Some Third Order Methods for Solving Systems of Nonlinear Equations

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1 Some Third Order Methods for Solving Systems of Nonlinear Equations Janak Raj Sharma Rajni Sharma International Science Index, Mathematical Computational Sciences waset.org/publication/1595 Abstract Based on Traub s methods for solving nonlinear equation fx = 0, we develop two families of third-order methods for solving system of nonlinear equations Fx = 0. The families include well-known existing methods as special cases. The stability is corroborated by numerical results. Comparison with well-known methods shows that the present methods are robust. These higher order methods may be very useful in the numerical applications requiring high precision in their computations because these methods yield a clear reduction in number of iterations. Keywords Nonlinear equations systems, Newton s method, fixed point iteration, order of convergence. I. INTRODUCTION Solving systems of nonlinear equations is a common important problem in science engineering [1], i.e. for a given nonlinear function F : D R n R n, to find a vector r = r 1,r 2,...,r n such that Fr = 0. This solution can be obtained as a fixed point of some function φ : R n R n by means of fixed point iteration x k1 = φx k, k = 0,1,... One of the basic procedures for approximating a solution of nonlinear equation fx = 0, is the quadratically convergent Newton s method defined as x k1 = x k fx k f, k = 0,1,... 1 x k For systems of nonlinear equations, Newton s method is given as see [2, 3] x k1 = x k J F x k 1 Fx k, 2 where J F x is the jacobian matrix of the function Fx J F x 1 is its inverse. For a system of n equations in n unknowns, the first Fréchet derivative is a matrix with n 2 evaluations while the second Fréchet derivative has n 3 evaluations. The methods like Halley Chebyshev [4, 5], despite their cubic convergence, are considered less practical from a computational point of view because of costly second derivative. Recently, many third order iterative methods have been derived analyzed for systems of nonlinear equations that do not require the computation of second Fréchet derivative. For Janak Raj Sharma is with Department of Mathematics, Sant Longowal Institute of Engineering Technology, Longowal , Sangrur, India. jrshira@yahoo.co.in Rajni Sharma is with Department of Applied Sciences, D.A.V. Institute of Engineering Technology, Jalhar , India. rajni daviet@yahoo.com example, Homeier [6] developed a modified Newton method of order three. Frontini et al. [7], Cordero et al. [8] Noor et al. [9] developed third order methods from quadrature formulae. Darvishi et al.[10] presented the methods of same order using Adomian decomposition technique. Hueso et al. [11] Lin et al. [12] also introduced iterative methods of same order for systems of nonlinear equations. Some authors have considered Traub s third order methods [2] for univariate case generalized to multivariate case. For example, Hernández [13] extended a member of the family given by Traub [2] { 2β 1f x k f } x k βux k x k1 = x k 2βf ux k, x k β 0, ux k = fx k f x k, 3 in Banach Space for β = 1/2. Recently, Babajee et al. [14] extended another member β = 1 of this family to systems of equations. Based on the ongoing work in this direction, here, we extend the family 3 to systems of equations for any β R. We consider another one-parameter family of same order given by Traub [2] x k1 = x k 2βfx k 2β 1f x k f x k βux k, 4 generalize this to systems of equations for any β R. The resulting family includes the methods by Homeier [6] Noor et al. [9] as particular cases. The paper is organized in 6 sections. Some basic results relevant to the present work are presented in Section 2. In Section 3, the schemes are developed behavior is analyzed. In Section 4, particular cases of the families are presented. In Section 5, new methods are compared with closest competitors in a series of numerical examples. Section 6, contains the concluding remarks. II. BASIC RESULTS AND NOTATIONS We consider the following results: A Let φx be a fixed point function with continuous partial derivatives of order p with respect to all components of x. The iterative method x k1 = φx k is of order p see [2] if φr = r; k φ j r i1 i2... ik = 0, for all 1 k p 1, 1 j,i 1,i 2,...,i k n; 1864

