- Email: [email protected]
A variant of Steffensen’s method of fourth-order convergence and its applications
Applied Mathematics and Computation 216 (2010) 1978–1983
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A variant of Steffensen’s method of fourth-order convergence and its applications Zhongli Liu a,*, Quan Zheng b, Peng Zhao b a b
College of Biochemical Engineering, Beijing Union University, Beijing 100023, China College of Sciences, North China University of Technology, Beijing 100144, China
a r t i c l e
i n f o
a b s t r a c t In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Nonlinear equations Newton’s method Steffensen’s method Derivative free Fourth-order convergence ODEs
1. Introduction Finding the root of a nonlinear equation f ðxÞ ¼ 0 is a classical problem in scientific computation. The most famous method is probably Newton’s method as follows (see [1]):
xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
n ¼ 0; 1; 2; . . . ;
ð1Þ
where x0 is an initial guess of the root. It uses two evaluations of the function and its derivative to achieve second-order convergence. Whereas, Steffensen’s method
xnþ1 ¼ xn
f 2 ðxn Þ ; f ðxn þ f ðxn ÞÞ f ðxn Þ
n ¼ 0; 1; 2; . . . ;
ð2Þ
is well known as a noticeable improvement of Newton’s method, since it is derivative free and maintains quadratic convergence (see Section 7.2.8 in Traub [2]). In order to control the approximation of the derivative and the stability of the iteration, a Steffensen’s type method (STM):
xnþ1 ¼ xn
f ðxn Þ ; ðf ðxn þ an jf ðxn Þjf ðxn ÞÞ f ðxn ÞÞ=an jf ðxn Þjf ðxn Þ
n ¼ 0; 1; 2; . . .
ð3Þ
has been proposed in Amat and Busquier [3]. And the recent paper [4] has extended it to Banach spaces, obtained its semilocal and local convergence theorems, and made its applications on boundary-value problems by the multiple shooting method. Meanwhile, Steffensen-secant method (SSM) deformed from Newton-secant has been proposed in [5] as follows:
* Corresponding author. E-mail address: [email protected] (Z. Liu). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.028
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
8 f 3 ðxn Þ < xnþ1 ¼ xn ; ½f ðxn þf ðxn ÞÞf ðxn Þ½f ðxn Þf ðx Þ n
f 2 ðxn Þ : x ¼ x ; n n f ðxn þf ðxn ÞÞf ðxn Þ
1979
ð4Þ
n ¼ 0; 1; 2; . . .
This method only uses three evaluations of the function in one step and arrives third-order convergence. Recently, a family of fourth-order methods free from any derivative, satisfying the highest convergence order conjectured by Kung and Traub, are established (see [6]):
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn f ðyn Þ : xnþ1 ¼ y ; n f ½xn ;yn þf ½yn ;zn f ½xn ;zn þaðyn xn Þðyn zn Þ
n ¼ 0; 1; 2; . . . ;
ð5Þ
where zn ¼ xn þ f ðxn Þ, and f ½; is divided difference. Other Steffensen-type methods and their applications are also discussed in Zheng et al. [7]. In the following of this paper, we suggest a new variant of derivative-free Steffensen’s method with the conjectured optimal fourth-order convergence, deduce its error equation and asymptotic convergence constant (Section 2), compare it with related methods for solving nonlinear equations and make applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs (Section 3), and finally make conclusions (Section 4). 2. The new method and its fourth-order convergence It’s clear that we have a fourth-order method:
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn : xnþ1 ¼ y f0ðyn Þ ; n f ðyn Þ
ð6Þ
n ¼ 0; 1; 2; . . . :
As we know by Newton’s interpolation formula that
f ðxÞ f ðxn Þ þ f ½xn ; yn ðx xn Þ þ f ½xn ; yn ; zn ðx xn Þðx yn Þ;
ð7Þ
0
we can have an approximation of f ðyÞ:
f 0 ðyn Þ f ½xn ; yn þ f ½xn ; yn ; zn ðyn xn Þ ¼ f ½xn ; yn þ f ½yn ; zn f ½xn ; zn :
ð8Þ
Substituting (8) into (6), we have
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn f ðyn Þ : xnþ1 ¼ y ; n f ½xn ;yn þf ½yn ;zn f ½xn ;zn
ð9Þ
n ¼ 0; 1; 2; . . . ;
