Improving homotopy analysis method for system of nonlinear algebraic equations

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1 Journal of Advanced Research in Applied Mathematics Vol., Issue. 4, 010, pp. -30 Online ISSN: Improving homotopy analysis method for system of nonlinear algebraic equations M.M. Hosseini, S.M. Hosseini Faculty of Mathematics, Yazd University, P.O. Box , Yazd, Iran. Abstract. In this paper, a new iterative method for solving the system of nonlinear equations is suggested based on Newton-Raphson method and homotopy analysis method. Comparison of the result obtained by present method with Newton-Raphson method and homotopy perturbation method reveals that the accuracy and fast convergence of the proposed method. Keywords: Homotopy analysis method; System of nonlinear equations; Newton-Raphson method. Mathematics Subject Classification 010: 65H10. 1 Introduction One of the oldest and most basic problems in mathematics is that of solving an nonlinear system of equations ϕ (X) = 0. This problem has motivated many theoretical developments including the fact that solution formulas do not in general exist. Thus, the development of algorithms for finding solutions has historically been an important enterprise. Newton-Raphson method 8 is the most popular technique for solving nonlinear system of equations. Many topics related to Newton s method still attract attention from researchers. As is well-known, a disadvantage of the method is that the initial approximation X 0, must be chosen sufficiently close to a true solution in order to guarantee their convergence. Finding a criterion for choosing X 0 is quite difficult and therefore effective and globally convergent algorithms are needed 3. In recent years, several iterative methods have been developed to solve the nonlinear system of equations ϕ (X) = 0, by using Adomian s decomposition method 1,7,13,14, homotopy perturbation method 16,17, quadrature formulas 3,6,9,10,1,15 and other techniques 11,31. In 199, Liao 9 employed the basic ideas of homotopy to propose a general method Correspondence to: M.M. Hosseini, Faculty of Mathematics, Yazd University,Yazd, Iran. hosse m@yazduni.ac.ir Received: 13 February 010, revised: 13 May 010, accepted: 6 July c 010 Institute of Advanced Scientific Research

2 M.M. Hosseini and S.M. Hosseini 3 for nonlinear problems, namely the homotopy analysis method. This method and its modifications and extensions have been successfully applied to solve many types of nonlinear problems,5,18-8,30. In 007, Abbasbandy et al., employed homotopy analysis method to solve some nonlinear algebraic equations. Furthermore, in, an efficient algorithm named Newton- homotopy analysis method was presented based on Newton-Raphson method and homotopy analysis method and applied to solve nonlinear algebraic equations. In this work, homotopy analysis method and Newton-homotopy analysis method are generalized to the system of nonlinear equations ϕ (X) = 0. Several examples are presented and compared to other methods, showing the accuracy and fast convergence of the proposed methods. The homotopy analysis method Consider the system of nonlinear equations: ϕ(x) = { f(x, y) = 0, g(x, y) = 0, X = (x, y) R (.1) where f, g : R R and ϕ : R R. Here, homotopy analysis method is proposed to solve Eq.(.1). Let q 0, 1 denotes an embedding parameter, and h, k nonzero auxiliary parameters. We construct a homotopy H : R R which satisfies H( X(q); q) = { (1 q) ( f( X(q)) f(x0 ) ) = qhf( X(q)), (1 q) ( g( X(q)) g(x 0 ) ) = qhg( X(q)), X(q) R, q 0, 1 (.) where X 0 = (x 0, y 0 ) is an initial approximation of Eq. (.1). Since h, k 0 it is obvious that and H ( X(0); 0 ) = ϕ ( X(0) ) ϕ (X0 ) = 0 (.3) H ( X(1); 1 ) = ϕ ( X(1) ) = 0. (.4) When q increases from 0 to 1 as (.3) is continuously deformed to original problem H ( ) ( ) X(1); 1 = ϕ X(1) = 0. The homotopy analysis method uses Taylor s theorem to expanding X(q) with respect to the q, one has where { x(q) = x0 + + X(q) = m=1 x mq m, ȳ(q) = y m=1 y mq m, q=0 (.5) x m = 1 d m x(q) m! dq m y m = 1 d m ȳ(q) (.6) q=0 m! dq m