2 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 p φ j r i1 i2... ip = 0, for at least one value of j,i 1,i 2,...,i p, 5 Exping φ j x in a Taylor series about r yields φ j x = φ j r where 1 3! i 1 =1 i 1 =1i 2 =1i 3 =1 φ j r i1 e i1 1 2! i 1 =1i 2 =1 2 φ j r i1 i2 e i1 e i2 3 φ j r i1 i2 i3 e i1 e i2 e i3..., 6 e il = x il r il, Thus if φx is an iteration function of order p, then by equation 5, we have φ j x r j 1 p!... i 1 =1i 2 =1 the quantity i p=1 p φ j r i1 i2... ip e i1 e i2...e ip. 7 1 p φ j r. p! i1 i2... ip is called the asymptotic error constant. B Let J ij x denotes the i,j entry of the matrix J F x the elements of J F x 1 are denoted by H ij x. Then, H ij xj jk x = δ ik, 8 where δ ik is Kronecker s delta, defined as: { 1 if i = k; δ ik = 0 if i =k. Let i q be arbitrary fixed. Then from 8, it follows that H ij x J jkx H ij x J jk x = 0. 9 q q Thus, we have H ij x J jk x = q = H ij x J jkx q H ij x 2 f j x k q. 10 Differentiating 9 partially with respect to x r, r being arbitrary fixed, one can easily obtain 2 H ij x H J jk x = ij x J jk x r q r q H ij x J jk x H ij x 2 J jk x. 11 q r r q C We use the following notations first two are introduced by Traub [2]: Z iqr x = H ij x 2 f j x, 12 q r Z iqrs x = Z iqk xz krs x [ n = H il x 2 f l x n ] H km x 2 f mx, 13 l=1 q k m=1 r s Z iqrs x = 3 f j x H ij x, 14 q r s where the last two subscripts of Z iqr x the last three of Z iqrs x may be permuted. The Z iqrs x can also be simplified to using 8 10 Z iqrs x = m=1 H im x 2 f m x. 15 q r s III. DEVELOPMENT OF METHODS For system of equations, the families 3 4 can be written as y k = x k βj F x k 1 Fx k, x k1 = x k 1 2β J Fx k 1 [2β 1J F x k J F y k ]J F x k 1 Fx k 16 y k = x k βj F x k 1 Fx k, x k1 = x k 2β[2β 1J F x k J F y k ] 1 Fx k. 17 Now, we shall prove convergence of The following lemmas, will be useful in the proof of main theorems: Lemma 1. Let λx = xβj F x 1 Fx, 18 λ i x = x i β H ij xf j x, 19 be the coordinate functions of λx for i = 1,2,...,n. Then, λ i r = r i, 20 i r q = 1βδ iq, 21 2 λ i r r q = βz iqr r. 22 Proof. Since f j r = 0, equation 19 implies that λx is fixed point function. i.e. λ i r = r i. 1865

3 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 Let i q be arbitrary fixed. Differentiating 19 partially with respect to x q, one obtains i x = i β H ij x f jx q q q H ij x β f j x. q = δ iq β H ij xj jq x β H ij x q f j x. 23 Setting x = r taking into account that f j r = 0. Then using 8, we have i r q = 1βδ iq. Now, we analyze the second derivative of λ i x. Differentiating 23 partially with respect to x r, r being arbitrary fixed, it follows that 2 λ i x r q = 0β β β H ij x 2 f j x r q H ij x f j x β r q H ij x f j x q r 2 H ij x r q f j x 24 Evaluating 24 in x = r using 10, we have 2 λ i r = β H ij r 2 f j r = βz iqr r. r q r q This completes the proof of lemma 1. Lemma 2. Let N j x be the jth entry of the vector Nx = J F x 1 Fx, that is N j x = H jk xf k x. 25 Then, N j r = 0, 26 N j r q = δ jq, 27 2 N j r r q = Z jqr r, 28 3 N j r s r q = Z jqsr rz jrsq rz jsrq r 2 Z jqrs r. 29 Proof. Since f j r = 0, it is clear that N j r = 0. Letj q be arbitrary fixed. Differentiating 25 partially with respect to x q, one obtains N j x q = H jk x f kx q At x = r, above equation yields N j r = H jk rj kq r = δ jq. q H jk x q f k x. 30 Differentiating 30 partially with respect to x