ( e j ) , whereck = tf/xi') -^'"" ^ — ^ ' ^

Proof Letci = xt - .v'be the error in Ath iterate. Usmg Taylor expansion of/(j:t) and f'(.Xk) about J : ' , we get

/(.v,) = fix') [ « + C2c| + C3c' + C4cJ + Cscj + Cjef + 0 ( e j ) ] , / ' ( « - ) = / ' ( J t * ) [ l + 2c2C<- + 3c3ef+4c4cJ + 5c5cJ + 6c6ef+ 0 ( c f ) ] .

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S. Singh. D. K Gupta Substituting f{xi^) and f'ixk) in (14), we get

» = \x' + 2ei - C2cf + {2cl - 2C3) el 4- ( - 4 c | + 7c2C3 - 3C4) ej + (Sc^ - 20c|c3 + 6c5 + 10C2C4 - 4C3) el + (-16c^ 4- 52c;c3

-33C2c| - 28c|c4 + 17C3C4 4- 13c2Cs - Scj) cJ + OielA . (16) Now expanding Taylor expansion of / ( y r ) about jr*, we get

/ ( » ) = fix') [ ( « - X-) + C2(yi - J * ) ' + C3(» - * • ) ' 4- C4(n - x')' +cs(jk - jr*)= 4- C6(» - ; c ' ) ' 4- 0(el)\ .

From (16), tlus gives

/<ys) = / ' ( ! * ) [2ct 4- 3c2cf + (-2cl + 6C3) el + ( 5 c | ~ 13c2r3 + Met) ej 4- [-nc\ + 42c|c3 - 18c| - 34C2C4 + 2 8 0 ) ej

+ (28c5 - 123c|c3 4- 103C2C| 4- 106c|c4 - 83C3C4 - 83c2C5 4- 59C6) cJ 4-0(,])].

Substituting f(xk). f'(xk) and / ( y j ) in (14) we get

Zi = ['' 4- ( 2 c | - 2C3) el + (-9cl + 17c2C3 - 8C4) ef 4- (sOc^ - 82c|c3 4- 24c5 4-50C2C4 - 22C5) el + (-88c^ 4- 314c|c3 - 202c2c] - 214c]c4 4- II5C3C4

4 - 1 2 7 C 2 C 3 - 5 2 c 6 ) c j 4 - 0 ( e J ) ] . (17) Again expanding the Taylor expansion of f(zk) about x', we get

/ ( Z i ) = / ' ( * * ) [{Zk - x') 4- C2(Zi - x'f 4- C3(Zi - x'f + C4(ZJ - x')"

+C5(Zk -x'f+ ce(zk -x'f 4- 0(cj)] . This gives

/(zj) = fix'] [(24 - 2C3) 4 4- ( - 9 c | 4- 17C2C, - 8C4) 4 + (30c3 - 82c|cj 4- 24c5 4-50C2C4 - 22C5) el + (-84cf + 306c|c3 - 198c^c^ - 214c|c4 + II5C3C4

+127c2Cs-52c6)cf + 0 ( e ; ) ] . (18) Using (13), this gives

fizk) - f[zk, ykl 4- f[Zk. Xk.xkKzk - yk)

= fix') [1 4- (4c^ - 6c2C3 4- 2C4) el + ( - 1 8 c ; + 43c|c3 - 8 c | - 25C2C4 4-8C3) ef + (60c^ - 194c|c3 + 100c2c| + I30c|c4 - 46C3C4 - 72c2C5 4-22C6) 4 4- ( - 2 1 8 c | 4- 751c;c3 - 5 4 1 c | c | 4- 24c| - 412c|c4

4-300C2C3C4 - 3 c | 4- 196c|cj - I6C3C5 - 73c2C6) ef 4- Oiel)]. (19)

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Iierative Methods of Higher Order for Nonlinear Equauo Now, using (17) and (18)-(19) m (14), we get

ek+i = xk+i -x" =4(cl-3clc3 + 2c2c| -I-cja -C3C4)e\ -I- 0(e\).

