An Improved Parameter Regula Falsi Method for Enclosing a Zero of a Function

Transcription

1 Applied Mathematical Sciences, Vol 6, 0, no 8, An Improved Parameter Regula Falsi Method for Enclosing a Zero of a Function Norhaliza Abu Bakar, Mansor Monsi, Nasruddin assan Department of Mathematics, Faculty of Science Universiti Putra Malaysia UPM Serdang, Selangor, Malaysia School of Mathematical Sciences, Faculty of Science and Technology Universiti ebangsaan Malaysia UM Bangi, Selangor, Malaysia n izaruffedge@yahoocom Abstract An improved parameter regula falsi method pl)-rf based on a direct generalization of the interval parameter regula falsi p-rf method is proposed in this paper This method is modified by using the midpoint of the current interval in the algorithm and the additional inner iterations l to improve its rate of convergence This modification on p-rf method is then verified on several test examples Based on the numerical results and CPU time of pl)-rf method, it is very clear that this method performs very well compared to the original method Mathematics Subject Classification: 65B99, 65G40 eywords: CPU time, interval analysis, parameter regula falsi, rate of convergence, zero of a function Introduction Apart from iterative procedures, the main tool to be used in this paper is interval analysis based upon the very simple idea of enclosing the zero of a function Researchers such as [5] and [] have shown that the iterative procedures which involved interval analysis approach is much convenience because the final interval of the iterative procedure always contain the zero of the function In other word, the zero is always bounded in the interval

2 348 N A Bakar, M Monsi and N assan Parameter regula falsi method p-rf is an extended interval version of Regula Falsi method RF) The modification of RF method had been done by various researchers such as [0] and [7] and only one by [9] on p-rf method p-rf method has higher-order of convergence even though only one function value used [4] We consider the equation fx) = 0 and the function f has a simple zero ξ in the initial interval 0) The rate of convergence of p-rf method [4] is at least p + p +4)p ) In the next section, we described briefly about p-rf method and then we present our modified method called pl)-rf method We also described the pl)-rf algorithm and then we compare the CPU time and the number of iterations obtained by pl)-rf method and p-rf method p-rf Algorithm The following is the algorithm [ for p-rf method [4] Step 0 : Given that: 0) = x 0),x 0) with conditions ] ; ξ 0) ; d k+)) = x k+) x k+) f x) =[h,h ]={f x) h f x) h },x ;0/ ) f x) =[k,k ]={f x) k f x) k } x ) Step : Set k =0,i = Step : x = m ) midpoint of ) Step 3 : k+) = {x f x ) } Step 4 : k := k + Step 5 : x = m ) Step 6 : k+,i ) = {x f x ) } Step 7 : Compute { k+,i ) = x x x k ) [ fx )+ fx ) fx k ) ) k+,i ) x ) k+,i ) x k )) ]} k+,i ) if f x ) 0 Otherwise, k+,i ) = k+,i ) Step 8 : z i) = m k+,i )) Step 9 : Compute

3 An improved parameter regula falsi method 349 { k+,i) = z i) z i) x [ fz i) )+ fz i) ) fx ) k+,i ) z i)) k+,i ) x ) ]} k+,i ) if f x ) 0 Otherwise, k+,i) = k+,i ) Step 0 : If i<pthen i := i + and go to 8 Step : k+) = k+,p) Step : If the width of the interval k+) greater than ɛ, that is d k+)) >ɛ, then go to 4 Step 3 : Stop Theorem Let the function f be twice be continuously differentiable in the interval and assume that f has a zero ξ in Let the conditions ) and ) be satisfied The parameter p is given as an integer number for which p Then, the sequence { } calculated according to algorithm p-rf method satisfies for p : ξ, k 0 and 0) ) ) with lim k = ξ d k+)) γd ) p) d k ) ) ; γ 0) where d k+)), d ), and d k )) are the width of the intervals k+), and k ) respectively Then the R-order of convergence of p- RF satisfies the inequality O R p RF),ξ) The proof of these theorem is available in [4] p + ) p +4 3) 3 The pl)-rf Algorithm Now, we present the algorithm[ of pl)-rf method Step 0 : Given that: 0) = x 0),x 0) with conditions ) and ) Step : Set k =0,i = and l = ] ; ξ 0) ; d k+)) = x k+) x k+)