r, we have 2 N j x r q = H jk x q H jk x 2 f k x r q f k x r H jk x r f k x q 2 H jk x r q f k x. 31 Setting x = r using 10, we obtain 2 N j r = H jk r 2 f k r = Z jqr r. r q r q Now, the third derivative of N j x with respect to x s, s being arbitrary fixed, is given by 3 N j x 3 f k x H jk x 2 f k x = H jk x s r q s r q s r q H jk x 2 f k x 2 H jk x f k x r s q s r q 2 H jk x f k x H jk x 2 f k x s q r q s r 3 H jk x 2 H jk x f k x f k x. 32 s r q r q s Setting x = r, then substituting 11 in 32, we get 3 N j r H jk r 2 f k r H jk r 2 f k r = s r q q s r r s q H jk r 2 f k r 3 f k r 2 H jk r. 33 s r q s r q Using equations 14 15, equation 33 yields 3 N j r s r q = Z jqsr rz jrsq rz jsrq r 2 Z jqrs r. This completes the proof of lemma 2. Theorem 1. Let F : D R n R n be sufficiently differentiable at each point of an open neighborhood of r R n, that is a solution of the system Fx = 0. Let us suppose that J F x is continuous nonsingular in r. Then, the sequence {x k } k 0 x 0 D obtained by using the iterative expression of method 16 converges to r with convergence order three. Proof. Let us consider r R n as a fixed point of iteration function ψ : D R n R n defined as ψx = x 1 2β J Fx 1 [2β 1J F xj F λx]nx. 34 The above equation can be written as 2βJ F xψx x[2β 1J F xj F λx]nx =

4 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 The ith component of 35 can be written as 2β J ijxψ jx x j [2β 1J ijxj ijλx]n jx = Setting x = r in 36 assuming that J F r is nonsingular. Then, using 20 26, we have ψ j r = r j. 37 Differentiating 36 partially with respect to x q, one obtains ψj x J ij x 2β J ij x δ jq 2β ψ j x x j q q [2β 1J ij xj ij λx] N jx J ij x 2β 1 q q J ij λx p1 x N j x = p 1 =1 p1 x q Let us substitute x = r apply 20, 21, 26, 27. Then, above equation yields ψ j r q = Differentiating 38 partially with respect to x r, we have 2β J ijx 2 ψ jx J ijx ψjx 2β δ jq r q r q J ijx ψjx 2 J ijx 2β δ jr 2β ψ jx x j q r r q [2β 1J ijx J ijλx] 2 N jx r q J ijx J 2β 1 ijλx p1 x Njx r p 1 =1 p1 x r q J ijx J 2β 1 ijλx p1 x Njx q p 1 =1 p1 x q r 2β 2 J ijx J 1 ijλx 2 λ p1 x N jx r q p 1 =1 p1 x r q 2 J ijλx p2 x p 1 =1 p 2 =1 p2 x p1 x r p 1 x ] N jx = q Setting x = r. Using 26, 27, 28 results of lemma 1 in above equation, we obtain 2 ψ j r r q = Now, we analyze the third derivative of 36. Therefore, differentiating 40 partially with respect to x s, evaluating the resulting expression in x = r using results of lemma 1, lemma 2 14, it can be easily proved that 2β 3 ψ j r s r q 3β 2 2β Z jqrs r 2β [ Z jqsr rz jrsq rz jsrq r ] = This yields 3 ψ j r = Z jqsr rz jrsq rz jsrq r s r q 1 3β Z jqrs r = So, by 5, we conclude that 16 presents a one-parameter family of third order methods for systems of equations. We denote this family by GTM1, as this family is generalization of Traub method 3. Substituting 43 into 6 noting that the summation is performed over all q, r s from 1 to n, yields e k1 j = 1 6 q=1 r=1 s=1 1 3β 2 [ 1 2 Z jqrsr ] Z jqrs r This completes the proof of theorem 1. q r s. 44 Remark 1. The error equation of 3 is given as see [2] [ e k1 1 f x 2 = 2 f x 1 1 3β f x ] 6 2 f x [ ] 3, 45 where x is the root of nonlinear equation fx = 0. Noting that the Jacobian matrix is a generalization of f that the inverse of jacobian matrix, whose elements are denoted by H ij, is generalization of 1/f, shows that i 1=1i 2=1 Z ji1i 2 r i 1 i 2, is generalization of f x /f x [ ] 2 i 1=1i 2=1i 3=1 Z ji1i 2i 3 r i 1 i 2 i 3, is generalization of f x /f x 2 [ ] 3. Also i 1=1i 2=1i 3=1 Z ji1i 2i 3 r i 1 i 2 i 3 is generalization of f x /f x [ ] 3. Thus, equation 44 is the generalization of 45. Theorem 2. Under the hypothesis of theorem 1, scheme 17 converges to r with convergence order three. Proof. Let us consider a solution r R n of Fx = 0 as a fixed point of the iteration function φ : R n R n defined as φx = x 2β[2β 1J F x J F λx] 1 Fx. 46 Equation 46 is equivalent to [2β 1J F x J F λx]φx x2βfx =