which is (5) with a ¼ 0. From (8), we also have
f 0 ðyn Þ f ½xn ; yn 1 þ f ½xf ½xn ;yn ;yn ;zn ðyn xn Þ n
1
f ½xn ;yn f ½xn ;yn ;zn ðyn xn Þ f ½xn ;yn
:
ð10Þ
Substituting (10) into (6), we have a new method of fourth-order convergence:
8 < yn ¼ xn f ½xf ðx;zn Þ ; n n : xnþ1 ¼ yn f ½xn ;yn f ½yn ;zn 2þf ½xn ;zn f ðyn Þ; f ½x ;y n
n
n ¼ 0; 1; 2; . . . :
ð11Þ
We state and prove the convergence theorem for the method (11) as follows. Theorem 2.1. Let f : D ! R be a sufficiently differentiable function with a simple root a 2 D; D R be an open set, x0 be close enough to a, then the new method (11) is fourth-order convergent, and satisfies the following error equation
enþ1 ¼
f 00 ðaÞð1 þ f 0 ðaÞÞ 8f 0 ðaÞ3
2 ½f 00 ðaÞ2 ð2 þ f 0 ðaÞÞ f 000 ðaÞf 0 ðaÞð1 þ f 0 ðaÞÞe4n þ Oðe5n Þ; 3
ð12Þ
where en ¼ xn a; n ¼ 0; 1; 2; . . . Proof. Let en ¼ yn a. By Taylor formula, noting that en ¼ xn a and f ðaÞ ¼ 0, we have
f 00 ðaÞ 2 f 000 ðaÞ 3 e þ en þ Oðe4n Þ; 2 n 6 f 00 ðaÞ 2 f 000 ðaÞ 3 zn a ¼ en þ f 0 ðaÞen þ e þ en þ Oðe4n Þ; 2 n 6
f ðxn Þ ¼ f 0 ðaÞen þ
ð13Þ ð14Þ
1980
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
then,
1 f ðzn Þ ¼ f 0 ðaÞð1 þ f 0 ðaÞÞen þ ½f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞð1 þ f 0 ðaÞÞ2 e2n 2 " # 0 1 0 f ðaÞ2 1 þ f ðaÞf 000 ðaÞ þ ð1 þ f 0 ðaÞÞ þ f 000 ðaÞð1 þ f 0 ðaÞÞ3 e3n þ Oðe4n Þ: 6 6 2
ð15Þ
According to the definition of divided differences, we find that
f ½zn ; xn ¼ ½f 0 ðaÞ2 en þ 12 ð3f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞf 0 ðaÞ2 Þe2n þ 16 f 0 ðaÞf 000 ðaÞ þ 12 f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ þ 16 f 000 ðaÞð1 þ f 0 ðaÞÞ3 16 f 000 ðaÞ e3n 00 ðaÞ
000 ðaÞ
e3 1 þ Oðe3n Þ h 00n 2 i 3 00 00 0 000 0 f 000 ðaÞ 2 þ f ðaÞð1þf6f 0ðaÞÞ en þ Oðe3n Þ: ¼ f 0 ðaÞ þ 2f ðaÞþf2 ðaÞf ðaÞ en þ f ðaÞ 4 ðaÞ ½f 0 ðaÞen þ f
2
e2n þ f
ð16Þ
6
So that
f ðxn Þ ð1 þ f 0 ðaÞÞf 00 ðaÞ 2 ¼ en þ Ae3n þ Oðe4n Þ; f ½zn ; xn 2f 0 ðaÞ
en ¼ en 000
00
2
00
2
f ðaÞ f ðaÞ f where A ¼ 3f f2fðaÞ 0 ðaÞ 0 ðaÞ þ 2f 0 ðaÞ2
f ðyn Þ ¼ f 0 ðaÞen þ Oðe4n Þ ¼
00 ðaÞ
2
00
0
000 ðaÞ
þ f ðaÞf6
f
00 ðaÞ2
4
ð17Þ
.
0
f ðaÞð1 þ f ðaÞÞ 2 en þ f 0 ðaÞAe3n þ Oðe4n Þ: 2
ð18Þ
Therefore, 0
f ½yn ; xn ¼
00 ðaÞ 000 e2n þ ðf 0 ðaÞA f 6ðaÞÞe3n 3 f 00 ðaÞð1þf 0 ðaÞÞ 2 en þ Aen 2f 0 ðaÞ
f 0 ðaÞen þ f ðaÞf2 en þ
¼ f 0 ðaÞ þ
f 00 ðaÞ f 00 ðaÞ2 þ f 00 ðaÞ2 f 0 ðaÞ 2 f 000 ðaÞ 2 en þ en þ Oðe3n Þ; en þ 2 4f 0 ðaÞ 6
ð19Þ
1 f ½zn ; yn ¼ f 0 ðaÞð1 þ f 0 ðaÞÞen þ ð2f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞf 0 ðaÞ2 Þe2n 2
! # f 000 ðaÞf 0 ðaÞ f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ ð1 þ f 0 ðaÞÞ3 f 000 ðaÞ 0 þ þ þ f ðaÞA e3n 6 2 6 000 1 f 00 ðaÞ 2 f ðaÞ ð1 þ f 0 ðaÞÞen 0 A e3n en þ 2f ðaÞ 6
¼ f 0 ðaÞ þ
f ½yn ; a ¼ f 0 ðaÞ þ
f 00 ðaÞð1 þ f 0 ðaÞÞ f 00 ðaÞ2 2 f 00 ðaÞ2 2 f 000 ðaÞð1 þ f 0ðaÞÞ2 2 en þ 0 e þ en þ en þ Oðe3n Þ; 2 4f ðaÞ n 2 6
f 00 ðaÞ f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ 2 en ¼ f 0 ðaÞ þ en þ Oðe3n Þ: 2 4f 0 ðaÞ
ð20Þ
ð21Þ
Furthermore,
f ½xn ; yn ; a ¼
f ½yn ; xn f ½yn ; a f 00 ðaÞ f 000 ðaÞ ¼ þ en þ Oðe2n Þ; en 2 6
ð22Þ 2
f ½xn ; yn ; zn ¼
f ½zn ; yn f ½yn ; xn f 00 ðaÞ f 000 ðaÞð1 þ f 0 ðaÞÞ f 000 ðaÞ ¼ þ en 0 en þ Oðe2n Þ: z n xn 2 6f ðaÞ 6f 0 ðaÞ
Finally, using (19), (21), (22), (23) and (17) in (11), we obtain the error equation as
f ½xn ; yn f ½xn ; yn ; zn ðyn xn Þ f ½yn ; aen f 2 ½xn ; yn f 2 ½xn ; yn f ½xn ; yn f ½yn ; a þ f ½yn ; af ½xn ; yn ; zn ðyn xn Þ ¼ en f 2 ½xn ; yn f ½x ; y f ½x ; y ; ae f ½xn ; yn ; zn f ½yn ; aen þ f ½xn ; yn ; zn f ½yn ; aen n n n n n ¼ en f 2 ½xn ; yn " # 2 2 00 000 0 000 e e f ðaÞ f ðaÞf ðaÞ f ðaÞ f 000 ðaÞð1 þ f 0 ðaÞÞ2 f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ ¼ n n2 þ þ Oðen Þ þ þ 6 6 4 4 6 f 0 ðaÞ
enþ1 ¼ en
¼ h
f 00 ðaÞð1 þ f 0 ðaÞÞ 8f 0 ðaÞ3
2 ½f 002 ðaÞð2 þ f 0 ðaÞÞ f 000 ðaÞf 0 ðaÞð1 þ f 0 ðaÞÞe4n þ Oðe5n Þ: 3
ð23Þ
1981
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
