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A One-step Steffensen-type Method with Super-cubic Convergence for Solving Nonlinear Equations1
Procedia Computer Science Volume 29, 2014, Pages 1870–1875 ICCS 2014. 14th International Conference on Computational Science
A one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations* 1
Zhongli Liu1† ,Quan Zheng2 College of Biochemical Engineering ,Beijing Union University, Beijing, China 2 College of Sciences, North China University of Technology, Beijing, China
Abstract In this paper, a one-step Steensen-type method of order 3.383 is designed and proved for solving nonlinear equations. This super-cubic convergence is obtained by self-accelerating second-order Steensen’s method twice with memory, but without any new function evaluations. The proposed method is very ecient and convenient, since it is still a derivative-free two-point method. Numerical examples confirm the theoretical results and high computational eciency. Keywords: Nonlinear equation, Newton’s method, Steensen’s method, Derivative free, Selfaccelerating
1 Introduction It is well-known in scientific computation that Newton’s method (NM, see [1]): xn +1 = xn −
f ( xn ) , n = 0,1,2, , f ′( xn )
(1)
is widely used for root-finding, where x0 is an initial guess of the root. However, when the derivative f ′ is unavailable or is expensive to be obtained, the derivative-free method is necessary. If the derivative f ′( xn ) is replaced by the divided difference f [ xn , xn + f ( xn )] = f ( xn + f ( xn )) − f ( xn ) in (1), Steffensen’s f ( xn )
method (SM, see [1]) is obtained as follows: f ( xn ) (2) xn +1 = xn − , n = 0,1,2, , f [ x, xn + f ( xn )] NM/SM converges quadratically and requires two function evaluations per iteration. The efficiency index of them is 2=1.414 .
* †
Supported by Beijing Natural Science Foundation (No.1122014) Corresponding author:E-mail:[email protected] (Z.-L. Liu)
1870
Selection and peer-review under responsibility of the Scientific Programme Committee of ICCS 2014 c The Authors. Published by Elsevier B.V.
A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Besides H.T.Kung and J.F.Traub conjectured that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1 (see[2]), Traub proposed a self-accelerating two-point method of order 2.414 with memory (see [3]): f ( xn ) ⎧ ⎪ xn +1 = xn − f [ x + β f ( x )] , ⎪ n n n ⎨ 1 ⎪β = − , ⎪⎩ n f [ xn , zn −1 ] = xn −1 + βn −1 f ( xn −1 ) , and β 0 = − sign(f ′( x0 )) or −1/ f [ x0 , x0 + f ( x0 )] , etc.
(3)
where zn −1 A lot of self-accelerating Steffensen-type methods were derived in the literature (see [1-7]). Steffensen-type methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in [1, 4, 5, 8]. Recently, by a new self-accelerating technique based on the second-order Newtonian interpolatory polynomial N 2 ( x) = f ( xn ) + f [ xn , zn −1 ]( x − xn ) + f [ xn , zn −1 , xn −1 ]( x − xn )( x − zn −1 ) , J. Dˇzuni´c and M.S. Petkovi´c proposed a cubically convergent Steffensenlike method (see [7]): f ( xn ) ⎧ ⎪ xn +1 = xn − f [ x , x + β f ( x )] , ⎪ n n n n ⎨ 1 ⎪β = − ), ⎪⎩ n f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ]
(4)
In this study, a one-step Steffensen-type method is proposed by doubly-self-accelerating in Section 2, its super-cubic convergence is proved in Section 3, numerical examples are demonstrated in Section 4.
