A one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations*

Zhongli Liu1† ,Quan Zheng2 1College 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 Ste ensen-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 Ste ensen’s method twice with memory, but without any new function evaluations. The proposed method is very e cient and convenient, since it is still a derivative-free two-point method. Numerical examples confirm the theoretical results and high computational e ciency. Keywords: Nonlinear equation, Newton’s method, Ste ensen’s method, Derivative free, Self-accelerating

1 Introduction It is well-known in scientific computation that Newton’s method (NM, see [1]):

1

( ) , 0,1,2, ,( )n

n nn

f xx x nf x+ = − =′

(1)

is widely used for root-finding, where 0x 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( )nf x′ is replaced by the divided difference ( ( )) ( )[ , ( )]

( )n n n

n n nn

f x f x f xf x x f xf x

+ −+ = in (1), Steffensen’s

method (SM, see [1]) is obtained as follows:

1( ) , 0,1,2, ,

[ , ( )]n

n nn n

f xx x nf x x f x+ = − =

+ (2)

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:liuzhongli2@163.com (Z.-L. Liu)

Procedia Computer Science

Volume 29, 2014, Pages 1870–1875

ICCS 2014. 14th International Conference on Computational Science

1870 Selection and peer-review under responsibility of the Scientific Programme Committee of ICCS 2014

doi: 10.1016/j.procs.2014.05.171 c© The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license.

Besides H.T.Kung and J.F.Traub conjectured that an iterative method based onm evaluations per iteration without memory would arrive at the optimal convergence of order 12m− (see[2]), Traub proposed a self-accelerating two-point method of order 2.414 with memory (see [3]):

1

1

( ) ,[ ( )]1 ,

[ , ]

nn n

n n n

nn n

f xx xf x f x

f x z

β

β

+

−

⎧ = −⎪ +⎪⎨⎪ = −⎪⎩

(3)

where 1 1 1 1( )n n n nz x f xβ− − − −= + , and 0 0sign( ( ))f xβ ′= − or 0 0 01/ [ , ( )]f x x f x− + , etc. 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

2 1( ) ( ) [ , ]( )n n n nN x f x f x z x x−= + − +

1 1 1[ , , ]( )( )n n n n nf x z x x x x z− − −− − , J. Dˇzuni´c and M.S. Petkovi´c proposed a cubically convergent Steffensen-like method (see [7]):

1

1 1 1 1

( ) ,[ , ( )]

1 ),[ , ] [ , ] [ , ]

nn n

n n n n

nn n n n n n

f xx xf x x f x

f x z f x x f x z

β

β

+

− − − −

⎧ = −⎪ +⎪⎨⎪ = −⎪ + −⎩

(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 1( ) ( ) [ , ]( )n n n nN x f x f x z x x= + − at points nx and

( )n n n nz x f xβ= + , we have 1 1( ) ( ) ( ),f x N x R x= +

where 1 1( ) ( ) ( ) [ , , ]( )( )n n n nR x f x N x f x z x x x x z= − = − − . So, with some [ , , ],n n nf x z xμ ≈ 2 ( ) ( ) [ , ]( )+ ( )( )n n n n n n nN x f x f x z x x x x x zμ= + − − −

should be better than 1( )N x to approximate ( )f x . Therefore, we suggest1

( )2

( )2,n n

N xn

N xnx x+

′= − i.e., a two-

parameter Steffensen’s method:

1( ) , 0,1,2, ,

[ , ] ( )n

n nn n n n

f xx x nf x z x zμ+ = − =

+ − (5)

where ( )n n n nz x f xβ= + , { }nβ and { }nμ are bounded constant sequences. The error equation of (5) is

2 31

( )

2 ( )[(1 ( )) ] ( ).n n n n n n

f a

f ae f a e O eβ μ β+

′′

′′= + − + By defining 0 0μ = and

11 [ , ]

