two step method without employing derivatives for solving a ...

Bulletin of Mathematics Vol. 04, No. 01 (2012), pp. 59–65.

TWO STEP METHOD WITHOUT EMPLOYING DERIVATIVES FOR SOLVING A NONLINEAR EQUATION

M. Imran, Agusni, A. Karma, S. Putra Abstract. We discuss an iterative method for finding root of a nonlinear equation employing central differences to avoid derivatives in the method. We show that this two step method is of order three. Numerical simulations show that the method is comparable with others third order methods.

1. INTRODUCTION Finding the root of a nonlinear equation, f (x) = 0, is a classic problem in numerical analysis. Recently, many iterative method have been developed to solve a nonlinear equation by combining two or more existing method [2, 6, 8, 10, 11]. Kasturiarachi [5] combines Newton’s method and secant method, that is x∗n = xn − xn+1 = xn −

f (xn ) f 0 (xn )

(1)

f 2 (xn ) . f 0 (xn )(f (xn ) − f (x∗n ))

(2)

Received 21-05-2012, Accepted 15-06-2012. 2000 Mathematics Subject Classification: 65D99, 65H05 Key words and Phrases: iterative method, Newton method, Steffensen like method, secant method, central difference.

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It is shown that this is a third order method. Combination of Secant method and Newton’s method has been discussed without analyzing the order of convergence of the method by Demidovich and Maron (See[3], Sec. 4.7). To avoid the derivative appearing in (1) and (2), Jain [4] approximates f 0 (xn ) =

f (xn + f (xn )) − f (xn ) f (xn )

so that equation(1) becomes x∗n = xn −

f 2 (xn ) f (xn + f (xn )) − f (xn )

(3)

which is Steffensen’s method, a good competitive for Newton’s method without employing a derivative in its formula. Using this strategy, he proposes the the third order method using three function evaluation per iteration as follows: f 2 (xn ) f (xn + f (xn )) − f (xn ) f 3 (xn ) . = xn − (f (xn + f (xn )) − f (xn ))(f (xn ) − f (x∗n ))

x∗n = xn − xn+1

(4) (5)

The aim of this paper is to propose an alternative method of order three without employing derivative in its formulae and to do some numerical comparisons with some available third order methods. 2. PRELIMINARY RESULTS Definition 2.1 (See [1]) A sequence of iterates {xn : n ≥ 0} is said to converge with order p ≥ 1 to a point α if |α − xn+1 | ≤ c|α − xn |p ,

n≥0

for some c > 0. If p = 1, the sequence is said to converge linearly to α. In that case, we require c < 1; the constant c is called the rate of linear convergence of xn to α. Definition 2.2 (See [11]) Let α be a root of the function f and suppose that xn−1 , xn , xn+1 are closer to the root α. Then the computational order of convergence p can be approximated using the formula p≈

ln |(xn+1 − α)/(xn − α)| ln |(xn − α)/(xn−1 − α)|

(6)

M. Imran et al. – Two-step Method

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3. PROPOSED METHOD If we approximate the derivative appearing in (1) and (2) with central difference, that is f 0 (xn ) =

f (xn + f (xn )) − f (xn − f (xn )) 2f (xn )

we have the following iterative formulae 2f 2 (xn ) f (xn + f (xn )) − f (xn − f (xn )) 2f 3 (xn ) . = xn − (f (xn + f (xn )) − f (xn − f (xn )))(f (xn ) − f (x∗n ))

x∗n = xn − xn+1

(7) (8)

We prove below that the iterative method (7) and (8) is of order 3. Theorem 3.1 Let f : D ⊂ R → R for an open interval D. Assume that f has first, second, and third derivatives in the interval D. If f has a simple root at α ∈ D and x0 is sufficiently close to α, then the new method defined by (7) and (8) satisfies the following error equation: en+1 = c22 e3n + O(e4n ), where en = xn − α and cj =

f (j) (α) , j! f 0 (α)

j = 2, 3.