3 4 Improving homotopy analysis method for system of nonlinear algebraic equations If the initial guess, and the auxiliary parameters h, k are so properly chosen that the series (.5) converges at q = 1, one has { limq 1 x(q) = x X = lim X(q) = m=1 x m, q 1 lim q 1 ȳ(q) = y m=1 y m, (.7) which must be one of the solutions of Eq.(.1). Taking the 1st-order derivative with respect to the q on both sides of equation (.) and setting q = 0, we have { x1 f x (X 0 ) + y 1 f y (X 0 ) = hf(x 0 ) x 1 g x (X 0 ) + y 1 g y (X 0 ) = kg(x 0 ) whose solution is x1 y 1 = 0 ) f y (X 0 ) g x (X 0 ) g y (X 0 ) 1 hf(x0 ) kg(x 0 ) Taking the second order derivative with respect to the q on both sides of equation (.) and then dividing them by! and finally setting q = 0, we have { x f x (X 0 ) + y f y (X 0 ) = 1 x 1 f xx (X 0 ) + x 1 y 1 f xy (X 0 ) + y1 f yy(x 0 ) A x g x (X 0 ) + y g y (X 0 ) = 1 x 1 g xx (X 0 ) + x 1 y 1 g xy (X 0 ) + y1 g yy(x 0 ) B (.8) where and and whose solution (.8) is x y = 0 ) f y (X 0 ) g x (X 0 ) g y (X 0 ) A = (h + 1) (x 1 f x (X 0 ) + y 1 f y (X 0 )) B = (k + 1) (x 1 g x (X 0 ) + y 1 g y (X 0 )) (.9) y x 1 f xx (X 0 ) + x 1 y 1 f xy (X 0 ) + y1 f yy(x 0 ) A x 1 g xx (X 0 ) + x 1 y 1 g xy (X 0 ) + y1 g yy(x 0 ) B 1 Substituting all the above terms into (.7), we can obtain the solution of Eq.(.1) as follows: 1 x0 X = + 0 ) f y (X 0 ) hf(x0 ) g x (X 0 ) g y (X 0 ) kg(x 0 ) 0 ) f y (X 0 ) g x (X 0 ) g y (X 0 ) 1 1 x 1 f xx (X 0 ) + x 1 y 1 f xy (X 0 ) + y1 f yy(x 0 ) A x 1 g xx (X 0 ) + x 1 y 1 g xy (X 0 ) + y1 g yy(x 0 ) B + 1 The above formulations allow us to suggest the following iterative methods for solving system of nonlinear equations (.1).

4 M.M. Hosseini and S.M. Hosseini 5 Algorithm 1. A.1( ). Put = h = k and for a given X 0, compute the approximate solution X i+1 by the iterative scheme xi+1 y i+1 = = xi y i xi y i + i ) f y (X i ) g x (X i ) g y (X i ) + i ) f y (X i ) g x (X i ) g y (X i ) 1 hf(xi ) kg(x i ) 1 f(xi ) g(x i ) (.10) (.11) Obviously, Newton-Raphson method 8 is special case of (.11) when = 1. The values of can be determined by plotting the so-called -curves, as suggested by Liao 7. Algorithm. A.( ). Put = h = k and for a given X 0, compute the approximate solution X i+1 by the iterative scheme ( 1 1 li xi+1 y i+1 m i = = i ) f y (X i ) g x (X i ) g y (X i ) xi y i li + m i 1 f(xi ) g(x i ) l i f xx (X i ) + l i m i f xy (X i ) + m i f yy(x i ) l i g xx (X i ) + l i m i g xy (X i ) + m i g yy(x i ) ( + 1), 1 i ) f y (X i ) g x (X i ) g y (X i ) (li f x (X i ) + m i f y (X i )) (m i fg x (X i ) + m i g y (X i )) Note that the above method for = 1 is the same as modified homotopy perturbation method which was presented in 16. The values of can be determined with by plotting the so-called -curves. ) 3 Newton-homotopy analysis method In (A.1( )) and (A.( )), we put = h = k as a fixed constant and it can be determined by -curves. However, in Newton-homotopy analysis method, we determine h, k by Newton-Raphson scheme in each step as follows. Let X 0 = (x 0, y 0 ) and 0 = (h 0, k 0 ) are the initial values for X = (x, y) and = (h, k), respectively. From (.10), we have where { xi+1 = x i + ha i + kb i y i+1 = y i + hc i + kd i a i = g y (X i )f(x i ) f x (X i )g y (X i ) f y (X i )g y (X i ), b f y (X i )g(x i ) i = f x (X i )g y (X i ) f y (X i )g y (X i ), g x (X i )f(x i ) c i = f x (X i )g y (X i ) f y (X i )g y (X i ), d i = f x (X i )g(x i ) f x (X i )g y (X i ) f y (X i )g y (X i ).

