On a few Iterative Methods for Solving Nonlinear Equations

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1 On a few Iterative Methods for Solving Nonlinear Equations Gyurhan Nedzhibov Laboratory of Matheatical Modelling, Shuen University, Shuen 971, Bulgaria e-ail: gyurhan@shu-bg.net Abstract In this study an unpopular ethod of quadrature forulas for receiving iterative ethods for solving nonlinear equations is applied. It is proved for the presented iterative ethods that the order of convergence is equal to two or three. The executed coparative nuerical experients show the efficiency of the presented ethods. Keywords: Iterative ethod, Newton s ethod, order of convergence, discrete odification. 1 Introduction In [1] and [] a ethod of quadrature forulas for solution of nonlinear equations is proposed. Here we continue this approach to receive new classes of iterative processes on the sae proble. In Section by using a new quadrature approach several iterative forulas are received. In Section 3 the discrete odifications of soe of these forulas are given. Analysis of convergence is ade in Section 4. Geoetrical interpretation is included in Section 5. The study finishes with soe nuerical experients Section 6) and conclusion Section 7). Several classes of iterative forulas obtained by using quadrature forulas We will bring out soe classes of iterative processes for solving the equation: fx) = 0. 1) Let consider the identity: x+1 fx +1 ) = fx ) + f η)dη. ) x In this identity we put the left side fx +1 ) = 0 i.e. suppose that x +1 is root of equation 1)) and approxiate the integral with different quadrature forulae, which correspond to different iterative functions..1. First faily of iterative functions Let approxiate and substitute the integral in ) with the quadrature forula: x+1 x f η)dη = x +1 x 1 i=0 f δ i ), where δ i = x + i x +1 x ). Further on we replace x +1 x = u which participate in δ i, where u = fx ). Then fro the f x ) equation ) we get: 0 = fx ) + x +1 x 1 i=0 f δ i ), where δ i = x i u. 1

2 Finally we express x +1 and obtain the iterative forula: fx ) ), u = fx ) f i 1) u f x ). 3) This is a class of iterative processesdepends fro N).It is easy to verify that when = 1 fro 3) Newton s iterative algorith is obtained. When = we obtain the forula:..second faily of iterative functions Now we shall use the quadrature forula: x+1 x f η)dη = x +1 x fx ) f x ) + f x 1 u ). 3.1) ) f δi 1 + δ i, where δ i = x + i u. By analogy such as in.1. we obtain the following class of iterative forulae: fx ) ), u = x fx ) f i 1) u f x ) 4) In particular we have: and fx ) f x 1 u ), for = 1; 4.1) fx ) f x 1 4 u ) + f x 3 4 u ), for = 4.).3.Third faily of iterative functions Let substitute the integral in ) with the quadrature forula of the trapezius: x+1 x f η)dη = x +1 x f x ) + 1 In this case we obtain the class of iterative processes: For = 1 we have: f δ i ) + f x +1 ) ), δ i = x + i x +1 x ). fx ) f x ) + 1 f x i u ) + f x u ). 5) This forula by is proposed by S.Weeraoon and T.G.I.Fernando in [1]. When = the iterative algorith is: fx ) f x ) + f x u ). 5.1) afx ) f x ) + bf x 1 u ) + f, where a = 4 and b =. 5.) x u ) A siilar iterative forula by V.I.Hasanov, I.G.Ivanov and G.Nedzhibov for a = 6 and b = 4 is observed in []. This one can be obtained fro equation ) if one use Sipson s quadrature forula. In [] the third order of convergence for this case is proved. The iterative processes 3.) and 4.1) also are investigated by J.F.T raub in [4].

3 3 Discrete odifications To reduce the nuber of the coputations of derivatives one ay use soe corresponding discrete odifications. Here we will get the discrete odifications of the forulas 3.1), 4.), 5.1). 3.1.Discrete odifications of the forula 3.1) After the substitution of the denoinator f x ) + f x 1 u ) with we receive the following discrete odification: 1 u fx ) + f x 1 u u fx ) fx ) f x 1 u ), where u = ux ) = fx ) f x ) )) 3.1.1) 3..Discrete odifications of the forula 4.) By analogical way as in 3.. we obtain the discrete odification of iterative forula 4.): u fx ) fx 1 4 u ) f x 3 4 u )), where u = ux ) = fx ) f x ) 4..1) 3.3.Discrete odifications of the forulas 5.1) and 5.) The discrete odifications of the forulas 5.1) and 5.) is the sae forula: u fx ) fx ) fx u ), where u = ux ) = fx ) f x ) 5.1.1) 4 Analysis of convergence Theore 1 Let f be a real function. Assue that fx) has first, second and third derivatives in the interval a, b). If fx) has a siple root an α a, b) and x 0 is sufficiently close to α, then the class of iterative ethods 3) has second order of convergence and the classes 4), 5) of iterative ethods has third order of convergence. Proof: a).first we shall discuss the convergence of the nown Newton s iterative forula. We denote: ϕ 1 x) = x fx) f x) = x u; δ 1 = ϕ 1 x) α and ε = x α, then δ 1 = ε fx) f x) = εf x) fx) = A 1, where A f 1 = εf x) fx), B 1 = f x) x) B 1 Let α be a siple root of fx), i.e. fα) = 0, f α) 0. The Taylor expansions gives: fx) = fα) + εf α) + ε f ) α) + Oε 3 ) and f x) = f α) + εf ) α) + Oε ). Then A 1 = εf x) fx) = ε f ) α) + Oε 3 ), and B 1 ) 1 = f α)) 1 + Oε). Therefore δ 1 = C ε + Oε 3 ) where This equation establishes the second order of convergence. b). Order of convergence of the class of iterative forulas 3). Denote: ϕ x) = x C = f ) α) f α). 6) fx) f x i 1 u) and δ = ϕ x) α = εs fx) = A, S B where S = f x i 1 u). Again using the Taylor expansion we get: f x i 1 u) = f α) + ε i 1 u) f ) α) + O ε ) i 1 u) = f α) + +1 i ε + Oε ) ) f ) α) + O ε i 1 u) ), 3