4 350 N A Bakar, M Monsi and N assan Step : x = m ) midpoint of ) Step 3 : k+) = {x f x ) } Step 4 : k := k + Step 5 : x = m ) Step 6 : k+,i ) = {x f x ) } Step 7 : Compute { k+,i ) = x x x k ) [ fx )+ fx ) fx k ) ) ]} k+,i ) x ) k+,i ) x k )) k+,i ) if f x ) 0 Otherwise, k+,i ) = k+,i ) Step 8 : z i) = m k+,i )) and z i ) = m k+,i )) Step 9 : Compute { k+,i) = z i) z i) z i ) [ fz i) )+ fz i) ) fz i ) ) k+,i ) z i)) k+,i ) z i )) ]} k+,i ) if f z i)) 0 Otherwise, k+,i) = k+,i ) Step 0 : If d k+,i)) <ɛ,goto8 Step : If i<pthen i := i + and go to 8 Step : k+) = k+,p) = k+,p,l ) Step 3 : Compute { k+,p,l) = z l) z l) z l ) [ fz l) )+ fz l) ) fz l ) ) ]} k+,p,l ) z l)) k+,p,l ) z l )) k+,p,l ) if f z l)) 0 Otherwise, k+,p,l) = k+,p,l ) Step 4 : If d k+,p,l)) <ɛ,goto8 Step 5 : If l<pthen l := l + and go to 3 Step 6 : k+) = k+,p,p) Step 7 : If d k+)) >ɛ,goto4 Step 8 : Stop

5 An improved parameter regula falsi method 35 4 The Rate of Convergence of pl)-rf Method The pl)-rf method which is based on p-rf method has its own speciality In this modification, we keep the computation process updated for every inner iteration i Therefore, we compute another midpoint z i ) to replaced the value of midpoint x in Step 9 refer to Section ) Not only that, we also introduce another inner iteration, l l ) This iteration process from l =, 3,,ptakes place after the computation of inner iteration i finished Then, we substitute the latest interval that is k+,p) with k+,p,l) Readers can track down the modification that we have done by comparing algorithm on Section and Section 3 The following theorem is very useful in order to determine the rate of convergence of a procedure I [4] Theorem Let I be an interval iteration procedure with the limit x and let ζi,x ) be the set of all sequences generated by I having the properties that lim k x = x and x x, k 0 If there exist a p and a constant γ such that for all {x } ζi,x ) and for a norm it holds that x k+) γ x p Then, it follows that the R-order of convergence of I satisfies the inequality O R I,x ) p or the R-order of convergence of I is at least p Based on the Theorem, then we have the following theorem for pl)-rf method Theorem 3 Let the function f be twice continuously differentiable in the interval and assume that f only has a simple real zero ξ in Furthermore, we have intervals and satisfying the conditions ) and ) Then, the sequence calculated from the pl)-rf method satisfies p ): ξ, k 0 4) 0) ) ) 5) lim k = ξ 6) Then, the R-order of convergence of the pl)-rf method is at least ) p +l )) + p +l )) +4 7)

6 35 N A Bakar, M Monsi and N assan where p ) ; l p) or O R pl) RF ),ξ) p +l )) + ) p +l )) +4 Proof : Of 4), 5) and 6) : Can be found in [4] Of 7) : The proof of d k+,)) 4 d ) d k )), can be found in [4] Next, we have d k+,)) d z ) z ) z ) [ ) f z ) + f z ) ) f z ) ) k+,) z )) k+,) z )) ]) z ) z ) = d f z ) ) f z ) ) k+,) z )) k+,) z )) ) z ) z ) By f z ) ) f z ) ) =f τ)) τ ), it follows that d k+,)) d k+,) z )) k+,) z )) ) = d k+,) z ))[ k+,) x ) + ) x z ))] By d AB) d A) B + A d B) [4] we have d k+,)) d k+,) z ))[ k+,) x ) + x z ))] ) { d k+,) z ))[ k+,) x ) + x z ))] )} Again, we use d AB) d A) B + A d B) and we have d k+,)) { d k+,) z ))) [ k+,) x ) + x z ))] + k+,) z )) [ d k+,) x ) + x z ))] )} { d k+,) z )) + k+,) z ) )) d k+,) x + x z ) ) + k+,) z ) [ d ) + )] ))}

7 An improved parameter regula falsi method 353 By using d ) and the fact that d x)n ) d )) n, we have d k+,)) { d k+,))) + ) k+,) z ) k+,) x + x z ) ) + k+,) z ) [ d d ),d )] + [ d ),d )]) ))} By using the same properties, we have d k+,)) { d k+,))) + ) k+,) z ) k+,) x + x z ) ) + k+,) z ) [ d d ), d )] ))} Then, we use property d A n ) n A n d A) and what we have is d k+,)) { d k+,)) + k+,) z ) ) k+,) x + x z ) ) + k+,) z ) [ ) d, d )] ))} 4d By using x ) d and x z ) ) d, we then have d k+,)) { d k+,)) + k+,) k+,) ) )) + d + k+,) k+,) )) } 6 d { d k+,)) + d k+,))) d ) + d )) + d k+,)) 6 d )} ))