2 The one-step Steffensen-type method By the first-order Newtonian interpolatory polynomial N1 ( x) = f ( xn ) + f [ xn , zn ]( x − xn ) at points xn and zn = xn + β n f ( xn ) , we have f ( x) = N1 ( x) + R1 ( x),
where R1 ( x) = f ( x) − N1 ( x) = f [ xn , zn , x]( x − xn )( x − zn ) . So, with some μn ≈ f [ xn , zn , x], N2 ( x) = f ( xn ) + f [ xn , zn ]( x − xn )+μn ( x − xn )( x − zn )
should be better than N1 ( x) to approximate f ( x) . Therefore, we suggest xn +1 = xn − N 2 ( xn ) , i.e., a twoN 2′ ( xn )
parameter Steffensen’s method: f ( xn ) (5) , n = 0,1,2, , f [ xn , zn ] + μ ( xn − zn ) and {μ n } are bounded constant sequences. The error equation of (5) is
xn +1 = xn −
where zn = xn + β n f ( xn ) , {β n } en +1 = [(1 + β n f ′(a))
f ′′( a ) 2 f ′( a )
− μn β n ]en2 + O(en3 ). By
defining μ0 = 0 and μn = 1 + β n f [ xn , zn ] f [ zn −1, xn , zn ](n > 0) β n f [ xn , z n ]
recursively as the iteration proceeds without any new evaluation to vanish the asymptotic convergence constant, we establish a self-accelerating Steffensen’s method with super quadratic convergence as follows: xn +1 = xn −
f ( xn ) , n = 0,1,2, , 1 f [ xn , zn ] + (1 + )( f [ zn −1, xn ] − f [ zn −1, zn ]) βn f [ xn , zn ]
(6)
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A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-self-accelerating as follows:
f ( xn ) ⎧ ⎪ xn +1 = xn − 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1 , zn ]) ⎪⎪ β n f [ xn , zn ] ⎨ ⎪ 1 ⎪βn = − f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ] ⎪⎩
(7)
3 Its super-cubic convergence 1 f (1 + c3en −1enz −1 ), where ck = f ′(a) Proof. By Taylor formula, we have f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ]
Lemma 3.1 βn ~ −
f
f ( xn ) − f ( xn −1 ) f ( xn ) − f ( zn −1 ) f ( zn −1 ) − f ( xn −1 ) + − xn − xn −1 xn − zn −1 zn −1 − xn −1
=
f ( xn ) − f ( xn −1 ) f ( xn ) − f ( zn −1 ) f ( zn −1 ) − f ( xn −1 ) + − en − en −1 en − enz −1 enz −1 − en −1
−
(a)
k ! f ′( a )
=
= f ′(a)[
(k )
, en = xn − a and enz = zn − a.
en − en −1 + c2 (en2 − en2−1 ) + c3 (en3 − en3−1 ) + en − enz −1 + c2 (en2 − (enz −1 )2 )+c3 (en3 − (enz −1 )3 ) + + en − en −1 en − enz −1
enz −1 − en −1 + c2 ((enz −1 )2 − en2−1 ) + c3 ((enz −1 )3 − en3−1 ) + ] ~ f ′(a)(1 − c3en −1enz−1 ). enz −1 − en −1
Then, the proof can be completed. Theorem 3.2 Let f : D → R be a suciently dierentiable function with a simple root a ∈ D , D ⊂ R be an open set, x0 be close enough to a , then (7) achieve the convergence of order 3.383. Proof. If zn converges to a with order p > 1 as: enz = Cnenp + o(enp ), and if xn converges to a with order r > 2 as: en +1 = Dnenr + o(enr ),
Then enz = Cn ( Dn −1enr −1 ) p + o(enrp−1 ) = Cn Dnp−1enrp−1 + o(enrp−1 ), 2
2
2
en +1 = Dn (Dn −1enr −1 )r + o(enr −1 ) = Dn Dnr−1enr −1 + o(enr −1 ). By Taylor formula and Lemma 3.1, we also have enz = (1 + βn f [ xn , a])en = −c3en −1Cn −1enp−1Dn −1enr −1 + o(enr −+1p +1 ) = −c3Cn −1Dn −1enr −+1p +1 + o(enr −+1p +1 ).
and en +1 = en −
1872
f [ xn , a]en 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
A one-step Steffensen-type method with super-cubic convergence ...
f [ xn , zn ] + (1 + = en
1
β n f [ xn , zn ]
f [ xn , zn ] + (1 +
Zhongli Liu
)( f [ zn −1, xn ] − f [ zn −1, zn ]) − f [ xn , a]
1
β n f [ xn , zn ]
)( f [ zn −1, xn ] − f [ zn −1, zn ])
1 ) f [ zn −1, xn , zn ]( − β n f [ xn , a]) β n f [ xn , zn ] 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
f [ xn , zn , a]enz + (1 + = en
f [ xn , zn , a](1 + β n f [ xn , a])en − (1 + β n f [ xn , zn ]) = en
1
f [ xn , zn ] + (1 +
f [ xn , a] f [ zn −1, xn , zn ]en f [ xn , zn ]
)( f [ z , x ] − f [ z
, z ])
n −1 n n −1 n β n f [ xn , zn ] f [ xn , zn , a] f [ xn , zn ] − f [ xn , a] f [ zn −1, xn , zn ] = en2 (1 + β n f [ xn , a]) 1 2 f [ xn , zn ] + (1 + ) f [ xn , zn ]( f [ zn −1, xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
f 2 [ xn , zn , a]enz − f [ xn , a] f [ zn −1, xn , zn , a]enz −1 1 f 2 [ xn , zn ] + (1 + ) f [ xn , zn ]( f [ zn −1, xn ] − f [ zn −1, zn ]) β n f [ xn , zn ] f ′′′(a) z en −1 + − f ′(a) z 2 3! = en (−c3en −1en −1 + ) f ′2 ( a ) + = en2 (1 + β n f [ xn , a])
= c32Cn2−1Dn2−1en2−r1+ 2 p +1 + o(en2−r 1+ 2 p +1 ) .