[ , ][ , , ]( 0)n n n n

f x zn n n

f x zn n nf z x z nβ

βμ −

+= >

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:

1

1 1

( ) , 0,1,2, ,1[ , ] (1 )( [ , ] [ , ])[ , ]

nn n

n n n n n nn n n

f xx x nf x z f z x f z z

f x zβ

+

− −

= − =+ + −

(6)

A one-step Steffensen-type method with super-cubic convergence ... Zhongli Liu

1871

Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-self-accelerating as follows:

1

1 1

1 1 1 1

( )1[ , ] (1 )( [ , ] [ , ])[ , ]

1[ , ] [ , ] [ , ]

nn n

n n n n n nn n n

nn n n n n n

f xx xf x z f z x f z z

f x z

f x z f x x f x z

β

β

+

− −

− − − −

⎧ = −⎪+ + −⎪⎪

⎨⎪

= −⎪+ −⎪⎩

(7)

3 Its super-cubic convergence

Lemma 3.1 3 1 11~ (1 ),( )

zn n nf c e e

f aβ − −− +

′where

( )( )

! ( ),

k

kf a

k f ac

′= n ne x a= − and .z

n ne z a= −

Proof. By Taylor formula, we have 1 1 1 1[ , ] [ , ] [ , ]n n n n n nf x z f x x f x z− − − −+ −

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

n n n n n n

n n n n n n

n n n n n nz z

n n n n n n

f x f x f x f z f z f xx x x z z x

f x f x f x f z f z f xe e e e e e

− − − −

− − − −

− − − −

− − − −

− − −= + −

− − −

− − −= + −

− − −

2 2 3 3 2 2 3 3

1 2 1 3 1 1 2 1 3 1

1 1

( ) ( ) ( ( ) )+ ( ( ) )( )[z z z

n n n n n n n n n n n nz

n n n n

e e c e e c e e e e c e e c e ef ae e e e

− − − − − −

− −

− + − + − + − + − − +′= +

− −

2 2 3 3

1 1 2 1 1 3 1 1

1 1

(( ) ) (( ) ) ]z z zn n n n n n

zn n

e e c e e c e ee e

− − − − − −

− −

− + − + − +−

− 3 1 1~ ( )(1 ).zn nf a c e e− −′ −

Then, the proof can be completed. Theorem 3.2 Let :f D R→ be a su ciently di erentiable function with a simple root a D∈ ,

D R⊂ be an open set, 0x be close enough to a , then (7) achieve the convergence of order 3.383. Proof. If nz converges to a with order 1p > as: = ( ),z p p

n n n ne C e o e+ and if nx converges to a with order 2r > as:

1= ( ),r rn n n ne D e o e+ +

Then 1 1 1 1 1 1= ( ) ( ) ( ),z r p rp p rp rp

n n n n n n n n ne C D e o e C D e o e− − − − − −+ = +

2 2 2

1 1 1 1 1 1 1= ( ) ( ) ( ).r r r r r rn n n n n n n n ne D D e o e D D e o e+ − − − − − −+ = +

By Taylor formula and Lemma 3.1, we also have = (1 [ , ])z

n n n ne f x a eβ+

13 1 1 1 1 1 1( )p r r pn n n n n nc e C e D e o e + +− − − − − −= − +

1 13 1 1 1 1( ).r p r pn n n nc C D e o e+ + + +− − − −= − +

and

1

1 1

[ , ]1[ , ] (1 )( [ , ] [ , ])[ , ]

n nn n

n n n n n nn n n

f x a ee ef x z f z x f z z

f x zβ

+

− −

= −+ + −

A one-step Steffensen-type method with super-cubic convergence ... Zhongli Liu

1872

1 1

1 1

1[ , ] (1 )( [ , ] [ , ]) [ , ][ , ]