4 fro 6) we have u = ε C ε, hence S = f x i 1 ) u = f α) f ) α) + Oε ). Therefore A = f ) α) ε + Oε 3 ) for B ) 1 = f α)) 1 + Oε), hence This equation establishes the second order of convergence. c). Observe the class of iterative forulas 4). Denote: ϕ 3 x) = x δ = C ε + Oε 3 ). 7) fx) f x i 1 u), δ 3 = ϕ 3 x) α = εs 3 fx) = A 3, S 3 B 3 where S 3 = f x i 1 u). We use Taylor expansion for: f x i 1 u) = f α) + ε i 1 u) f ) α) + ε i 1 = f α) + i)+1 f ) α)ε + i 1 C f ) α) + u) f 3) α) 1)+1 + Oε 3 ) ) f 3) α) ) ε, fro 6) we have u = ε C ε, hence S 3 = f α) + f ) α) ε + f ) α) C + f 3) α) ) ) i) + 1 ε + Oε 3 ). We use B 3 ) 1 = f α)) 1 + Oε), hence by not difficult coputation we obtain: )) δ 3 = C 3 + C i) + 1) 1 ε 3 + Oε 4 ); C 3 = f 3) α) 3!f α). 8) This equation establishes the third order of convergence. d). Observe the class of iterative forulas 5), denote: ϕ 4 x) = x fx) S 4 and δ 4 = ϕ 4 x) α = εs 4 fx) S 4 = A 4 B 4, where S = f α) + 1 f x i 1 u) + f x u). We use Taylor expansion for: T i = f x i ) u = f α) + ε i ) u f ) α) + ε i ) u f 3) α) + O Substitute in ε i ) ) 3 u. S 4 = f α) + εf ) α) + ε f 3) α) + 1 T i + f α) + ε u)f ) α) =... =f α) + εf ) α) + C f ) α) + f 3) α) i ) )) ε + Oε 3 ). We use that B 4 ) 1 = f α)) 1 + Oε), finally we obtain: ) ) 1 δ 4 = C 3 + C 3 3 i) 3) 1 ε 3 + Oε 4 ). 9) This equation establishes the third order of convergence. The theore is proved. 4

5 Corollary If the condition of the Theore 1 is held then the iterative forulae 4.1) we have: δ = C C ) 3 ε 3 + Oε 4 ), where C 3 = f 3) α) 4 3!f α). Proof:We use Theore 1 in case = 1 at equation 8). Corollary 3 If the condition of the Theore 1 is held then the iterative forulae 5.1) we have: δ = C + 1 ) C 3 ε 3 + Oε 4 ), where C 3 = f 3) α) 3!f α). Proof:We use Theore 1 in case = 1 at equation 9). Theore 4 If the condition of the Theore 1 is held then the iterative forulae 5.1.1) has third order of convergence. Proof: Denote fx) ϕ 5 x) = x u fx) fx u)) = x u Using Taylor expansion we get: 1 + ) fx u), δ 5 = ε ϕ 5 x). fx) fx u)) fx u) = x u α)f α) + Ox u α) ) = C ε f α) + Oε 4 ) = ε f ) α) + Oε 4 ) and fx) fx u) = εf α)+oε ). Therefore fx u) fx) fx u) = C ε+oε 3 ). Fro u = ε C ε obtain δ 5 = ε u1 + C ε) = C ε ε C ε )C ε + Oε 4 ) = C ε 3 + Oε 4 ). This equation establishes the third order of convergence at iterative process 5.1.1). The theore is proved. 5 Geoetrical interpretation. This paragraph gives geoetrical interpretation of soe of the above iterative processes. For the classes of iterative forulas 3),4) and 5) can be said that each next approxiation x +1 of the root α is obtained as an intersection point of Ox l, where straight line l passes through the point x u, fx u )) and tanl, Ox) is equals to corresponding su of the value of the derivatives of the functionin denoinator). We have denoted x +1 an intersection point of Ox t where the straight line t is the tangent in point x, fx )) in F ig.1 and F ig.). 5.1.Iterative functions 4..1) At iterative process 4..1) each next approxiation x +1 is an intersection point of Ox l 1, where l 1 is a straight line through the point x u, fx u )) and parallel to the straight line through the points x 1 4 u, fx 1 4 u )) and x 3 4 u, fx 3 4 u ))- the point x 1) +1 in F ig.). 5..Iterative functions 5.1) At iterative algorith 5.1) each next approxiation x +1 is an intersection point of Ox l, where l is a straight line through the point x u, fx u )) and parallel to the tangent in point x 1 u, fx 1 u )) the point x +1 in F ig.1). 5.3.Iterative functions 5.1.1) The iterative process 5.1.1) nown by the nae Newton-Secant ethod is observed in [4] it deterines each next approxiation x +1 as an intersection point of Ox l 3, where the line l 3 is through the points x, fx )) and x u, fx u )) - the point x +1 in F ig.). 5