So, comparing the exponents of en −1 in expressions of enz and en +1 for (7), we obtain the same system of two equations: ⎧ rp = r + p + 1, ⎨ 2 ⎩r = 2r + 2 p + 1.
From its non-trivial solution r = 4/(3 3
44 + 2) + 3 27
44 + 2 +1 ≈ 3.383 and p ≈ 1.839 , we prove that 27
the convergence of (7) is of order 3.383. Without any additional function evaluations, the eciency indices of (3), (4) and (7) are 1 + 2 = 1.554, 3 = 1.732 and
3.383 = 1.839, respectively.
4 Numerical examples Related one-step methods only using two function evaluations per iteration are showed in the following numerical examples. The proposed method is a derivative-free two-point method with high computational eciency. Example 1. The numerical results of NM, SM, (3), (4) and (7) in Table 1 agree with the theoretical analysis. The computational order of convergence is defined by COC =
log( en / en −1 ) log( en −1 / en − 2 )
.
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A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Table 1. f ( x) = x2 − e− x − 3x + 1, a = 0, x0 = 0.2 Methods NM SM
n
4
5
.35561e-5
.15808e-11
.31235e-24
.12195e-49
COC
COC
2.25256 .28174e-1 1.21776 .28174e-1 1.21776
2.01691 .51325e-3 2.04376 .15996e-4 3.81335
2.00030 .16476e-6 2.00830 .13132e-12 2.49109
2.00000 .16966e-13 2.00009 .43283e-32 2.40945
xn − a
.28174e-1
.16560e-6
.11521e-21
.39821e-67
1.21776
6.14536
2.89776
2.99925
.28174e-1
.43010e-7
.21604e-27
.23153e-94
1.21776
6.83322
3.49004
3.29917
2.00000 .17989e-27 2.00000 .38442e-79 2.41512 .16444e203 3.00000 .20021e321 3.39052
xn − a
COC
(7)
3
.53279e-2
COC
(4)
2
xn − a xn − a
(3)
1
xn − a COC
6 .15890e100 2.00000 .20226e-55 2.00000 .99936-193 2.41406 .11580e612 3.00000 .69689e1090 3.38434
Example 2. The numerical results of NM, SM, (3), (4) and (7) are in Table 2 for the following nonlinear functions: f1 ( x) = 0.5(e x − 2 − 1), a = 2, x0 = 2.5, 2
f 2 ( x) = ex + sin x − 1, a = 0, x0 = 0.25, f3 ( x) = e− x
2
+ x+2
− 1, a = −1, x0 = −0.85,
−x
f 4 ( x) = e − arctan x − 1, a = 0, x0 = −0.
