NEW ITERATIVE METHODS BASED ON SPLINE FUNCTIONS FOR SOLVING NONLINEAR EQUATIONS

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1 Bulletin of Mathematical Analysis and Applications ISSN: , URL: Volume 3 Issue 4(011, Pages NEW ITERATIVE METHODS BASED ON SPLINE FUNCTIONS FOR SOLVING NONLINEAR EQUATIONS (COMMUNICATED BY MARTIN HERMANN HADI TAGHVAFARD Abstract. Two new iterative methods for solving nonlinear equations are presented using a new quadrature rule based on spline functions. Analysis of convergence shows that these methods have third-order convergence. Their practical utility is demonstrated by numerical examples to show that these methods are more efficient than that of Newton s. 1. Introduction Solving nonlinear equations is one of the most predominant problems in numerical analysis. Newton s method is the most popular one in solving such equations. Some historical points on this method can be found in [14]. Recently, some methods have been proposed and analyzed for solving nonlinear equations [1,, 4, 6, 7, 8]. These methods have been suggested by using quadrature formulas, decomposition and Taylor s series [3, 8, 9, 13]. As we know, quadrature rules play an important and significant role in the evaluation of integrals. One of the most well-known iterative methods is Newton s classical method which has a quadratic convergence rate. Some authors have derived new iterative methods which are more efficient than that of Newton s [5, 9, 1, 13]. This paper is organized as follows. Section provides some preliminaries which are needed. Section 3 is devoted to suggest two iterative methods by using a new quadrature rule based on spline functions. These are implicit-type methods. To implement these methods, we use Newton s and Halley s method as a predictor and then use these new methods as a corrector. The resultant methods can be considered as two-step iterative methods. In section 4, it will be shown that these two-step iterative methods are of third-order convergence. In section 5, a comparison between these methods with that of Newton s is made. Several examples are given to illustrate the efficiencies and advantages of these methods. Finally, section 6 will close the paper. 000 Mathematics Subject Classification. 6A18, 65D05, 65D07, 65D15, 65D3. Key words and phrases. Newton s method, Halley s method, Iterative method, Spline functions, Order of convergence. c 011 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted April 5, 011. Published August 9,

2 3 HADI TAGHVAFARD We use the following definition [13]:. Preliminaries Definition 1. Let α R and x n R, n = 0, 1,,.... Then the sequence x n is said to be convergence to α if lim x n α = 0. n If there exists a constant c > 0, an integer n 0 0 and p 0 such that for all n > n 0, we have x n+1 α c x n α p, then x n is said to be convergence to α with convergence order at least p. If p = or p = 3, the convergence is said to be quadratic or cubic, respectively. Notation 1. The notation e n = x n α is the error in the n th iteration. The equation e n+1 = ce p n + O(e p+1 n, is called the error equation. By substituting e n = x n α for all n in any iterative method and simplifying, we obtain the error equation for that method. The value of p obtained is called the order of this method. We consider the problem of numerical determine a real root α of nonlinear equation f(x = 0, f : D R R. (.1 The known numerical method for solving equation (.1 is the classical Newton s method given by f(x n f, n = 0, 1,,... (x n where x 0 is an initial approximation sufficiently close to α. The convergence order of the classical Newton s method is quadratic for simple roots [3]. If the second order derivative f(x n is available, the third-order convergence rate can be achieved by using the Halley s method [5]. Its iterative formula is f(x n f (x n (f (x n f(x n f, n = 1,, 3,.... (x n 3. Iterative method For several reasons, we know that cubic spline functions are popular spline functions. They are smooth functions with which to fit data, and when used for interpolation, they do not have the oscillatory behavior that is the characteristic of high-degree polynomial interpolation. We have derived a new quadrature rule which is based on spline interpolation [11]. In this section, we review it briefly and then present the iterative methods. Let us suppose = {x i i = 0, 1, } be a uniform partition of the interval [a, b] by knots a = x 0 < a+b = x 1 < b = x and h = x 1 x 0 = x x 1 and y i = f(x i ; then the cubic spline function S (x which interpolates the values of the function f at the knots x 0, x 1, x and satisfies S (a = S (b = 0 is readily characterized by their moments, and these moments of interpolating cubic spline function can