6 6 Nuerical experients We have done nuerical experients for different functions and initial points. All progras were written in Matlab. We copare five iterative procedures for coputing the root of nonlinear equations. Table 1. Function x 0 iter Root fx) MN V 1 V V 3 NS α ) x 1) 0 i=0 xi = ) x ) = ND 3 ND ) xi = ) expx) i=0 xi ) x 4) + x Accuracyfx n )) tie 500 f MN V 1 V V 3 NS MN V 1 V V 3 NS 1.8e e ) 1.3e e e e-1 5.1e e e e e e e e e ) 5.1e e e e e e-10 ND 0 ND 1.4e ND 4.39 ND e e e e ) 1.7e e e-09-1.e e e-07-9.e e e e e-10-1.e e-08.e e e e e e e ) -1.6e e e e-10-4.e e e e e-10-5.e e e e-1-7.7e e ) e e e e e e We introduce the notations: x 0 - a initial point; iter - nuber of iterations; tie the execution tie for 500 ties execution; ND- Not defined; MN- Newton s iterative forulae; V 1-Iterative forulae 5.1); V - Iterative forulae 5.) for a = 4 and b = 4 ; V 3- Iterative forulae 4.1); NS- Iterative forulae 5.1.1). 7 Conclusion We have considered a few classes of iterative forulas and the discrete odifications for soe of the. 6

7 In conclusion it should be said that fro practical point of view the iterative algoriths 4.1) and 5.1.1) show the best results and that is shown in the nuerical experients. The reason for reduction of the execution tie in these two algoriths is that the nuber of coputation of the values of the derivative of the function is the sallest. The iterative process 5.1.1) nown by the nae of Newton-Secant ethod is a coposition of Newton s and Secant s ethods. Thus this ethod unites their positive qualities such as: at each step of the iteration the value of the derivative of the function is coputed only once; in the beginning one initial point is chosen; the process is convergent if the first and the second derivatives of the function do not change the signs); third order of convergence does not require coputing the second or derivatives fro higher order of the function. Have to be denote for the classes of iterative forulas 3),4) and 5) that: with growing the value of it grow the nuber of coputation of the derivative of the function, too. Thus for higher values of the index of efficiency is reduces sensitivity. For that reason the ore interesting cases are when is sall < 3). In conclusion have to say that in practical view the best results shows the iterative algoriths 4.1) and 5.1.1)- that is shown in the nuerical experients. The reason for reduce the execution tie by this two algoriths is the less nuber of coputation the value of derivative of the function. The iterative process 5.1.1) nown with nae Newton-Secant ethod is coposition of Newton s and Secant ethods. Thus this ethod unites the positive qualities of the, lie: at each step of the iteration is copute only one tie the value of derivatives of the function; in the beginning we chose one initial point; the process is convergesif the first and the second derivatives of the function are not change the signs); third order of convergence without require to copute the second or higher derivatives of the function. Acnowledgent. This wor was supported by the Shuen University under contract 4/ The author would lie to than prof. Milo Petov for helpful suggestions. 7

8 References [1] Weeraoon, S., Fernando, T.G.I. 000) A Variant of Newton s Method with Accelerated Third- Order Convergence, Applied Matheatics Letters, 13, [] Hasanov, V.I., Ivanov, I.G., Nedzhibov, G. A New Modification of Newton s Method, In: Application of Matheatics in Engineering 7, Proc. of the XXVII Suer School Sozopol 01, pp , Heron Press, Sofia 00. [3] Sendov, B., Popov, V. 1976) Nuerical Methods, Part I, Naua i Izustvo, Sofia, in Bulgarian) [4] Traub, J.F. 1964) Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, New Jersey. 8