Table 2. Numerical results for solving f i ( x), i = 1, 2,3, 4. Methods f1 : e6 COC
f 2 : e6 COC
f 3 : e6 COC
f 3 : e6 COC
NM .19785e-40 2.0000 .32328e-44 2.0000 .18813e-51 2.0000 .35988e-79 2.0000
SM .88156e-29 2.0000 .42920e-26 2.0000 .15758e-18 2.0000 .96290e-84 2.0000
(3) .50439e-84 2.4141 .19843e-85 2.4141 .12013e-86 2.4140 .16834e-248 2.4161
(4) .19314e-313 3.0000 .57587e-282 3.0000 .34524e-286 3.0000 .21536e-597 3.0000
(7) .75162e-578 3.3831 .13494e-706 3.3825 .27679e-677 3.3796 .25291e-1154 3.3831
Example 3. Consider solving the following nonlinear ODE by finite dierence method: ⎧⎪ x′′(t ) + x3 / 2 (t )=0, t ∈ (0,1), ⎨ =0. ⎪⎩ x(0) = x(1
Taking nodes ti = ih, where h =
1 and N = 10, we have a system of nine nonlinear equations: N
⎧ 2 x1 − h2 x13 / 2 − x2 = 0, ⎪ 2 3/ 2 ⎨− xi −1 + 2 xi − h xi − xi +1 = 0, i = 2,3, 8, ⎪ − x + 2 x − h2 x3 / 2 = 0. 9 9 ⎩ 8
For an example, SM is carried out as follows:
⎧ xn +1 = xn − J ( xn , H n )−1 F ( xn ), ⎪ N 1 −1 ⎨ J ( xn , H n ) = ( F ( xn + H ne ) − F ( xn ), , F ( xn + H ne ) − F ( xn )) H n , ⎪ H = diag(f (x ), f (x ), f (x )). 1 n 2 n N n ⎩ n
1874
A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
And other methods are carried out by using similar approximations of the divided dierences. The numerical results are in Table 3, where x0 = (40,80,100,120,140,130,100,80, 40)′, x* = (33.57391205, 65.2024509, 91.5660200, 109.1676243, 115.3630336, 109.1676243, 91.5660200, 65.2024509, 33.57391205)′ . Table 3 The finite dierence method for solving x′′ + x3/ 2 = 0, x(0) = x(1 =0 Methods NM
n xn − x∗ f ( xn )
SM
f ( xn ) f ( xn )
xn − x
.66590e-14
.26493e-29
.41936e-60
2
.37036e-12
.38446e-27
.41429e-57
.37077
.31892e-2
.56743e-6
.18275e-13
.18970e-28
.20442e-58
2
4.8552
.11027e-1
.58355e-8
.33191e-23
.55918e-60
.90270e-149
.37077
.54534e-3
.28807e-9
.16384e-24
.27602e-61
.16919e-150
4.8552
.11260e-2
.15165e-13
.37078e-46
.54192e-144
.16919e-437
.37077
.55632e-4
.74858e-15
.18302e-47
.26750e-145
.83514e-439
4.8552
.35305e-4
.21872e-18
.51682e-61
.3380e-190
.61416e-576
.37077
.75225e-5
.35743e-19
.45250e-62
.10709e-190
.71775e-577
∗ 2 2
xn − x∗ f ( xn )
.33390e-6 .11495e-4
2
f ( xn )
(7)
.23685e-2 .64055e-1
2
xn − x∗
(4)
.24453 4.8552
2
xn − x∗
(3)
2
1 .40882e-1
2 2
2 .47895e-1
3 .67632e-5
4 .13490e-12
5 .53672e-28
6 .84957e-59
5 Conclusion The proposed method is a derivative-free two-point method with high computational effciency. Its convergence order is 3.383 and its effciency index is 3.383 = 1.839 . By numerical experiments, we can see that the suggested method is suitable to solving nonlinear equations and can be used for solving boundary-value problems of nonlinear ODEs as well. The future work can be to combine the current method with the multiple shooting method for solving BVPs of nonlinear ODEs, since the proposed method is a derivative-free.
References [1]
J.M. Ortega, W.G. Rheinboldt,( 1970) Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.
[2]
H.T. Kung, J.F. Traub, (1974) Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21, 634-651.
[3]
J.F. Traub, (1964)Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs.
[4]
Q. Zheng, J. Wang, P. Zhao, L. Zhang, (2009)A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214,10-16.
[5]
Q. Zheng, P. Zhao, L. Zhang, W. Ma, (2010)Variants of Steffensen-secant method and applications, Appl. Math. Comput. 216 ,3486-3496.
[6]
M.S. Petkovi´c, S. Ili´c, J. Dˇzuni´c, (2010) Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 217,1887-1895.
[7]
J. Dˇzuni´c, M.S. Petkovi´c, (2012)A cubically convergent Steffensen-like method for solving nonlinear equations, Appl. Math. Let. 1881-1886.
[8]
Alarc´on, S. Amat, S. Busquier, D. J. L´opez, (2008)A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216, 243-250.
[9]
Z.-L. Liu, Q. Zheng, P. Zhao, (2010)A variant of Steffensen's method of fourth-order convergence and its applications, Applied Mathematics and Computation, 216,1978-1983.