5 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 Letφ i x,i = 1,2,...,n be the coordinate functions ofφx. Then, ith component of equation 47 is [2β1J ijx J ijλx]φ jx x j2βf ix = From above equation, it is clear that if β = 0, J F r is assumed to be nonsingular f j r = 0, then φx is fixed point function, i.e. φ j r = r j. 49 Now, differentiating equation 48 partially with respect to x q, we have [ 2β 1 n p 1 =1 J ijx q J ijλx p1 x ] p1 x φ jx x j q φjx 2β 1 J ijx J ijλx q δ jq 2β fix q = Evaluating 50 in x = r using 49, it can be easily seen that φ j r q = Now by partial differentiation of 50 with respect to x r, we have J ijx 2β 1 q J ijλx p1 x φjx δ jr p 1 =1 p1 x q r 2β 2 J ijx 1 r q J ijλx 2 λ p1 x φ jx x j p 1 =1 p1 x r q 2 J ijλx p2 x p 1 x p 1 =1 p 2 =1 p2 x p1 x r q φ jx x j 2β 1 J ijx J ijλx 2 φ jx 2β 2 f ix r q r q J ijx J 2β 1 ijλx p1 x r p 1 =1 p1 x r φjx δ jq = q Setting x = r then substituting 20, 21, in above equation, it can be proved that 2 φ j r r q = Differentiating equation 52 partially with respect to x s, evaluating the resulting expression in x = r. Applying 13, 14, 49, 51, 53 results of lemma 1 it can be shown that which implies 2β 3 φ j r s r q 3β 2 2β Z jqrs r β [ Z jqsr rz jrsq rz jsrq r ] = 0, 54 3 φ j r s r q = Therefore, we conclude that 17 presents a one-parameter family of third order methods for systems of equations. We denote this family by GTM2. Substituting 55 into 6, the error equation for family 17 is given as e k1 j = 1 6 q=1 r=1 s=1 1 3β 2 [ 1 4 Z jqrsr ] Z jqrs r This completes the proof of theorem 2. q r s. 56 Remark 2. The error equation in case of 4 is given as [2] [ e k1 1 f x 2 = 4 f x 1 1 3β f x ] 6 2 f x [ ] 3, 57 Thus, by the similar argument as in remark 1 the equation 56 is generalization of 57. IV. PARTICULAR CASES A Particular Cases of GTM1 For β = 1/2, we get third order method developed by Hernández [13] y k = x k 1/2J F x k 1 Fx k, x k1 =x k J F x k 1 J F y k 2J F x k J F x k 1 Fx k. 58 If β = 1, we obtain third order method developed by Babajee et al. [14] y k = x k J F x k 1 Fx k, x k1 = x k 1 2 J Fx k 1 J F y k 3J F x k J F x k 1 Fx k. 59 B Particular Cases of GTM2 The value β = 1/2, gives third order method developed by Homeier [6] which is given as y k = x k 1/2J F x k 1 Fx k, x k1 = x k J F y k 1 Fx k

6 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 If we let β = 2/3, equation 17 generates third order method developed by Noor et al. [9]. y k = x k 2/3J F x k 1 Fx k, x k1 = x k 4J F x k 3J F y k 1 Fx k. 61 V. NUMERICAL EXAMPLES Here we present some examples to test the performance of present families GTM1 GTM2 by taking β = The comparison is carried out with Newton s method NM with the special members of the families; Hernández method HM, formula 58 Homeier method HMM, formula 60. To check the theoretical order of convergence, we obtain the computational order of convergence ρ using the formula [15] ρ ln xk1 r / x k r ln x k r / x k 1 r. 62 We consider the following systems of nonlinear equations: a { x 1 e x2 cosx 2 = 0, 3x 1 x 2 sinx 2 = 0. b { x 2 1 2x 1 x = 0, x 2 1 4x = 0. c { x 2 1 x = 0, x 2 1 x = 0. d { x 2 1 x 2 2 3logx 1 = 0, 2x 2 1 x 1 x 2 5x 1 1 = 0. e { e x1 x 1 x 2 1 = 0, sinx 1 x 2 x 1 x 2 1 = 0. f { x 1 2x 2 3 = 0, 2x 2 1 x = 0. g h x 2 1 x 2 2 x 3 3 = 9, x 1 x 2 x 3 = 1. x 1 x 2 x 2 3 = 0. x 2 x 3 x 4 x 2 x 3 = 0, x 1 x 3 x 4 x 1 x 3 = 0, x 1 x 2 x 4 x 1 x 2 = 0, x 1 x 2 x 1 x 3 x 2 x 3 1 = 0. i { x i x i1 1 = 0, i = 1,2,...,98 x 99 x 1 1 = 0. whose solutions, respectively are: a r = 0,0 t. b r = , t. c r = , t s = , t. d r = , t. e r = 0,1 t. f r = , t. g r = , , t, s = , , t h r = , , , t, s = , , , t. i r = 1,1,...,1 t s = 1, 1,... 