3 NEW ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS 33 be calculated as the solution of a system of linear equations. We can obtain the following representation of the cubic spline function in terms of its moments [10]: S (x = α i + β i (x x i + γ i (x x i + δ i (x x i 3, (3.1 for x [x i, x i+1 ], i = 0, 1, where α i = y i, β i = y i+1 y i h M i M i+1 h, 6 γ i = M i+1 M i 6h M 0 = M = 0, M 1 = 3 + b [f(a f(a h + f(b]. Now from (3.1 we obtain, b a and thus a+b b S (xdx = S (xdx + S (xdx a+b a = h [ f(a + f( a + b ] + f(b h3 4 [M 0 + M 1 + M ] = h [ 3f(a + 10f( a + b ] 8 + 3f(b. b a f(tdt b a 16 { 3f(a + 10f ( a + b } + 3f(b. (3. We use the above quadrature rule to approximate integrals and use it to obtain an iterative method. In order to do this, let α D be a simple zero of sufficiently differentiable function f : D R R for an open interval D and x 0 is sufficiently close to α. To derive the iterative method, we consider the computation of the indefinite integral on an interval of integration arising from Newton s theorem f(x = f(x n + x x n f (tdt. (3.3 Using the quadrature rule (3. to approximate the right integral of (3.3, x f (tdt = x x { ( } n 3f (x n + 10f xn + x + 3f (x, (3.4 x n 16 and looking for f(x = 0. From (3.3 and (3.4 we obtain x = x n 3f (x n + 10f ( x+x n + 3f (x. (3.5 This fixed point formulation enables us to suggest the following implicit iterative method (, (3.6 3f (x n + 10f xn +x n+1 + 3f (x n+1 which requires the (n+1th iterate x n+1 to calculate the (n+1th iterate itself. To obtain the explicit form, we make use of the Newton s iterative step to compute the (n + 1th iterate x n+1 on the right-hand side of (3.6, namely replacing f( xn+xn+1 with f( x n+y n, where y n = x n f(x n f (x n following explicit method: is the Newton iterate and obtain the

4 34 HADI TAGHVAFARD Algorithm 1. For a given x 0, compute the approximate solution x n+1 by the following iterative scheme where 3f (x n + 10f ( x n+y n, n = 0, 1,, f (y n y n = x n f(x n f (x n. We can replace x n+1 in the right hand-side of (3.6 by Halley s method. Therefore we suggest the following algorithm: Algorithm. For a given x 0, compute the approximate solution x n+1 by the following iterative scheme where 3f (x n + 10f ( x n +y n y n = x n + 3f (y n, n = 0, 1,,... (3.7 f(x n f (x n (f (x n f(x n f (x n. ( Analysis of convergence In this section, we prove the convergence of Algorithm. In a similar way, one can prove the convergence of Algorithm 1. Theorem 4.1. Let α D be a simple zero of sufficiently differentiable function f : D R R for an open interval D. If x 0 is sufficiently close to α, then the two-step iterative method defined by Algorithm converges cubically to α in a neighborhood of α and it satisfies the error equation ( 7 e n+1 = 16 c e 3 n + O(e 4 n, where c k = 1 f k (α k! f (α, k = 1,, 3,... and e n = x n α. Proof. Let α be a simple zero of f, i.e. f(α = 0 and f (α 0. Since f is sufficiently differentiable, by using Taylor s expansion of f(x n, f (x n and f (x n about α, we obtain f(x n = f (α(x n α + 1! f (α(x n α + 1 3! f (α(x n α ! f (4 (α(x n α ! f (5 (α(x n α 5 + O((x n α 6 = f (α [ e n + c e n + c 3 e 3 n + c 4 e 4 n + c 5 e 5 n + O(e 6 n ], (4.1 f (x n = f (α [ 1 + c e n + 3c 3 e n + 4c 4 e 3 n + 5c 5 e 4 n + O(e 5 n ], (4. f (x n = f (α [ c + 6c 3 e n + 1c 4 e n + 0c 5 e 3 n + O(e 4 n ], (4.3 where c k = 1 f k (α k! f (α, k = 1,, 3,... and e n = x n α. Form (4.1, (4. and (4.3 one obtains f(x n f (x n = f (α[e n + 3c e n + (4c 3 + c e 3 n + O(e 4 n] (4.4 f (x n = f (α[1 + 4c e n + (6c 3 + 4c e n + (1c c 3 + 8c 4 e 3 n + O(e 4 n](4.5 f(x n f (x n = f (α[c e n + (6c 3 + c e n + (8c c 3 + 1c 4 e 3 n + O(e 4 n] (4.6