1875
A one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations* 1
Zhongli Liu1† ,Quan Zheng2 College of Biochemical Engineering ,Beijing Union University, Beijing, China 2 College of Sciences, North China University of Technology, Beijing, China
Abstract In this paper, a one-step Steensen-type method of order 3.383 is designed and proved for solving nonlinear equations. This super-cubic convergence is obtained by self-accelerating second-order Steensen’s method twice with memory, but without any new function evaluations. The proposed method is very ecient and convenient, since it is still a derivative-free two-point method. Numerical examples confirm the theoretical results and high computational eciency. Keywords: Nonlinear equation, Newton’s method, Steensen’s method, Derivative free, Selfaccelerating
1 Introduction It is well-known in scientific computation that Newton’s method (NM, see [1]): xn +1 = xn −
f ( xn ) , n = 0,1,2, , f ′( xn )
(1)
is widely used for root-finding, where x0 is an initial guess of the root. However, when the derivative f ′ is unavailable or is expensive to be obtained, the derivative-free method is necessary. If the derivative f ′( xn ) is replaced by the divided difference f [ xn , xn + f ( xn )] = f ( xn + f ( xn )) − f ( xn ) in (1), Steffensen’s f ( xn )
method (SM, see [1]) is obtained as follows: f ( xn ) (2) xn +1 = xn − , n = 0,1,2, , f [ x, xn + f ( xn )] NM/SM converges quadratically and requires two function evaluations per iteration. The efficiency index of them is 2=1.414 .
* †
Supported by Beijing Natural Science Foundation (No.1122014) Corresponding author:E-mail:[email protected] (Z.-L. Liu)
1870
Selection and peer-review under responsibility of the Scientific Programme Committee of ICCS 2014 c The Authors. Published by Elsevier B.V.
A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Besides H.T.Kung and J.F.Traub conjectured that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1 (see[2]), Traub proposed a self-accelerating two-point method of order 2.414 with memory (see [3]): f ( xn ) ⎧ ⎪ xn +1 = xn − f [ x + β f ( x )] , ⎪ n n n ⎨ 1 ⎪β = − , ⎪⎩ n f [ xn , zn −1 ] = xn −1 + βn −1 f ( xn −1 ) , and β 0 = − sign(f ′( x0 )) or −1/ f [ x0 , x0 + f ( x0 )] , etc.
(3)
where zn −1 A lot of self-accelerating Steffensen-type methods were derived in the literature (see [1-7]). Steffensen-type methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in [1, 4, 5, 8]. Recently, by a new self-accelerating technique based on the second-order Newtonian interpolatory polynomial N 2 ( x) = f ( xn ) + f [ xn , zn −1 ]( x − xn ) + f [ xn , zn −1 , xn −1 ]( x − xn )( x − zn −1 ) , J. Dˇzuni´c and M.S. Petkovi´c proposed a cubically convergent Steffensenlike method (see [7]): f ( xn ) ⎧ ⎪ xn +1 = xn − f [ x , x + β f ( x )] , ⎪ n n n n ⎨ 1 ⎪β = − ), ⎪⎩ n f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ]
(4)
In this study, a one-step Steffensen-type method is proposed by doubly-self-accelerating in Section 2, its super-cubic convergence is proved in Section 3, numerical examples are demonstrated in Section 4.