1 t. All computations are done using MATHEMATICA [16] software, where the solutions r s are computed with a precision of 200 decimal digits. The stopping criterion used is x k1 x k Fx k < For every method, we analyze the number of iterations needed to converge to the solution computational order of convergence. The results are displayed in Table 1. It can be observed that the numerical results are in accordance with the theory developed in paper. We can also observe that the present methods show good stability require less number of iterations k than NM to reach the required solution. VI. CONCLUSIONS We have presented two one-parameter families of third order methods for solving systems of equations. These families are the generalization of known univariate families by Traub. Analysis of convergence is supplied in Theorem 1 2. Numerical results are presented performance is compared with well known methods. These numerical results overwhelmingly support the theoretical results we have derived. Similar experimentations have been performed on number of other problems results are found at par with those presented here. Finally, we conclude the paper with the remark that there is no clear winner among the presented methods in the sense that in some situation one method may be the winner while in some different the other method may be the winner. 1869

7 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 TABLE I COMPARISON OF THE PERFORMANCE OF METHODS Fx x 0 k ρ Sol. NM HM HMM GTM1 GTM2 GTM1 GTM2 NM HM HMM GTM1 GTM2 GTM1 GTM2 β = 0.1 β = 1 β = 0.1 β = 1 a 1.5,.1 t r.3,.5 t r b 3,2 t r 1.6,0 t r c.7,1.2 t r 1, 2 t s d.91, 2 t r 1.5, 1.5 t r e.9,.9 t r 0.1,0.2 t r f.9,.5 t r 1.5,1 t r g 2,1.5,1.9 t r, s 3,.5,2 t r h 1, 1, 1, 1 t s 2,2,2,0 t r i 2,2,...,2 t r 4, 4,... 4 t s *NM, HMM GTM2 converge to r, while HM GTM1 converge to s. 1870

8 International Science Index, Mathematical Computational Sciences waset.org/publication/1595 REFERENCES [1] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, [2] J.F. Traub, Iterative Methods for the solution of equations, Prentice- Hall, Englewood Cliffs, NJ, [3] C.T. Kelley, Solving nonlinear equations with Newton s method, SIAM, Philadelphia, PA, [4] S. Amat, S. Busquier J.M. Gutiérrez, Geometrical constructions of iterative functions to solve nonlinear equations, Journal of Computational Applied Mathematics [5] J.M. Gutiérrez, M.A. Hernández, A family of Chebyshev-Halley type methods in Banach spaces, Bulletin of the Australian Mathematical Society [6] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariable case, Journal of Computational Applied Mathematics [7] M. Frontini, E. Sormani,Third-order methods from quadrature formulae for solving systems of nonlinear equations, Applied Mathematics Computuation [8] A. Cordero, J.R. Torregrosa, Variants of Newton s method using fifthorder quadrature formulas, Applied Mathematics Computuation [9] M.A. Noor, M. Wassem, Some iterative methods for solving a system of nonlinear equations, Applied Mathematics Computuation [10] M.T. Darvishi, A. Barati, A third-order Newton-type method to solve systems of nonlinear equations, Applied Mathematics Computuation [11] J.L. Hueso, E. Martínez, J.R. Torregrosa, Third order iterative methods free from second derivative for nonlinear systems, Applied Mathematics Computuation [12] Y. Lin, L. Bao, X. Jia, Convergence analysis of a variant of the Newton method for solving nonlinear equations, Computers Mathematics with Applications [13] M.A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, Journal of Optimization Theory Applications [14] D.K.R. babajee, M.Z. Dauhoo, M.T. Darvishi, A.Karami, A. Barati, Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, Journal of Computational Applied Mathematics [15] S. Weerakoon, T.G.I. Ferno, A variant of Newton s method with accerated third-order convergence, Applied Mathematics Letters [16] S. Wolfram, The Mathematica Book, fifth ed., Wolfram Media,