Remark. Newton’s method, Steffensen’s method and Ren’s methods have the asymptotic convergence constants 2 00 00 0 00 ðaÞ 0 f ðaÞ f 00 ðaÞ2 f 000 ðaÞ a respectively (see [1,2,4]). And the asymptotic converC NM ¼ 2ff 0ðaÞ ; C SM ¼ ð1þf2fðaÞÞf and C RM ¼ ð1þf 2fðaÞÞ 6f 0 ðaÞ 0 ðaÞ 0 ðaÞ þ f 0 ðaÞ ðaÞ 4f 0 ðaÞ2 0
00
2
ðaÞ gence constant of Jain’s method (see [3]) is C JM ¼ ð1þf4fðaÞÞf , which is newly deduced by us. 0 ðaÞ2
3. Numerical examples The new method (11) is demonstrated by solving some nonlinear equations and by comparison with Newton’s method (NM), Steffensen’s method (SM), Steffensen’s type method (STM), Steffensen-secant method (SSM) and Ren’s method (RM) in Tables 1 and 2, where STM is of an ¼ 1=2n , RM0 is RM with a ¼ 0 and RM1 is RM with a ¼ 1; C is the asymptotic convergence constant of a method in its error equation, and the number of evaluations of the functions and its derivatives in each method is nf = 12. Example 1. Consider the nonlinear function: f ðxÞ ¼ x2 þ ex 3x 1. Example 2. We have numerical results in Table 2 for the following nonlinear functions:
Table 1 Numerical results for f ðxÞ ¼ x2 þ ex 3x 1; a ¼ 0 and x0 ¼ 0:5. Method
C
NM 0.375
SM 1.125
STM 0.375
SSM 0.422
RM0 0.334
RM1 0.510
(11) 0.176
n
1
2
3
4
5
6
jen j
.13051 0.52207
.59838e2 0.35126
.13385e4 0.37382
.67187e10 0.37499
.16928e20 0.37500
.10746e41 0.37500
f ðxn Þ
.54801
.23989e1
.53541e4
.26875e9
.67712e20
.42984e41
jen j
.18294 0.73178
.31640e1 0.94534
.10907e2 1.08952
.13368e5 1.12374
.20105e11 1.12499
.45477e23 1.12499
f ðxn Þ
.68255
.12506
.43610e2
.53474e5
.80423e11
.18191e22
jen j en e2n1
.14540 0.58158
.40308e2 0.19067
.60308e5 0.37118
.13639e10 0.37499
.69757e22 0.37499
.18248e44 0.37500
f ðxn Þ
.55037
.16148e1
.24123e4
.54556e10
.27903e21
.72991e44
jen j
.42263e1 0.33811
.32943e4 0.43638
.15083e13 0.42186
.14476e41 0.42187
f ðxn Þ
.17174
.13177e3
.60332e13
.57904e41
jen j
.95756e2 0.15321
.28675e8 0.34105
.22582e34 0.33398
.86851e139 0.33399
f ðxn Þ
.38440e1
.11470e7
.90328e34
.34740e138
jen j
.10839e1 0.17343
.68608e8 0.49699
.11294e32 0.50976
.82952e132 0.50976
en e2n1
en e2n1
en e3n1
en e4n1
en e4n1
f ðxn Þ
.43182e1
.27443e7
.45178e32
.33181e131
jen j
.40256e2 0.6441e1
.46586e10 0.17738
.28794e42 0.17578
.82600e169 0.17579
.16126e1
.18635e9
.33117e41
.33040e168
en e4n1
f ðxn Þ
Table 2 Numerical results for fi ðxÞ; i ¼ 1; 2; 3; 4.
f1 : jxn C jf1 ðxn Þj f2 : jxn C jf2 ðxn Þj f3 : jxn C jf4 ðxn Þj f4 : jxn C jf4 ðxn Þj
aj
aj
aj
aj
NM
SM
STM
SSM
RM0
RM1
(11)
.19785e40 0.500 .98952e41 .13664e35 1.667 .40993e35 .29768e70 0.167 .89305e70 .54043e50 2.500 .10808e49
.88155e29 0.750 .44077e29 .21078e15 4.667 .63235e15 .31158e40 0.333 .93476e40 .19219e60 0.250 .38438e60
.40351e39 0.500 .20175e39 .24128e25 1.667 .72386e25 .30274e58 0.167 .90822e58 .12050e70 0.250 .24101e70
.72136e44 0.375 .36068e44 .43806e35 5.444 .13142e34 .64051e67 0.555e1 .19215e66 .21554e71 0.625e1 .43109e71
.16412e165 0.938 .82061e166 .31543e84 16.074 .94631e84 .15139e155 0.185e1 .45419e155 .67397e215 0.365e1 .13479e214
.25854e75 2.344 .12927e75 .69356e76 22.296 .20807e75 .48575e123 0.204 .14573e122 .17486e182 0.885e1 .34971e182
.11976e134 0.281 .59882e135 .27102e81 22.426 .81307e81 .10378e171 0.926e2 .31135e171 .94899e263 0.208e1 .18979e262
1982
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
1 x2 ðe 1Þ; a ¼ 2; x0 ¼ 2:5; 2 x2 þxþ2 f2 ðxÞ ¼ e 1; a ¼ 1; x0 ¼ 0:7; f1 ðxÞ ¼
f3 ðxÞ ¼ lnðx2 x þ 1Þ 4 sinðx 1Þ; a ¼ 1; x0 ¼ 0:5; f4 ðxÞ ¼ ex arctan x 1; a ¼ 0; x0 ¼ 0:5: The numerical results agree with the theoretical analysis on the error equation and its asymptotic convergence constant in the two tables. Now, we show its applications on systems of nonlinear equations in Table 3 and boundary-value problems of nonlinear ODEs in Table 4 by using the Steffensen-type method in Rn space as follows:
8 1 > < xkþ1 ¼ xk Jðxk ; Hk Þ Fðxk Þ; Jðxk ; Hk Þ ¼ ðFðxk þ Hk e1 Þ Fðxk Þ; . . . ; Fðxk þ Hk ek Þ Fðxk ÞÞH1 k ; > : Hk ¼ diagðf1 ðxk Þ; f2 ðxk Þ; . . . ; fn ðxk ÞÞ; k ¼ 0; 1; 2; . . .