2 The one-step Steffensen-type method By the first-order Newtonian interpolatory polynomial N1 ( x) = f ( xn ) + f [ xn , zn ]( x − xn ) at points xn and zn = xn + β n f ( xn ) , we have f ( x) = N1 ( x) + R1 ( x),
where R1 ( x) = f ( x) − N1 ( x) = f [ xn , zn , x]( x − xn )( x − zn ) . So, with some μn ≈ f [ xn , zn , x], N2 ( x) = f ( xn ) + f [ xn , zn ]( x − xn )+μn ( x − xn )( x − zn )
should be better than N1 ( x) to approximate f ( x) . Therefore, we suggest xn +1 = xn − N 2 ( xn ) , i.e., a twoN 2′ ( xn )
parameter Steffensen’s method: f ( xn ) (5) , n = 0,1,2, , f [ xn , zn ] + μ ( xn − zn ) and {μ n } are bounded constant sequences. The error equation of (5) is
xn +1 = xn −
where zn = xn + β n f ( xn ) , {β n } en +1 = [(1 + β n f ′(a))
f ′′( a ) 2 f ′( a )
− μn β n ]en2 + O(en3 ). By
defining μ0 = 0 and μn = 1 + β n f [ xn , zn ] f [ zn −1, xn , zn ](n > 0) β n f [ xn , z n ]
recursively as the iteration proceeds without any new evaluation to vanish the asymptotic convergence constant, we establish a self-accelerating Steffensen’s method with super quadratic convergence as follows: xn +1 = xn −
f ( xn ) , n = 0,1,2, , 1 f [ xn , zn ] + (1 + )( f [ zn −1, xn ] − f [ zn −1, zn ]) βn f [ xn , zn ]
(6)
1871
A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-self-accelerating as follows:
f ( xn ) ⎧ ⎪ xn +1 = xn − 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1 , zn ]) ⎪⎪ β n f [ xn , zn ] ⎨ ⎪ 1 ⎪βn = − f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ] ⎪⎩
(7)
3 Its super-cubic convergence 1 f (1 + c3en −1enz −1 ), where ck = f ′(a) Proof. By Taylor formula, we have f [ xn , zn −1 ] + f [ xn , xn −1 ] − f [ xn −1 , zn −1 ]
Lemma 3.1 βn ~ −
f
f ( xn ) − f ( xn −1 ) f ( xn ) − f ( zn −1 ) f ( zn −1 ) − f ( xn −1 ) + − xn − xn −1 xn − zn −1 zn −1 − xn −1
=
f ( xn ) − f ( xn −1 ) f ( xn ) − f ( zn −1 ) f ( zn −1 ) − f ( xn −1 ) + − en − en −1 en − enz −1 enz −1 − en −1
−
(a)
k ! f ′( a )
=
= f ′(a)[
(k )
, en = xn − a and enz = zn − a.
en − en −1 + c2 (en2 − en2−1 ) + c3 (en3 − en3−1 ) + en − enz −1 + c2 (en2 − (enz −1 )2 )+c3 (en3 − (enz −1 )3 ) + + en − en −1 en − enz −1
enz −1 − en −1 + c2 ((enz −1 )2 − en2−1 ) + c3 ((enz −1 )3 − en3−1 ) + ] ~ f ′(a)(1 − c3en −1enz−1 ). enz −1 − en −1
Then, the proof can be completed. Theorem 3.2 Let f : D → R be a suciently dierentiable function with a simple root a ∈ D , D ⊂ R be an open set, x0 be close enough to a , then (7) achieve the convergence of order 3.383. Proof. If zn converges to a with order p > 1 as: enz = Cnenp + o(enp ), and if xn converges to a with order r > 2 as: en +1 = Dnenr + o(enr ),
Then enz = Cn ( Dn −1enr −1 ) p + o(enrp−1 ) = Cn Dnp−1enrp−1 + o(enrp−1 ), 2
2
2
en +1 = Dn (Dn −1enr −1 )r + o(enr −1 ) = Dn Dnr−1enr −1 + o(enr −1 ). By Taylor formula and Lemma 3.1, we also have enz = (1 + βn f [ xn , a])en = −c3en −1Cn −1enp−1Dn −1enr −1 + o(enr −+1p +1 ) = −c3Cn −1Dn −1enr −+1p +1 + o(enr −+1p +1 ).
and en +1 = en −
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f [ xn , a]en 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
A one-step Steffensen-type method with super-cubic convergence ...
f [ xn , zn ] + (1 + = en
1
β n f [ xn , zn ]
f [ xn , zn ] + (1 +
Zhongli Liu
)( f [ zn −1, xn ] − f [ zn −1, zn ]) − f [ xn , a]
1
β n f [ xn , zn ]
)( f [ zn −1, xn ] − f [ zn −1, zn ])
1 ) f [ zn −1, xn , zn ]( − β n f [ xn , a]) β n f [ xn , zn ] 1 f [ xn , zn ] + (1 + )( f [ zn −1 , xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
f [ xn , zn , a]enz + (1 + = en
f [ xn , zn , a](1 + β n f [ xn , a])en − (1 + β n f [ xn , zn ]) = en
1
f [ xn , zn ] + (1 +
f [ xn , a] f [ zn −1, xn , zn ]en f [ xn , zn ]
)( f [ z , x ] − f [ z
, z ])
n −1 n n −1 n β n f [ xn , zn ] f [ xn , zn , a] f [ xn , zn ] − f [ xn , a] f [ zn −1, xn , zn ] = en2 (1 + β n f [ xn , a]) 1 2 f [ xn , zn ] + (1 + ) f [ xn , zn ]( f [ zn −1, xn ] − f [ zn −1, zn ]) β n f [ xn , zn ]
f 2 [ xn , zn , a]enz − f [ xn , a] f [ zn −1, xn , zn , a]enz −1 1 f 2 [ xn , zn ] + (1 + ) f [ xn , zn ]( f [ zn −1, xn ] − f [ zn −1, zn ]) β n f [ xn , zn ] f ′′′(a) z en −1 + − f ′(a) z 2 3! = en (−c3en −1en −1 + ) f ′2 ( a ) + = en2 (1 + β n f [ xn , a])
= c32Cn2−1Dn2−1en2−r1+ 2 p +1 + o(en2−r 1+ 2 p +1 ) .