5 NEW ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS 35 Substituting (4.4, (4.5 and (4.6 into (3.8, we get Now, from (4.7 we have y n = x n x n + y n f(x n f (x n (f (x n f(x n f (x n = α + (c c 3 e 3 n + O(e 4 n (4.7 = x n f(x n f (x n (f (x n f(x n f (x n = α + 1 e n + 1 (c c 3 e 3 n + O(e 4 n (4.8 Substituting (4.7 and (4.8 into the Taylor s expansion of f(y n, f (y n and f ( x n+y n about α and get f(y n = f (α(y n α + 1! f (α(y n α + 1 3! f (α(y n α ! f (4 (α(y n α 4 + O((y n α 5 = f (α [ (c c 3 e 3 n + ( 3c 3 + 6c c 3 3c 4 e 4 n + O(e 5 n ], (4.9 f (y n = f (α[1 + c (y n α + c 3 (y n α + c 4 (y n α 3 + O((y n α 4 ] = f (α[1 + (c 3 c c 3 e 3 n + O(e 4 n], (4.10 f ( x n + y n = f (α[1 + c e n + (c 3 c c 3 e 3 n + O(e 4 n] (4.11 By substituting (4., (4.10 and (4.11 into (3.7, we get and thus 3f (x n + 10f ( x n+y n + 3f (y n ( 7 = α + 16 c e 3 n + O(e 4 n. e n+1 = ( 7 16 c e 3 n + O(e 4 n. This means that the method defined by algorithm is cubically convergent. Theorem 4.. Let α D be a simple zero of sufficiently differentiable function f : D R R for an open interval D. If x 0 is sufficiently close to α, then the two-step iterative method defined by Algorithm 1 converges cubically to α in a neighborhood of α and it satisfies the error equation ( 3 e n+1 = 8 c c 1 3 c 3 e 3 n + O(e 4 n, where c n = 1 f n (α n! f (α, n = 1,, 3,... and e n = x n α. Proof. Similar to the proof of Theorem 4.1.

6 36 HADI TAGHVAFARD 5. Numerical examples In this section, we employ the methods obtained in this paper to solve some nonlinear equations and compare them with the Newton s method (NM. We use the stopping criteria x n+1 x n < ϵ and f(x n+1 < ϵ, where ϵ = 10 14, for computer programs. All programs are written in Matlab. In Table 1, the number of iterations (NI and function evaluations (NFE is given so that the stopping criterion is satisfied. In table, the comparison of the number of operations needed for obtaining solutions for examples 1 and is given. We use the following test functions and display the approximate zeros x found up to the 14 th decimal place. Example1 f 1 (x = x 3 x + 3, x = Example f (x = x 3 + 4x 10, x = Example3 f 3 (x = cos(x x, x = Example4 f 4 (x = xe x sin (x + 3 cos(x + 5, x = Table 1. Comparison of number of iterations and function evaluations of different iterative methods. NI NFE f(x x 0 NM Alg.1 Alg. NM Alg.1 Alg. f 1 (x f (x f 3 (x π/ f 4 (x Table. Comparison of number of operations of different iterative methods. + or or f(x x 0 NM Alg.1 Alg. NM Alg.1 Alg. f 1 (x f (x In Tables 1 and, it is shown that in some cases, algorithms 1 and are better than the Newton s method for solving nonlinear equations. It is observe that in some cases the new methods require less iteration and function evaluation than that of Newton s. Moreover, as you see in Table, the new methods require less number of operations than Newton s method. 6. Conclusion We derived two iterative methods based on spline functions for solving nonlinear equations. Convergence proof is presented in details for algorithm. In Theorem 4.1 and 4., we proved that the order of convergence of these methods is three. Analysis of efficiency showed that these methods are preferable to the well-known Newton s method. From numerical examples, we showed that these methods have great practical utilities.

7 NEW ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS 37 Acknowledgement The author would like to thank the referees for their constructive suggestions and criticisms that led to an improved presentation. References [1] S. Abbasbandy, Improving Newton?-Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145, (003, 887?-893. [] S. Amat, S. Busquier, J.M. Gutierrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157, ( ?-05. [3] K. E. Atkinson, An introduction to numerical analysis, nd ed., John Wiley & Sons, New York, [4] E. Babolian, J. Biazar, Solution of nonlinear equations by adomian decomposition method, Appl. Math. Comput. 13, (00 167?-17. [5] H. Bateman, Halley s method for solving equations, The American Mathematical Monthly, Vol. 45, No. 1, (1938, [6] F. Costabile, M.I. Gualtieri, S.S. Capizzano, An iterative method for the solutions of nonlinear equations, Calcolo 30, ( ?-34. [7] M. Frontini, Hermite interpolation and a new iterative method for the computation of the roots of non-linear equations, Calcolo 40, ( ?-119. [8] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149, (004, 771?-78. [9] A.Y. Ozban, Some new variants of Newton s method, Appl. Math. Lett. 17, (004, 677?-68. [10] J. Stoer, R. Bulrisch, Introduction to Numerical Analysis, Second edition, Springer Verlag, [11] H. Taghvafard, A New Quadrature Rule Derived from Spline Interpolation with Error Analysis, Int. J. Comput. Math. Sci., 5(3, (011, 119?-14. [1] J. Traub, Iterative methods for the solution of equations, Prentice?Hall, Englewood Cliffs, New Jersey, [13] S. Weerakoon, T. G. I. Fernando, A Variant of Newton s Method with Accelerated Third- Order Convergence, Appl. Math. Lett., 13, (000, 87?-93. [14] T. Yamamoto, Historical development in convergence analysis for Newton s and Newton-like methods, J. Comput. Appl. Math., 14, (000, 1 3. Hadi Taghvafard Department of Mathematics, Shiraz University, Shiraz, Iran. address: taghvafard@gmail.com