ð24Þ
where FðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; . . . ; fn ðxÞÞ0 , and x ¼ ðx1 ; x2 ; . . . ; xn Þ0 . Example 3. For a system of nonlinear equations:
(
ðx1 1Þ4 þ ex2 x22 þ 3x2 þ 1 ¼ 0;
ð25Þ
4 sinðx1 1Þ lnðx21 x1 þ 1Þ x22 ¼ 0:
x0 ¼ ð0:8; 0:1Þ0 and x ð1:271384307950; 0:880819073102Þ0 . We have numerical results for the methods in Table 3, where STM with an ¼ 108 minðkF1i kÞkFk. Example 4. Consider to solve a boundary-value problem of ODEs as the following:
Table 3 The results for a system of nonlinear equations by the methods. Method
k
1
2
3
4
5
6
SM
kxk x k2 kFðxk Þk2
4.61909e1 6.70188e1
7.12980e2 1.21111e1
1.48194e3 3.07302e3
9.77722e7 1.85146e6
9.27307e13 6.94215e13
6.73973e13 9.65273e26
STM
kxk x k2 kFðxk Þk2
2.33251e1 7.81566e1
1.28406e2 2.94176e2
6.51851e5 1.22801e4
1.77131e9 3.41658e9
6.73973e13 2.50598e18
6.73973e13 1.36567e36
SSM
kxk x k2 kFðxk Þk2
1.47459e1 2.60665e1
5.87560e4 1.08318e3
2.97785e11 5.64369e11
6.73974e13 7.74837e33
RM0
kxk x k2 kFðxk Þk2
6.41762e1 1.33829
1.30813e1 2.36817e1
2.37921e5 4.11493e5
6.73973e13 3.86236e20
(11)
kxk x k2 kFðxk Þk2
6.47338e2 1.52073e1
1.68203e5 3.14660e5
6.73974e13 2.14176e19
6.73973e13 3.66306e59
Table 4 The results of the multiple shooting method by the methods. Method
n
1
2
3
4
SM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
5.5970e2 3.0898e2 6.9725e2
1.4912e4 6.4125e5 1.4348e4
9.9365e11 3.2621e6 4.9799e6
6.4395e15 3.2621e6 4.9799e6
STM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
3.8583e2 2.7290e2 4.5231e2
1.1509e4 7.3974e5 1.1617e4
1.9887e9 3.2629e6 4.9812e6
2.5438e16 3.2621e6 4.9798e6
SSM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
1.3479e3 4.6681e4 2.1077e3
4.1452e12 3.2621e6 4.9798e6
1.2412e16 3.2621e6 4.9798e6
RM0
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
1.0177e1 3.8732e2 1.2041e1
7.8405e4 1.1790e4 5.1796e4
7.6978e9 3.2635e6 4.9868e6
(11)
kFð~ sn Þk2 kyðtÞ yn k2 0 ky ðtÞ y0n k2
4.1682e3 1.9092e3 3.3783e3
5.9014e11 3.2621e6 4.9798e6
2.7756e16 3.2621e6 4.9798e6
6.0809e16 3.2621e5 4.9798e5
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
(
y00 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y02 ;
yð0Þ ¼ 1; yð1Þ ¼ 12 ðe þ e1 Þ:
1983
ð26Þ
Generally, for a boundary-value problem of the type: y00 ðtÞ ¼ f ðt; yðtÞ; y0 ðtÞÞ; yðaÞ ¼ ya ; yðbÞ ¼ yb , we can describe briefly the multiple shooting method. Considering N subintervals, t i ¼ a þ i ðb aÞ=N ¼ a þ i h; 0 6 i 6 N; in each interval ½ti ; t iþ1 , the function yðt; s0 ; s1 ; . . . ; sN1 Þ is calculated in recursive form, by solving the initial-value problems y00 ðtÞ ¼ f ðt; yðtÞ; y0 ðtÞÞ; yðt i Þ ¼ yðt i ; s0 ; s1 ; . . . ; si1 Þ; y0 ðt i Þ ¼ si , with yðt0 Þ ¼ ya ; y0 ðt0 Þ ¼ s0 . The system of the initial-value problem of ODEs is solved by the classical Runge-Kutta method. We should deal with the system of nonlinear equations F : RN ! RN in the following form
8 F 1 ðs0 ; s1 ; . . . ; sN1 Þ ¼ s1 y0 ðt1 ; s0 Þ; > > > > > F 2 ðs0 ; s1 ; . . . ; sN1 Þ ¼ s2 y0 ðt2 ; s0 ; s1 Þ; > > < .. . > > > 0 > > > F N1 ðs0 ; s1 ; . . . ; sN1 Þ ¼ sN1 y ðt N1 ; s0 ; s1 ; . . . ; sN2 Þ; > : F N ðs0 ; s1 ; . . . ; sN1 Þ ¼ yb y0 ðt N ; s0 ; s1 ; . . . ; sN2 ; sN1 Þ:
ð27Þ
For solving (26), let N ¼ 4, the initial guess s0 ¼ 12 ðe þ e1 Þ 1 and an ¼ 108 minðkF1 kÞkFk in STM. we compute the error until i we obtain convergence ðkFð~ sn Þk2 < 1014 Þ. The numerical results by the multiple shooting method are obtained in Table 4. 4. Conclusions By theoretical analysis and numerical experiments, we confirm that the new variant of Steffensen’s method is free from any derivative and only uses three evaluations of the function per iteration to achieve fourth-order convergence for solving a 1=w simple rootpof ffiffiffi nonlinear functions. As the efficiency index is p , the efficiency ffiffiffi index of Newton’s method and Steffensen’s p 3 method is 2 ¼ 1:4142, the efficiency index of Steffensen-secant method is 3 ¼ 1:4422, and the efficiency index of Ren’s pffiffiffi method and the suggested method (11) is 3 4 ¼ 1:5874. The suggested method is very suitable to solve nonlinear equations, and is able to be applied on solving systems of nonlinear equations and boundary-value problems of nonlinear ODEs as well. Acknowledgements The work is supported by the Scientific Research Project of Beijing Union University (No. 2010) and in part by Natural Science Foundation of Beijing (No. 1072009). References [1] [2] [3] [4]
J.M. Ortega, W.G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations, Appl. Math. Comput. 177 (2006) 819–823. V. Alarcón, S. Amat, S. Busquier, D.J. López, A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216 (2008) 243–250. [5] P. Jain, Steffensen type methods for solving non-linear equations, Appl. Math. Comput. 194 (2007) 527–533. [6] H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206–210. [7] Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214 (2009) 10–16.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A variant of Steffensen’s method of fourth-order convergence and its applications Zhongli Liu a,*, Quan Zheng b, Peng Zhao b a b
College of Biochemical Engineering, Beijing Union University, Beijing 100023, China College of Sciences, North China University of Technology, Beijing 100144, China
a r t i c l e
i n f o
a b s t r a c t In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Nonlinear equations Newton’s method Steffensen’s method Derivative free Fourth-order convergence ODEs
1. Introduction Finding the root of a nonlinear equation f ðxÞ ¼ 0 is a classical problem in scientific computation. The most famous method is probably Newton’s method as follows (see [1]):
xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
n ¼ 0; 1; 2; . . . ;
ð1Þ
where x0 is an initial guess of the root. It uses two evaluations of the function and its derivative to achieve second-order convergence. Whereas, Steffensen’s method
xnþ1 ¼ xn
f 2 ðxn Þ ; f ðxn þ f ðxn ÞÞ f ðxn Þ
n ¼ 0; 1; 2; . . . ;
ð2Þ
is well known as a noticeable improvement of Newton’s method, since it is derivative free and maintains quadratic convergence (see Section 7.2.8 in Traub [2]). In order to control the approximation of the derivative and the stability of the iteration, a Steffensen’s type method (STM):
xnþ1 ¼ xn
f ðxn Þ ; ðf ðxn þ an jf ðxn Þjf ðxn ÞÞ f ðxn ÞÞ=an jf ðxn Þjf ðxn Þ
n ¼ 0; 1; 2; . . .