So, comparing the exponents of en −1 in expressions of enz and en +1 for (7), we obtain the same system of two equations: ⎧ rp = r + p + 1, ⎨ 2 ⎩r = 2r + 2 p + 1.
From its non-trivial solution r = 4/(3 3
44 + 2) + 3 27
44 + 2 +1 ≈ 3.383 and p ≈ 1.839 , we prove that 27
the convergence of (7) is of order 3.383. Without any additional function evaluations, the eciency indices of (3), (4) and (7) are 1 + 2 = 1.554, 3 = 1.732 and
3.383 = 1.839, respectively.
4 Numerical examples Related one-step methods only using two function evaluations per iteration are showed in the following numerical examples. The proposed method is a derivative-free two-point method with high computational eciency. Example 1. The numerical results of NM, SM, (3), (4) and (7) in Table 1 agree with the theoretical analysis. The computational order of convergence is defined by COC =
log( en / en −1 ) log( en −1 / en − 2 )
.
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A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
Table 1. f ( x) = x2 − e− x − 3x + 1, a = 0, x0 = 0.2 Methods NM SM
n
4
5
.35561e-5
.15808e-11
.31235e-24
.12195e-49
COC
COC
2.25256 .28174e-1 1.21776 .28174e-1 1.21776
2.01691 .51325e-3 2.04376 .15996e-4 3.81335
2.00030 .16476e-6 2.00830 .13132e-12 2.49109
2.00000 .16966e-13 2.00009 .43283e-32 2.40945
xn − a
.28174e-1
.16560e-6
.11521e-21
.39821e-67
1.21776
6.14536
2.89776
2.99925
.28174e-1
.43010e-7
.21604e-27
.23153e-94
1.21776
6.83322
3.49004
3.29917
2.00000 .17989e-27 2.00000 .38442e-79 2.41512 .16444e203 3.00000 .20021e321 3.39052
xn − a
COC
(7)
3
.53279e-2
COC
(4)
2
xn − a xn − a
(3)
1
xn − a COC
6 .15890e100 2.00000 .20226e-55 2.00000 .99936-193 2.41406 .11580e612 3.00000 .69689e1090 3.38434
Example 2. The numerical results of NM, SM, (3), (4) and (7) are in Table 2 for the following nonlinear functions: f1 ( x) = 0.5(e x − 2 − 1), a = 2, x0 = 2.5, 2
f 2 ( x) = ex + sin x − 1, a = 0, x0 = 0.25, f3 ( x) = e− x
2
+ x+2
− 1, a = −1, x0 = −0.85,
−x
f 4 ( x) = e − arctan x − 1, a = 0, x0 = −0.