Thus, the PM2 is of sixth order. Its efficiency index is also 64 = 1.5651. D

4 Numerical Examples

In this section, a number of numerical examples are worked out to demonstrate the efficacy of the proposed methods. All the numerical computations have been performed in MATLAB using double precision arithmetic. The roots are approximated correct up to 15 decimal places. The performance measures used are the number of iterations (NN) and the total number of function evaluations (NFE). The stopping criteria used are |J:,+I — X,\ < tol and

|/(xi)| < tol, where tol — 10"'^, x, and x^+i are the ith and (( -I- l)th iterates. The results obtained by PMl and PM2 given by (8) and (14) are compared with those obtained by Newton's mediod (NM), Chun's metiiod (CM), Kou's method (KM), Neta's method (MEM), and the method of Kou and Li (KLM) given by (2), (5), (9), (3) and (4), respectively. The following examples are considered,

: x^ -f 4JC^ - 10, X* = 1,365230013414097;

: sin^(x) - jc^ -h 1, JC* = 1.404491648215341:

: xe"-' - sm^(jr) -|- 3cos(;c) 4 - 5 , x' = -1.207647827130919;

: ^^^+'^-30 _ 1, x' = 3;

• x^ - e'^ ^ 3x + 2, x' = 0.257530285439861;

: ll;r" - I , X* = 0,804133097503664;

: g-'^+-<+2 - cos(jr -I- 1) + jr^ -H 1, x " = - 1;

: x^ - x'^ - I, x' = 1465571231876768;

: x'° - 1, ;r* - 1;

= X - 0.9995 sin(-c) - 0.01, x* = 0.389977774946362.

(a) (b) (c) (d) (e) (fl

(S)

(h) (i) 0)

Mx) f2ix) fl(x)

M')

fsi«) fiix) fix)

M')

Mx) foix)

Tablet Comparisonof number of itera lions (NN) Funclions xo NN

NM CM P M l KM P M l NEM KLM

Mx) /!(•) hix) Mx) Mx) Mx) Mx) Mx) Mx) /IO(J:)

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396

I^bleZ Companso Functions xo

n of number of function evaluations (NFE) NFE

NM CM P M l KM

S Singh. D. K Gupia

PM2 NEM KLM f,ix)

Mx) fiix) Mx) fsix) hix) fiix)

/<(«)

hix) fmix)

Table 1 shows the comparison of the number of iterations (NN). In Table 2. the number of function evaluations (NFE) are compared. Computational order of convergence p IS displayed in Table 3 which is defined as

^ ln(|.irt+i -Xk\l\xk -Xk-\\)

\R{\xk -Xk-\\l\xk-i -Xk-2\y

DIV and—denote die divergence of the method and no function evaluations ate counted.

One can easily see from Tables 1-3 that our methods (8) and (14) give improved results in terms of computational speed and efficiency. Moreover, our mediods eiflier behaves similarly or better on companson with existing sixth order methods for considered examples.

l^ble 3 Comparison of computadonal order of convergence ip)

ftix) hix) Mx) fsix) Mx) f6ix) Mx) M-') ftix) fMx)

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Iterative Methods of Higher Order for Nonlinear Equauons

5 C o n c l u s i o n s

Two n e w t h r e e - s t e p h i g h e r - o r d e r i t e r a u v e m e t h o d s b y m o d i f y i n g t w o t h h d - o r d e r t w o - s t e p m e t h o d s for s o l v i n g n o n l i n e a r e q u a t i o n s a r e p r o p o s e d . T h i s is d o n e b y i n t r o d u c i n g a t h i r d step in first o f t h e m a n d a p p r o x i m a t i n g its d e r i v a t i v e b y l i n e a r i n t e r p o l a t i o n m a k i n g it a sixdi-order m e t h o d . I n a s i m i l a r m a n n e r , a third s t e p i s a d d e d i n a s e c o n d t h i r d - o r d e r m e t h o d but u s d e r i v a t i v e is a p p r o x i m a t e d b y d i v i d e d d i f f e r e n c e s u p t o t h e s e c o n d o r d e r l e a d i n g it also t o a s i x t h - o r d e r m e t h o d . C o n v e r g e n c e a n a l y s i s o f b o t h t h e m e t h o d s is e s t a b l i s h e d . This e n h a n c e s t h e e f f i c i e n c y i n d i c e s o f t h e n e w m e t h o d s f r o m 1.442 t o 1.565. A n u m b e r of n u m e r i c a l e x a m p l e s a r e w o r k e d o u t t o d e m o n s t r a t e t h e e f f i c i e n c y o f t h e p r o p o s e d m e t h - ods T h e r e s u l t s o b t a i n e d a r e c o m p a r e d w i t h s o m e e x i s t i n g m e t h o d s It is o b s e r v e d t h a t t h e m e d i o d s give i m p r o v e d r e s u l t s i n t e r m s of c o m p u t a t i o n a l s p e e d a n d e f f i c i e n c y .

Acknowledgments The authors fhank the referees for their valuable comments which have improved the presentaiion of the paper. The authors thankfully acknowledge the financial assistance provided by the Council of Scientific and Industrial Research (CSIR), New Delhi. India,

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