ð3Þ
has been proposed in Amat and Busquier [3]. And the recent paper [4] has extended it to Banach spaces, obtained its semilocal and local convergence theorems, and made its applications on boundary-value problems by the multiple shooting method. Meanwhile, Steffensen-secant method (SSM) deformed from Newton-secant has been proposed in [5] as follows:
* Corresponding author. E-mail address: [email protected] (Z. Liu). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.028
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
8 f 3 ðxn Þ < xnþ1 ¼ xn ; ½f ðxn þf ðxn ÞÞf ðxn Þ½f ðxn Þf ðx Þ n
f 2 ðxn Þ : x ¼ x ; n n f ðxn þf ðxn ÞÞf ðxn Þ
1979
ð4Þ
n ¼ 0; 1; 2; . . .
This method only uses three evaluations of the function in one step and arrives third-order convergence. Recently, a family of fourth-order methods free from any derivative, satisfying the highest convergence order conjectured by Kung and Traub, are established (see [6]):
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn f ðyn Þ : xnþ1 ¼ y ; n f ½xn ;yn þf ½yn ;zn f ½xn ;zn þaðyn xn Þðyn zn Þ
n ¼ 0; 1; 2; . . . ;
ð5Þ
where zn ¼ xn þ f ðxn Þ, and f ½; is divided difference. Other Steffensen-type methods and their applications are also discussed in Zheng et al. [7]. In the following of this paper, we suggest a new variant of derivative-free Steffensen’s method with the conjectured optimal fourth-order convergence, deduce its error equation and asymptotic convergence constant (Section 2), compare it with related methods for solving nonlinear equations and make applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs (Section 3), and finally make conclusions (Section 4). 2. The new method and its fourth-order convergence It’s clear that we have a fourth-order method:
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn : xnþ1 ¼ y f0ðyn Þ ; n f ðyn Þ
ð6Þ
n ¼ 0; 1; 2; . . . :
As we know by Newton’s interpolation formula that
f ðxÞ f ðxn Þ þ f ½xn ; yn ðx xn Þ þ f ½xn ; yn ; zn ðx xn Þðx yn Þ;
ð7Þ
0
we can have an approximation of f ðyÞ:
f 0 ðyn Þ f ½xn ; yn þ f ½xn ; yn ; zn ðyn xn Þ ¼ f ½xn ; yn þ f ½yn ; zn f ½xn ; zn :
ð8Þ
Substituting (8) into (6), we have
8 < yn ¼ xn f ðxn Þ ; f ½xn ;zn f ðyn Þ : xnþ1 ¼ y ; n f ½xn ;yn þf ½yn ;zn f ½xn ;zn
ð9Þ
n ¼ 0; 1; 2; . . . ;
which is (5) with a ¼ 0. From (8), we also have
f 0 ðyn Þ f ½xn ; yn 1 þ f ½xf ½xn ;yn ;yn ;zn ðyn xn Þ n
1
f ½xn ;yn f ½xn ;yn ;zn ðyn xn Þ f ½xn ;yn
:
ð10Þ
Substituting (10) into (6), we have a new method of fourth-order convergence:
8 < yn ¼ xn f ½xf ðx;zn Þ ; n n : xnþ1 ¼ yn f ½xn ;yn f ½yn ;zn 2þf ½xn ;zn f ðyn Þ; f ½x ;y n
n
n ¼ 0; 1; 2; . . . :
ð11Þ
We state and prove the convergence theorem for the method (11) as follows. Theorem 2.1. Let f : D ! R be a sufficiently differentiable function with a simple root a 2 D; D R be an open set, x0 be close enough to a, then the new method (11) is fourth-order convergent, and satisfies the following error equation
enþ1 ¼
f 00 ðaÞð1 þ f 0 ðaÞÞ 8f 0 ðaÞ3
2 ½f 00 ðaÞ2 ð2 þ f 0 ðaÞÞ f 000 ðaÞf 0 ðaÞð1 þ f 0 ðaÞÞe4n þ Oðe5n Þ; 3
ð12Þ
where en ¼ xn a; n ¼ 0; 1; 2; . . . Proof. Let en ¼ yn a. By Taylor formula, noting that en ¼ xn a and f ðaÞ ¼ 0, we have
f 00 ðaÞ 2 f 000 ðaÞ 3 e þ en þ Oðe4n Þ; 2 n 6 f 00 ðaÞ 2 f 000 ðaÞ 3 zn a ¼ en þ f 0 ðaÞen þ e þ en þ Oðe4n Þ; 2 n 6
f ðxn Þ ¼ f 0 ðaÞen þ
ð13Þ ð14Þ
1980
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
then,
1 f ðzn Þ ¼ f 0 ðaÞð1 þ f 0 ðaÞÞen þ ½f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞð1 þ f 0 ðaÞÞ2 e2n 2 " # 0 1 0 f ðaÞ2 1 þ f ðaÞf 000 ðaÞ þ ð1 þ f 0 ðaÞÞ þ f 000 ðaÞð1 þ f 0 ðaÞÞ3 e3n þ Oðe4n Þ: 6 6 2
ð15Þ
According to the definition of divided differences, we find that
f ½zn ; xn ¼ ½f 0 ðaÞ2 en þ 12 ð3f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞf 0 ðaÞ2 Þe2n þ 16 f 0 ðaÞf 000 ðaÞ þ 12 f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ þ 16 f 000 ðaÞð1 þ f 0 ðaÞÞ3 16 f 000 ðaÞ e3n 00 ðaÞ
000 ðaÞ
e3 1 þ Oðe3n Þ h 00n 2 i 3 00 00 0 000 0 f 000 ðaÞ 2 þ f ðaÞð1þf6f 0ðaÞÞ en þ Oðe3n Þ: ¼ f 0 ðaÞ þ 2f ðaÞþf2 ðaÞf ðaÞ en þ f ðaÞ 4 ðaÞ ½f 0 ðaÞen þ f
2
e2n þ f
ð16Þ
6
So that
f ðxn Þ ð1 þ f 0 ðaÞÞf 00 ðaÞ 2 ¼ en þ Ae3n þ Oðe4n Þ; f ½zn ; xn 2f 0 ðaÞ
en ¼ en 000
00
2
00
2
f ðaÞ f ðaÞ f where A ¼ 3f f2fðaÞ 0 ðaÞ 0 ðaÞ þ 2f 0 ðaÞ2
f ðyn Þ ¼ f 0 ðaÞen þ Oðe4n Þ ¼
00 ðaÞ
2
00
0
000 ðaÞ
þ f ðaÞf6
f
00 ðaÞ2
4
ð17Þ
.