1[ , ] (1 )( [ , ] [ , ])[ , ]

n n n n n n nn n n

n

n n n n n nn n n

f x z f z x f z z f x af x ze

f x z f z x f z zf x z

β

β

− −

− −

+ + − −

=+ + −

1

1 1

1[ , , ] (1 ) [ , , ]( [ , ])[ , ]1[ , ] (1 )( [ , ] [ , ])[ , ]

zn n n n n n n n

n n nn

n n n n n nn n n

f x z a e f z x z f x af x ze

f x z f z x f z zf x z

ββ

β

−

− −

+ + −

=+ + −

1

1 1

[ , ][ , , ](1 [ , ]) (1 [ , ]) [ , , ][ , ]

1[ , ] (1 )( [ , ] [ , ])[ , ]

nn n n n n n n n n n n n

n nn

n n n n n nn n n

f x af x z a f x a e f x z f z x z ef x ze

f x z f z x f z zf x z

β β

β

−

− −

+ − +

=+ + −

2 1

21 1

[ , , ] [ , ] [ , ] [ , , ](1 [ , ]) 1[ , ] (1 ) [ , ]( [ , ] [ , ])[ , ]

n n n n n n n nn n n

n n n n n n n nn n n

f x z a f x z f x a f z x ze f x af x z f x z f z x f z z

f x z

β

β

−

− −

−= +

+ + −

22 1 1

21 1

[ , , ] [ , ] [ , , , ](1 [ , ]) 1[ , ] (1 ) [ , ]( [ , ] [ , ])[ , ]

z zn n n n n n n n

n n n

n n n n n n n nn n n

f x z a e f x a f z x z a ee f x af x z f x z f z x f z z

f x z

β

β

− −

− −

−= +

+ + −

1

23 1 1 2

( )( )3!( )( )

zn

zn n n

f af a ee c e e

f a

−

− −

′′′′− +

= − +′ +

2 2 2 2 2 1 2 2 13 1 1 1 1( )r p r p

n n n nc C D e o e+ + + +− − − −= + .

So, comparing the exponents of 1ne − in expressions of zne and +1ne for (7), we obtain the same

system of two equations:

2

1,2 2 1.

rp r pr r p

= + +⎧⎨

= + +⎩

From its non-trivial solution 3 344 444/(3 2) 2 +1 3.38327 27

r = + + + ≈ and 1.839p ≈ , we prove that

the convergence of (7) is of order 3.383. Without any additional function evaluations, the e ciency 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 e ciency.

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

1

1 2

log( / )COC .

log( / )n n

n n

e ee e

−

− −

=

A one-step Steffensen-type method with super-cubic convergence ... Zhongli Liu

1873

Table 1. 20( ) 3 1, 0, 0.2xf x x e x a x−= − − + = =

Methods n 1 2 3 4 5 6 NM

nx a− .53279e-2 .35561e-5 .15808e-11 .31235e-24 .12195e-49 .15890e-100

COC 2.25256 2.01691 2.00030 2.00000 2.00000 2.00000 SM nx a− .28174e-1 .51325e-3 .16476e-6 .16966e-13 .17989e-27 .20226e-55 COC 1.21776 2.04376 2.00830 2.00009 2.00000 2.00000 (3) nx a− .28174e-1 .15996e-4 .13132e-12 .43283e-32 .38442e-79 .99936-193 COC 1.21776 3.81335 2.49109 2.40945 2.41512 2.41406 (4)

nx a− .28174e-1 .16560e-6 .11521e-21 .39821e-67 .16444e-203

.11580e-612

COC 1.21776 6.14536 2.89776 2.99925 3.00000 3.00000 (7)

nx a− .28174e-1 .43010e-7 .21604e-27 .23153e-94 .20021e-321

.69689e-1090

COC 1.21776 6.83322 3.49004 3.29917 3.39052 3.38434

Example 2. The numerical results of NM, SM, (3), (4) and (7) are in Table 2 for the following nonlinear functions:

21 0( ) 0.5( 1), 2, 2.5,xf x e a x−= − = =

2

2 0( ) sin 1, 0, 0.25,xf x e x a x= + − = = 2 2

3 0( ) 1, 1, 0.85,x xf x e a x− + += − = − = −

4 0( ) arctan 1, 0, 0.xf x e x a x−= − − = = −

Table 2. Numerical results for solving ( ), 1,2,3,4.if x i = Methods NM SM (3) (4) (7) 1 6:f e .19785e-40 .88156e-29 .50439e-84 .19314e-313 .75162e-578 COC 2.0000 2.0000 2.4141 3.0000 3.3831 2 6:f e .32328e-44 .42920e-26 .19843e-85 .57587e-282 .13494e-706 COC 2.0000 2.0000 2.4141 3.0000 3.3825 3 6:f e .18813e-51 .15758e-18 .12013e-86 .34524e-286 .27679e-677 COC 2.0000 2.0000 2.4140 3.0000 3.3796 3 6:f e .35988e-79 .96290e-84 .16834e-248 .21536e-597 .25291e-1154 COC 2.0000 2.0000 2.4161 3.0000 3.3831 Example 3. Consider solving the following nonlinear ODE by finite di erence method:

3/ 2( ) ( )=0, (0,1),

(0) (1 =0.x t x t tx x′′⎧ + ∈⎪

⎨=⎪⎩

Taking nodes ,it ih= where 1hN

= and 10,N = we have a system of nine nonlinear equations:

2 3 / 21 1 2

2 3 / 21 1

2 3 / 28 9 9

2 0,2 0, 2,3, 8,2 0.

i i i i

x h x xx x h x x ix x h x− +

⎧ − − =⎪− + − − = =⎨⎪ − + − =⎩

For an example, SM is carried out as follows:

11

1 1

1 2

( , ) ( ),( , ) ( ( ) ( ), , ( ) ( )) ,

diag( ( ), ( ), ( )).

n n n n nN

n n n n n n n n n

n n n N n

x x J x H F xJ x H F x H e F x F x H e F x HH f x f x f x

−+

−

⎧ = −⎪

= + − + −⎨⎪ =⎩

A one-step Steffensen-type method with super-cubic convergence ... Zhongli Liu

1874

And other methods are carried out by using similar approximations of the divided di erences. The numerical results are in Table 3, where 0 (40,80,100,120,140,130,100,80,40) ,x ′= *x = (33.57391205,65.2024509, 91.5660200,109.1676243, 115.3630336,109.1676243, 91.5660200, 65.2024509, 33.57391205)′ .

Table 3 The finite di erence method for solving 3/ 2 0, (0) (1 =0x x x x′′+ = = Methods n 1 2 3 4 5 6 NM

2nx x∗− .40882e-1 .47895e-1 .67632e-5 .13490e-12 .53672e-28 .84957e-59 2

( )nf x .24453 .23685e-2 .33390e-6 .66590e-14 .26493e-29 .41936e-60 SM

2nx x∗− 4.8552 .64055e-1 .11495e-4 .37036e-12 .38446e-27 .41429e-57 2

( )nf x .37077 .31892e-2 .56743e-6 .18275e-13 .18970e-28 .20442e-58 (3)

2nx x∗− 4.8552 .11027e-1 .58355e-8 .33191e-23 .55918e-60 .90270e-149 2

( )nf x .37077 .54534e-3 .28807e-9 .16384e-24 .27602e-61 .16919e-150 (4)

2nx x∗− 4.8552 .11260e-2 .15165e-13 .37078e-46 .54192e-144 .16919e-437 2

( )nf x .37077 .55632e-4 .74858e-15 .18302e-47 .26750e-145 .83514e-439 (7)

2nx x∗− 4.8552 .35305e-4 .21872e-18 .51682e-61 .3380e-190 .61416e-576 2

( )nf x .37077 .75225e-5 .35743e-19 .45250e-62 .10709e-190 .71775e-577

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

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A one-step Steffensen-type method with super-cubic convergence ... Zhongli Liu

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