5 6 Improving homotopy analysis method for system of nonlinear algebraic equations After computing X i+1, we want to renew h i, k i by Newton-Rafson scheme on ψ i ( ) = { F ( ) = f(xi+1, y i+1 ) = 0, G( ) = g(x i+1, y i+1 ) = 0, = (h, k) R. Hence i+1 = i ψ i( i ) 1 ψ i ( i ), i = 0, 1,,, where ψ i ( i) is the Jacobian matrix at the point i = (h i, k i ) and 0 = ψ 0(0) 1 ψ 0 (0), Therefore, this method does not need to choose = (h, k) and has better numerical behavior. Some results are listed in Tables Numerical examples In this section, we present some numerical results to illustrate the efficiency of the new methods proposed in this paper. We present some numerical test results for various wellknown schemes in Tables -5. Compared were modified homotopy perturbation method (MHPM) 16, Newton-Raphson method (N-R) 8 and the Newton-Homotopy analysis method (N-HAM) which introduced in the present contribution. All computations are done by Maple 1 with 64 digits precision. We accept an approximate solution rather than the exact root, depending on the precision (ε) of the computer. The following stopping criteria is used for computer programmes: (i) X i+1 X i < ε, and (ii) ϕ(x i ) < ε. For every method, we analyze the number of iterations needed to converge to the solution. Furtermore, the computional order of convergence p is approximated by means of p ln ( X i+1 X i X i+1 X i ) ln ( X i X i 1 X i 1 X i ). Example 4.1. System of nonlinear equations { sin (x) + cos(y) = 1 3e x cos(y) + sin (x) = 1 Starting with the same initial value X 0 = ( π 6, π 3 )t and using Algorithm, A.( ), we get a number of roots in Table 1. Here stopping criteria is X i+1 X i < Note that, using Newton s method, after 5 iterations we get approximate solution X 5 = (6.7035, ) t with ϕ(x 5 ) <

6 M.M. Hosseini and S.M. Hosseini 7 Table 1. Multiple-roots given by HAM, A.( ) X IN IN ϕ(x 5 ) -1 ( , ) t ( , ) t ( , ) t ( , ) t ( , ) t Example 4.. Consider the following system of nonlinear equations1: { x + y = 0, e x 1 + y 3 = 0, with exact solution (1, 1) t and the same initial value X 0 = (.5,.5) t, Table. Example 4.3. Consider the following system of nonlinear equations: { x 4 y xy + x y 1 = 0, ye x + x y e 1 = 0, with exact solution (1, 1) t and the same initial value X 0 = (5.4, 4.) t, Table 3. Example 4.4. Consider the following system of nonlinear equations3: { x x y = 0, x + 4y 4 = 0, with exact solution ( , ) t value X 0 = (4, 6) t, Table 4. and the same initial Example 4.5. Consider the following system of nonlinear equations: { x + 3 log(x) y = 0, x xy x + 1 = 0, with exact solution ( , ) t and the same initial value X 0 = (8, 8) t, Table 5.

7 8 Improving homotopy analysis method for system of nonlinear algebraic equations Table, for example. Table 3, for example3. Method IN p ϕ(x IN ) Method IN p ϕ(x IN ) N-R N-R MHPM MHPM N-HAM N-HAM Table 4, for example4. Table 5, for example5. Method IN p ϕ(x IN ) Method IN p ϕ(x IN ) N-R N-R MHPM MHPM N-HAM N-HAM Numerical results for Examples and comparison Example 4.6. Consider the following system of nonlinear equations4: 3x cos(yz) 0.5 = 0, x 81(y + 0.1) + sin(z) = 0, e xy + 0z + 10π 3 3 = 0 with exact solution (0, 0.5, ) t and the same initial value (5, 5, 3) t. Note that, using Newton-Raphson method, after 10 iterations we get approximate solution (0, 0.5, ) t with ϕ(x 10 ) < Table 3. Numerical results given by N-HAM for 4.6 IN ϕ(x IN ) In Tables and 3, IN denotes the number of iteration and denotes the computational order of convergence, defined in Conclusion In this paper, homotopy analysis method and Newton-homotopy analysis method have been applied to solve the system of nonlinear equations. The Newton-homotopy analysis method was compared in performance to the homotopy perturbation method and Newton-Raphson method. The proposed method is both effective and convenient. Furthermore, it has been shown that the homotopy perturbation method for solving nonlinear system of equations is exactly especial case of that given by homotopy analysis method.

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