0
f ðaÞð1 þ f ðaÞÞ 2 en þ f 0 ðaÞAe3n þ Oðe4n Þ: 2
ð18Þ
Therefore, 0
f ½yn ; xn ¼
00 ðaÞ 000 e2n þ ðf 0 ðaÞA f 6ðaÞÞe3n 3 f 00 ðaÞð1þf 0 ðaÞÞ 2 en þ Aen 2f 0 ðaÞ
f 0 ðaÞen þ f ðaÞf2 en þ
¼ f 0 ðaÞ þ
f 00 ðaÞ f 00 ðaÞ2 þ f 00 ðaÞ2 f 0 ðaÞ 2 f 000 ðaÞ 2 en þ en þ Oðe3n Þ; en þ 2 4f 0 ðaÞ 6
ð19Þ
1 f ½zn ; yn ¼ f 0 ðaÞð1 þ f 0 ðaÞÞen þ ð2f 0 ðaÞf 00 ðaÞ þ f 00 ðaÞf 0 ðaÞ2 Þe2n 2
! # f 000 ðaÞf 0 ðaÞ f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ ð1 þ f 0 ðaÞÞ3 f 000 ðaÞ 0 þ þ þ f ðaÞA e3n 6 2 6 000 1 f 00 ðaÞ 2 f ðaÞ ð1 þ f 0 ðaÞÞen 0 A e3n en þ 2f ðaÞ 6
¼ f 0 ðaÞ þ
f ½yn ; a ¼ f 0 ðaÞ þ
f 00 ðaÞð1 þ f 0 ðaÞÞ f 00 ðaÞ2 2 f 00 ðaÞ2 2 f 000 ðaÞð1 þ f 0ðaÞÞ2 2 en þ 0 e þ en þ en þ Oðe3n Þ; 2 4f ðaÞ n 2 6
f 00 ðaÞ f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ 2 en ¼ f 0 ðaÞ þ en þ Oðe3n Þ: 2 4f 0 ðaÞ
ð20Þ
ð21Þ
Furthermore,
f ½xn ; yn ; a ¼
f ½yn ; xn f ½yn ; a f 00 ðaÞ f 000 ðaÞ ¼ þ en þ Oðe2n Þ; en 2 6
ð22Þ 2
f ½xn ; yn ; zn ¼
f ½zn ; yn f ½yn ; xn f 00 ðaÞ f 000 ðaÞð1 þ f 0 ðaÞÞ f 000 ðaÞ ¼ þ en 0 en þ Oðe2n Þ: z n xn 2 6f ðaÞ 6f 0 ðaÞ
Finally, using (19), (21), (22), (23) and (17) in (11), we obtain the error equation as
f ½xn ; yn f ½xn ; yn ; zn ðyn xn Þ f ½yn ; aen f 2 ½xn ; yn f 2 ½xn ; yn f ½xn ; yn f ½yn ; a þ f ½yn ; af ½xn ; yn ; zn ðyn xn Þ ¼ en f 2 ½xn ; yn f ½x ; y f ½x ; y ; ae f ½xn ; yn ; zn f ½yn ; aen þ f ½xn ; yn ; zn f ½yn ; aen n n n n n ¼ en f 2 ½xn ; yn " # 2 2 00 000 0 000 e e f ðaÞ f ðaÞf ðaÞ f ðaÞ f 000 ðaÞð1 þ f 0 ðaÞÞ2 f 00 ðaÞ2 ð1 þ f 0 ðaÞÞ ¼ n n2 þ þ Oðen Þ þ þ 6 6 4 4 6 f 0 ðaÞ
enþ1 ¼ en
¼ h
f 00 ðaÞð1 þ f 0 ðaÞÞ 8f 0 ðaÞ3
2 ½f 002 ðaÞð2 þ f 0 ðaÞÞ f 000 ðaÞf 0 ðaÞð1 þ f 0 ðaÞÞe4n þ Oðe5n Þ: 3
ð23Þ
1981
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
Remark. Newton’s method, Steffensen’s method and Ren’s methods have the asymptotic convergence constants 2 00 00 0 00 ðaÞ 0 f ðaÞ f 00 ðaÞ2 f 000 ðaÞ a respectively (see [1,2,4]). And the asymptotic converC NM ¼ 2ff 0ðaÞ ; C SM ¼ ð1þf2fðaÞÞf and C RM ¼ ð1þf 2fðaÞÞ 6f 0 ðaÞ 0 ðaÞ 0 ðaÞ þ f 0 ðaÞ ðaÞ 4f 0 ðaÞ2 0
00
2
ðaÞ gence constant of Jain’s method (see [3]) is C JM ¼ ð1þf4fðaÞÞf , which is newly deduced by us. 0 ðaÞ2
3. Numerical examples The new method (11) is demonstrated by solving some nonlinear equations and by comparison with Newton’s method (NM), Steffensen’s method (SM), Steffensen’s type method (STM), Steffensen-secant method (SSM) and Ren’s method (RM) in Tables 1 and 2, where STM is of an ¼ 1=2n , RM0 is RM with a ¼ 0 and RM1 is RM with a ¼ 1; C is the asymptotic convergence constant of a method in its error equation, and the number of evaluations of the functions and its derivatives in each method is nf = 12. Example 1. Consider the nonlinear function: f ðxÞ ¼ x2 þ ex 3x 1. Example 2. We have numerical results in Table 2 for the following nonlinear functions:
Table 1 Numerical results for f ðxÞ ¼ x2 þ ex 3x 1; a ¼ 0 and x0 ¼ 0:5. Method