Proof: Let α be a simple root of f (x) = 0, and xn = α + en . Denote Fi = f (i) (α), i = 1, 2, 3

so that

cj =

Fj , i = 2, 3. j!F1

Taylor expansion of f (xn ), f (xn + f (xn )) and f (xn − f (xn )) about x = α, which is a zero of f , is f (xn ) = F1 (en + c2 e2n + c3 e3n + O(e4n )),  h  f (xn + f (xn )) = F1 (1 + F1 )en + 3F2 2 + c2 + F12F2 e2n   i + 2F3 3 + c2 F2 + c3 + F12F3 + F36F1 e3n + O(e4n ) ,

(9)

(10)

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and h   F1 F2 2 f (xn − f (xn )) = F1 (−1 + F1 )en + −3F + c + e2n 2 2 2  i  F1 F3 F3 F1 3 + O(e4 ) . (11) 3 − c F + c + + e + −2F 2 2 3 n n 3 2 6 Using (9)–(11), we obtain after simplifying 2f 2 (xn ) = f (xn + f (xn )) − f (xn )

(2c3 +c22 )e3n +2c2 e2n +en +O(e4n ) c

1+3c2 en +(2c22 + 2F2 + 1

F1 F3 c +4c3 − 2F31 )+O(e4n ) 6

(12)

Recalling (1 + x)(1 − x + x2 − x3 + x4 − x5 + · · · + O(xn )) = 1 + O(xn )

(13)

and after some algebra we can express (12) as 2f 2 (xn ) f (xn +f (xn ))−f (xn )

= en − c2 e2n + − 2c3 −

F1 F3 6

+ 2c22 −

c2 2F1




e3n + O(e4n ). (14)

On substituting (14) into (7) and simplifying yields x∗n = α + c2 e2n − − 2c3 −

F1 F3 c2
3 + 2c22 − e + O(e4n ). 6 2F1 n

(15)

(xn ) Expanding f (x∗n ) about x = α, computing f (xnf)−f (x∗n ) , noting (13) and simplifying, we end up with   f (xn ) 2 + c2 + F1 F3 − c3 e2 = 1 + c e + 2c − 2c ∗ 2 n 3 n 2 2F1 6 2F1 f (xn )−f (xn )   2 c 2 c3 + − 2c32 + c2 F61 F3 − c2F + c2 c3 + 2F21 e3n + O(e4n ). (16) 1

Furthermore, adding (14) and (16), and simplifying, we obtain 2f 3 (xn ) (f (xn +f (xn ))−f (xn −f (xn )))(f (xn )−f (x∗n ))

= en − c22 e3n + O(e4n ).

On substituting (17) into (8), we obtain, the following equation error en+1 = c22 e3n + O(e4n ). This ends the proof.

(17)

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4. NUMERICAL COMPARISONS In this section we present numerical comparisons of proposed method (SS), Newton method (NM), Newton Secant method (NS) and Jain method (JM) to solve a nonlinear equation. We see the number of iteration of each method for a given accuracy and the COC of each method. We stop the iteration process if one of the following criteria satisfied (a) f 0 (xn ) = 0,

for NM and NS methods.

(b)

Maximum iteration=1000.

(c)

If |f (xn+1 )| < 10−10 .

(d)

If|xn+1 − xn | < xn+1 × 10−10 .

We use the following test functions: f1 (x) = x3 − 3x2 − 5, 4

3

α = 3.42598875736162212607, [5] 2

f2 (x) = x − 3x − 54x − 150x − 100, f3 (x) = 0.5 + sin(x), f4 (x) = x exp (2x),

α = −1.0, [7]

α = −0.52359877559829887307, [7] α = 0.0.

In the Table 1, NA stands for the method is not applicable, NC is the method is not convergent to the given root, and the start * in the number of iteration in the third nonlinear function indicates that the method is convergent to a different root. From Table 1, we see that for the first and fourth functions, in terms of the number of iteration, SS-method method is better than others mentioned methods. In most cases, The SS-method is better than JM-method, in term of the number of iteration. All the mention methods cannot differentiate between the roots and the asymptotic trend of the function as occurs for initial guesses −4.0, −2.0, −1.0 in the third nonlinear function. Overall SS-method is comparable with JM-method. Acknowledgements This research was supported by Laboratorium Research Program funded by DIPA Universitas Riau under contract number: 71/UN.19.2/PL/2011.