Table 2. Numerical results for solving f i ( x), i = 1, 2,3, 4. Methods f1 : e6 COC
f 2 : e6 COC
f 3 : e6 COC
f 3 : e6 COC
NM .19785e-40 2.0000 .32328e-44 2.0000 .18813e-51 2.0000 .35988e-79 2.0000
SM .88156e-29 2.0000 .42920e-26 2.0000 .15758e-18 2.0000 .96290e-84 2.0000
(3) .50439e-84 2.4141 .19843e-85 2.4141 .12013e-86 2.4140 .16834e-248 2.4161
(4) .19314e-313 3.0000 .57587e-282 3.0000 .34524e-286 3.0000 .21536e-597 3.0000
(7) .75162e-578 3.3831 .13494e-706 3.3825 .27679e-677 3.3796 .25291e-1154 3.3831
Example 3. Consider solving the following nonlinear ODE by finite dierence method: ⎧⎪ x′′(t ) + x3 / 2 (t )=0, t ∈ (0,1), ⎨ =0. ⎪⎩ x(0) = x(1
Taking nodes ti = ih, where h =
1 and N = 10, we have a system of nine nonlinear equations: N
⎧ 2 x1 − h2 x13 / 2 − x2 = 0, ⎪ 2 3/ 2 ⎨− xi −1 + 2 xi − h xi − xi +1 = 0, i = 2,3, 8, ⎪ − x + 2 x − h2 x3 / 2 = 0. 9 9 ⎩ 8
For an example, SM is carried out as follows:
⎧ xn +1 = xn − J ( xn , H n )−1 F ( xn ), ⎪ N 1 −1 ⎨ J ( xn , H n ) = ( F ( xn + H ne ) − F ( xn ), , F ( xn + H ne ) − F ( xn )) H n , ⎪ H = diag(f (x ), f (x ), f (x )). 1 n 2 n N n ⎩ n
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A one-step Steffensen-type method with super-cubic convergence ...
Zhongli Liu
And other methods are carried out by using similar approximations of the divided dierences. The numerical results are in Table 3, where x0 = (40,80,100,120,140,130,100,80, 40)′, x* = (33.57391205, 65.2024509, 91.5660200, 109.1676243, 115.3630336, 109.1676243, 91.5660200, 65.2024509, 33.57391205)′ . Table 3 The finite dierence method for solving x′′ + x3/ 2 = 0, x(0) = x(1 =0 Methods NM
n xn − x∗ f ( xn )
SM
f ( xn ) f ( xn )
xn − x
.66590e-14
.26493e-29
.41936e-60
2
.37036e-12
.38446e-27
.41429e-57
.37077
.31892e-2
.56743e-6
.18275e-13
.18970e-28
.20442e-58
2
4.8552
.11027e-1
.58355e-8
.33191e-23
.55918e-60
.90270e-149
.37077
.54534e-3
.28807e-9
.16384e-24
.27602e-61
.16919e-150
4.8552
.11260e-2
.15165e-13
.37078e-46
.54192e-144
.16919e-437
.37077
.55632e-4
.74858e-15
.18302e-47
.26750e-145
.83514e-439
4.8552
.35305e-4
.21872e-18
.51682e-61
.3380e-190
.61416e-576
.37077
.75225e-5
.35743e-19
.45250e-62
.10709e-190
.71775e-577
∗ 2 2
xn − x∗ f ( xn )
.33390e-6 .11495e-4
2
f ( xn )
(7)
.23685e-2 .64055e-1
2
xn − x∗
(4)
.24453 4.8552
2
xn − x∗
(3)
2
1 .40882e-1
2 2
2 .47895e-1
3 .67632e-5
4 .13490e-12
5 .53672e-28
6 .84957e-59
5 Conclusion The proposed method is a derivative-free two-point method with high computational effciency. Its convergence order is 3.383 and its effciency index is 3.383 = 1.839 . By numerical experiments, we can see that the suggested method is suitable to solving nonlinear equations and can be used for solving boundary-value problems of nonlinear ODEs as well. The future work can be to combine the current method with the multiple shooting method for solving BVPs of nonlinear ODEs, since the proposed method is a derivative-free.
References [1]
J.M. Ortega, W.G. Rheinboldt,( 1970) Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.
[2]
H.T. Kung, J.F. Traub, (1974) Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21, 634-651.
[3]
J.F. Traub, (1964)Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs.
[4]
Q. Zheng, J. Wang, P. Zhao, L. Zhang, (2009)A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214,10-16.
[5]
Q. Zheng, P. Zhao, L. Zhang, W. Ma, (2010)Variants of Steffensen-secant method and applications, Appl. Math. Comput. 216 ,3486-3496.
[6]
M.S. Petkovi´c, S. Ili´c, J. Dˇzuni´c, (2010) Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 217,1887-1895.
[7]
J. Dˇzuni´c, M.S. Petkovi´c, (2012)A cubically convergent Steffensen-like method for solving nonlinear equations, Appl. Math. Let. 1881-1886.
[8]
Alarc´on, S. Amat, S. Busquier, D. J. L´opez, (2008)A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216, 243-250.
[9]
Z.-L. Liu, Q. Zheng, P. Zhao, (2010)A variant of Steffensen's method of fourth-order convergence and its applications, Applied Mathematics and Computation, 216,1978-1983.
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