C
NM 0.375
SM 1.125
STM 0.375
SSM 0.422
RM0 0.334
RM1 0.510
(11) 0.176
n
1
2
3
4
5
6
jen j
.13051 0.52207
.59838e2 0.35126
.13385e4 0.37382
.67187e10 0.37499
.16928e20 0.37500
.10746e41 0.37500
f ðxn Þ
.54801
.23989e1
.53541e4
.26875e9
.67712e20
.42984e41
jen j
.18294 0.73178
.31640e1 0.94534
.10907e2 1.08952
.13368e5 1.12374
.20105e11 1.12499
.45477e23 1.12499
f ðxn Þ
.68255
.12506
.43610e2
.53474e5
.80423e11
.18191e22
jen j en e2n1
.14540 0.58158
.40308e2 0.19067
.60308e5 0.37118
.13639e10 0.37499
.69757e22 0.37499
.18248e44 0.37500
f ðxn Þ
.55037
.16148e1
.24123e4
.54556e10
.27903e21
.72991e44
jen j
.42263e1 0.33811
.32943e4 0.43638
.15083e13 0.42186
.14476e41 0.42187
f ðxn Þ
.17174
.13177e3
.60332e13
.57904e41
jen j
.95756e2 0.15321
.28675e8 0.34105
.22582e34 0.33398
.86851e139 0.33399
f ðxn Þ
.38440e1
.11470e7
.90328e34
.34740e138
jen j
.10839e1 0.17343
.68608e8 0.49699
.11294e32 0.50976
.82952e132 0.50976
en e2n1
en e2n1
en e3n1
en e4n1
en e4n1
f ðxn Þ
.43182e1
.27443e7
.45178e32
.33181e131
jen j
.40256e2 0.6441e1
.46586e10 0.17738
.28794e42 0.17578
.82600e169 0.17579
.16126e1
.18635e9
.33117e41
.33040e168
en e4n1
f ðxn Þ
Table 2 Numerical results for fi ðxÞ; i ¼ 1; 2; 3; 4.
f1 : jxn C jf1 ðxn Þj f2 : jxn C jf2 ðxn Þj f3 : jxn C jf4 ðxn Þj f4 : jxn C jf4 ðxn Þj
aj
aj
aj
aj
NM
SM
STM
SSM
RM0
RM1
(11)
.19785e40 0.500 .98952e41 .13664e35 1.667 .40993e35 .29768e70 0.167 .89305e70 .54043e50 2.500 .10808e49
.88155e29 0.750 .44077e29 .21078e15 4.667 .63235e15 .31158e40 0.333 .93476e40 .19219e60 0.250 .38438e60
.40351e39 0.500 .20175e39 .24128e25 1.667 .72386e25 .30274e58 0.167 .90822e58 .12050e70 0.250 .24101e70
.72136e44 0.375 .36068e44 .43806e35 5.444 .13142e34 .64051e67 0.555e1 .19215e66 .21554e71 0.625e1 .43109e71
.16412e165 0.938 .82061e166 .31543e84 16.074 .94631e84 .15139e155 0.185e1 .45419e155 .67397e215 0.365e1 .13479e214
.25854e75 2.344 .12927e75 .69356e76 22.296 .20807e75 .48575e123 0.204 .14573e122 .17486e182 0.885e1 .34971e182
.11976e134 0.281 .59882e135 .27102e81 22.426 .81307e81 .10378e171 0.926e2 .31135e171 .94899e263 0.208e1 .18979e262
1982
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
1 x2 ðe 1Þ; a ¼ 2; x0 ¼ 2:5; 2 x2 þxþ2 f2 ðxÞ ¼ e 1; a ¼ 1; x0 ¼ 0:7; f1 ðxÞ ¼
f3 ðxÞ ¼ lnðx2 x þ 1Þ 4 sinðx 1Þ; a ¼ 1; x0 ¼ 0:5; f4 ðxÞ ¼ ex arctan x 1; a ¼ 0; x0 ¼ 0:5: The numerical results agree with the theoretical analysis on the error equation and its asymptotic convergence constant in the two tables. Now, we show its applications on systems of nonlinear equations in Table 3 and boundary-value problems of nonlinear ODEs in Table 4 by using the Steffensen-type method in Rn space as follows:
8 1 > < xkþ1 ¼ xk Jðxk ; Hk Þ Fðxk Þ; Jðxk ; Hk Þ ¼ ðFðxk þ Hk e1 Þ Fðxk Þ; . . . ; Fðxk þ Hk ek Þ Fðxk ÞÞH1 k ; > : Hk ¼ diagðf1 ðxk Þ; f2 ðxk Þ; . . . ; fn ðxk ÞÞ; k ¼ 0; 1; 2; . . .