REFERENCES

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M. Imran et al. – Two-step Method

Table 1: Comparisons of the number of iterations and the COC of the discused methods f (x)

x3 − 3x2 − 5

x4 − 3x3 − 54x2 − 150x − 100

0.5 + sin(x)

x exp (2x)

x0 −2.0 −0.9 0.0 1.5 5.0 −1.9 −1.6 −1.2 −0.4 0.0 −1.0 −0.5 0.0 1.0 1.5 −4.0 −2.0 −1.0 −0.3 0.3

Number of Iterations NM NS JM SS 15 10 34 16 11 > 1000 22 13 NA NA 18 16 10 5 16 12 6 4 5 5 9 6 9 8 7 4 6 6 5 4 4 4 5 4 5 5 6 4 5 5 5 3 4 3 3 2 2 2 4 3 3 3 7* 4 4* 5* 6* 6* 3* 7* NC NC NC NC NC NC NC NC NC NC NC NC 8 5 7 5 6 4 4 4

NM 2.00 2.00 – 2.00 2.00 2.00 2.08 2.00 2.00 2.00 2.00 1.99 1.99 2.00 2.00 – – – 2.00 2.00

COC NS JM 2.86 3.06 – 3.01 – 2.95 3.01 2.99 2.99 2.86 3.12 3.20 3.12 3.13 3.13 3.30 2.99 3.09 2.96 2.65 2.48 3.03 INF 2.5 INF 2.4 3.17 3.51 2.49 2.62 – – – – – – 2.90 3.00 2.97 3.00

SS 3.15 3.20 3.23 3.35 3.24 3.35 3.29 3.49 3.24 3.54 3,41 INF INF 3.01 INF – – – 3.02 3.01

References [1] Atkinson, K. E. 1989. An Introduction to Numerical Analysis. Second Edition. John Wiley & Son, New York. [2] Chen M. and Chang T.S. 2008. Higher Order Two-Step Methods for Root Finding. Appl. Comput. Math. 7. 206–213 [3] Demidovich, B. P., and Maron, I. A., 1981. Computational Mathematics (English Translation). Third Edition. Mirs Publisher,Moscow. [4] Jain, P. 2007. Steffensen type methods for solving non-linear equations. Appl. Maths. Comp. 194: 527–533. [5] Kasturiarachi, A. B. 2002. Leap frogging Newton’s method. Int. J. Math. Educ. Sci. Technol. 33: 521–527.

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[6] Kumar S. Kanwan V. and Singh S. 2010. Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations. Applied Mathematics , 1, 153-158 [7] Nerinckx D. and Haegenans D. 1976. A Comparison of Non-linear Equation Solver. J. Comput. Appl. Math. 2: 145–148. [8] Noor, M.A., Ahmad F., and Javeed S. 2006. Two-Step Iterative Methods for Nonlinear Equations. Appl. Math. Comput. 181: 1068-1075. [9] Traub,J.F. 1964. Iterative Methods for Solution of Equations. Prentice Hall. Englewood Cliffs, New Jersey [10] Ujevic, N. 2006. A Method for Solving Nonlinear Equations, Appl. Math. Comput. 174: 1416-1426 [11] Weerakoon, S. and Fernando, T. G. I. 2000. A variant of Newton’s Method With Accelerated Third-Order Convergence. Applied Mathematics Letters. 13: 87–93.

M. Imran: Laboratorium Matematika Terapan, Jurusan Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Riau, Kampus Binawidya Pekanbaru (28293), Indonesia.

E-mail: [email protected] Agusni: Laboratorium Matematika Terapan, Jurusan Matematika, Fakultas Matem-

atika dan Ilmu Pengetahuan Alam, Universitas Riau, Kampus Binawidya Pekanbaru (28293), Indonesia.

E-mail: [email protected] A. Karma: Laboratorium Matematika Terapan, Jurusan Matematika, Fakultas

Matematika dan Ilmu Pengetahuan Alam, Universitas Riau, Kampus Binawidya Pekanbaru (28293), Indonesia.

E-mail: [email protected] S. Putra: Laboratorium Matematika Terapan, Jurusan Matematika, Fakultas

Matematika dan Ilmu Pengetahuan Alam, Universitas Riau, Kampus Binawidya Pekanbaru (28293), Indonesia.

E-mail: [email protected]