ð24Þ
where FðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; . . . ; fn ðxÞÞ0 , and x ¼ ðx1 ; x2 ; . . . ; xn Þ0 . Example 3. For a system of nonlinear equations:
(
ðx1 1Þ4 þ ex2 x22 þ 3x2 þ 1 ¼ 0;
ð25Þ
4 sinðx1 1Þ lnðx21 x1 þ 1Þ x22 ¼ 0:
x0 ¼ ð0:8; 0:1Þ0 and x ð1:271384307950; 0:880819073102Þ0 . We have numerical results for the methods in Table 3, where STM with an ¼ 108 minðkF1i kÞkFk. Example 4. Consider to solve a boundary-value problem of ODEs as the following:
Table 3 The results for a system of nonlinear equations by the methods. Method
k
1
2
3
4
5
6
SM
kxk x k2 kFðxk Þk2
4.61909e1 6.70188e1
7.12980e2 1.21111e1
1.48194e3 3.07302e3
9.77722e7 1.85146e6
9.27307e13 6.94215e13
6.73973e13 9.65273e26
STM
kxk x k2 kFðxk Þk2
2.33251e1 7.81566e1
1.28406e2 2.94176e2
6.51851e5 1.22801e4
1.77131e9 3.41658e9
6.73973e13 2.50598e18
6.73973e13 1.36567e36
SSM
kxk x k2 kFðxk Þk2
1.47459e1 2.60665e1
5.87560e4 1.08318e3
2.97785e11 5.64369e11
6.73974e13 7.74837e33
RM0
kxk x k2 kFðxk Þk2
6.41762e1 1.33829
1.30813e1 2.36817e1
2.37921e5 4.11493e5
6.73973e13 3.86236e20
(11)
kxk x k2 kFðxk Þk2
6.47338e2 1.52073e1
1.68203e5 3.14660e5
6.73974e13 2.14176e19
6.73973e13 3.66306e59
Table 4 The results of the multiple shooting method by the methods. Method
n
1
2
3
4
SM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
5.5970e2 3.0898e2 6.9725e2
1.4912e4 6.4125e5 1.4348e4
9.9365e11 3.2621e6 4.9799e6
6.4395e15 3.2621e6 4.9799e6
STM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
3.8583e2 2.7290e2 4.5231e2
1.1509e4 7.3974e5 1.1617e4
1.9887e9 3.2629e6 4.9812e6
2.5438e16 3.2621e6 4.9798e6
SSM
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
1.3479e3 4.6681e4 2.1077e3
4.1452e12 3.2621e6 4.9798e6
1.2412e16 3.2621e6 4.9798e6
RM0
kFð~ sn Þk2 kyðtÞ yn k2 ky0 ðtÞ y0n k2
1.0177e1 3.8732e2 1.2041e1
7.8405e4 1.1790e4 5.1796e4
7.6978e9 3.2635e6 4.9868e6
(11)
kFð~ sn Þk2 kyðtÞ yn k2 0 ky ðtÞ y0n k2
4.1682e3 1.9092e3 3.3783e3
5.9014e11 3.2621e6 4.9798e6
2.7756e16 3.2621e6 4.9798e6
6.0809e16 3.2621e5 4.9798e5
Z. Liu et al. / Applied Mathematics and Computation 216 (2010) 1978–1983
(
y00 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y02 ;
yð0Þ ¼ 1; yð1Þ ¼ 12 ðe þ e1 Þ:
1983
ð26Þ
Generally, for a boundary-value problem of the type: y00 ðtÞ ¼ f ðt; yðtÞ; y0 ðtÞÞ; yðaÞ ¼ ya ; yðbÞ ¼ yb , we can describe briefly the multiple shooting method. Considering N subintervals, t i ¼ a þ i ðb aÞ=N ¼ a þ i h; 0 6 i 6 N; in each interval ½ti ; t iþ1 , the function yðt; s0 ; s1 ; . . . ; sN1 Þ is calculated in recursive form, by solving the initial-value problems y00 ðtÞ ¼ f ðt; yðtÞ; y0 ðtÞÞ; yðt i Þ ¼ yðt i ; s0 ; s1 ; . . . ; si1 Þ; y0 ðt i Þ ¼ si , with yðt0 Þ ¼ ya ; y0 ðt0 Þ ¼ s0 . The system of the initial-value problem of ODEs is solved by the classical Runge-Kutta method. We should deal with the system of nonlinear equations F : RN ! RN in the following form
8 F 1 ðs0 ; s1 ; . . . ; sN1 Þ ¼ s1 y0 ðt1 ; s0 Þ; > > > > > F 2 ðs0 ; s1 ; . . . ; sN1 Þ ¼ s2 y0 ðt2 ; s0 ; s1 Þ; > > < .. . > > > 0 > > > F N1 ðs0 ; s1 ; . . . ; sN1 Þ ¼ sN1 y ðt N1 ; s0 ; s1 ; . . . ; sN2 Þ; > : F N ðs0 ; s1 ; . . . ; sN1 Þ ¼ yb y0 ðt N ; s0 ; s1 ; . . . ; sN2 ; sN1 Þ:
ð27Þ
For solving (26), let N ¼ 4, the initial guess s0 ¼ 12 ðe þ e1 Þ 1 and an ¼ 108 minðkF1 kÞkFk in STM. we compute the error until i we obtain convergence ðkFð~ sn Þk2 < 1014 Þ. The numerical results by the multiple shooting method are obtained in Table 4. 4. Conclusions By theoretical analysis and numerical experiments, we confirm that the new variant of Steffensen’s method is free from any derivative and only uses three evaluations of the function per iteration to achieve fourth-order convergence for solving a 1=w simple rootpof ffiffiffi nonlinear functions. As the efficiency index is p , the efficiency ffiffiffi index of Newton’s method and Steffensen’s p 3 method is 2 ¼ 1:4142, the efficiency index of Steffensen-secant method is 3 ¼ 1:4422, and the efficiency index of Ren’s pffiffiffi method and the suggested method (11) is 3 4 ¼ 1:5874. The suggested method is very suitable to solve nonlinear equations, and is able to be applied on solving systems of nonlinear equations and boundary-value problems of nonlinear ODEs as well. Acknowledgements The work is supported by the Scientific Research Project of Beijing Union University (No. 2010) and in part by Natural Science Foundation of Beijing (No. 1072009). References [1] [2] [3] [4]
J.M. Ortega, W.G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations, Appl. Math. Comput. 177 (2006) 819–823. V. Alarcón, S. Amat, S. Busquier, D.J. López, A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216 (2008) 243–250. [5] P. Jain, Steffensen type methods for solving non-linear equations, Appl. Math. Comput. 194 (2007) 527–533. [6] H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206–210. [7] Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214 (2009) 10–16.