8 354 N A Bakar, M Monsi and N assan Simplifying the inequality, we have d k+,)) = { 3 d k+,))) )) d + 6 d k+,)) d )) } = { d k+,)) d ) ) + 6 d k+,)) d } )) = 8 d k+,)) d ) ) d k+,)) = 4 d k+,)) d ) Therefore, d k+,)) 4 d k+,)) d ) = 4 4 d ) d k ))) d ) d k+,)) 7 d ) 3 ) d k ) The solution to this simple recursion is the relation d k+,i)) ) i ) 7 i d ) i ) d k ) d k+,i)) β i d ) i d k ) ), with ) i ) 7 i β i =, i =,,,p Using k+) = k+,p), we get d k+)) β p d ) p ) d k ) β p 0 ) Next, we let k+,p,) = k+,p) = k+) and we have d k+,p,)) d )

9 An improved parameter regula falsi method 355 Then, it follows that d k+,p,)) d k+,p,)) d k+,p,)) d z l) z l) z l ) [ ) f z l) + f z l) ) f z l ) ) k+,p,) z l)) k+,p,) z l )) ]) By using the same approach as above, then we will get the following result d k+,p,)) 4 d k+,p,)) d ) 7 ) = 4 p p d ) p )) d k ) d ) ) p 7 p+ = 4 d ) p + ) d k ) By using the same mathematical induction on l, then we have d ) k+,p,l)) p 7 4 l ) p+l ) d ) p+l ) ) d k ) or we can write it as where d k+,p,l)) γ l p d ) p+l ) d k ) ) ) p 7 γp l =4 l ) p+l ), l p For l = p, then k+) = k+,p,p) and we get the relation d k+)) γ l p d ) p+l ) d k ) ) 8) where ) p 7 γp l =4 l ) p+l ), l p Therefore, based on this result and [4] the R-order of convergence of pl)-rf method is O R pl) RF),ξ) ) p +l )) + p +l )) +4

10 356 N A Bakar, M Monsi and N assan where p, l p If we compare the R-order of convergence of pl)-rf method and p-rf method, so it is clear that pl)-rf method is more effective Let p = 5, then lets substitute the value of p into 3) first ence, O R p RF),ξ) p + ) p +4 = 5+ ) 5 +4 = 5+ ) 9 O R p RF),ξ) 5+ ) 9 Next, substituting p = 5 into 3) gives O R pl) RF),ξ) ) p +l )) + p +l )) +4 = ) 5) + l )) + 5) + l )) +4 = ) 8 + l)+ 8 + l) +4 O R pl) RF),ξ) ) 8 + l)+ 8 + l) +4 Thus O R pl) RF) O R p RF) for l 5 5 Numerical Results We now compare the performance of pl)-rf method with p-rf method The comparisons are based on the CPU Central Processing Unit) time and number of iterations which are presented in Table and Table respectively The following table contains the test functions used to test the performance of both methods where the zero of each function contain in the final interval We also present the final interval results that contain the zeros of the functions 6 and 7 produced by p-rf method and pl)-rf method in Table 3 and Table 4 respectively The stopping criterion used is d k+)) <ɛ=0 5 for p =5 and l 5 The procedure p-rf and pl)-rf have been implemented using Matlab R007a [] in associate with Intlab []

11 An improved parameter regula falsi method 357 No Function, fx) 0) Final interval xe 5 + e 5x [4] [0, ] [ , ] x 3 x 5 [3] [, 3] [ , ] 3 x 3 +4x 0 [6] [, ] [ , ] 4 e x 4x [3] [4, 45] [ , ] 5 xe x 0 [] [0, 0] [ , ] 6 x )x 4 + ) [8] [0, 0] [ , ] 7 x x 9 ) [4] [0, ] [ , ] Function CPU time seconds) p-rf method pl)-rf method Table : Comparison of CPU time for both methods after converge Function Number of iterations p-rf method pl)-rf method k i i =,,p) k i i p) Table : Comparison of number of iterations for both methods after converge

12 358 N A Bakar, M Monsi and N assan Figure : Comparison of CPU time and number of iterations of both methods p-rf Method pl)-rf Method [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] Already converge [ , ] [ , ] Table 3: The results of p-rf and pl)-rf methods with fx) =x ) x 4 + )

13 An improved parameter regula falsi method 359 p-rf Method pl)-rf Method [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] Already converge [ , ] [ , ] Table 4: The results of p-rf and pl)-rf methods with fx) =x x 9 )

14 360 N A Bakar, M Monsi and N assan 6 Discussion and Conclusion We can see that in Table, the CPU time of pl)-rf method is lesser than p-rf method Table contains the number of iteration k which applied for the whole steps of both algorithms and inner iteration i in Steps 8-0 of both methods after the algorithms stopped by using the given convergence criterion In certain cases the number of iterations k are the same for both methods, but the number of inner iterations i for pl)-rf are lower This indicate that pl)- RF method have reached the stopping criterion and converges to the zero of the function much faster For a clearer view on the CPU time and number of iterations of both methods, we represent the results in a form of bar charts as in Figure a) and Figure b) respectively Furthermore, Table 3 and Table 4 are the computer output of the functions 6 and 7 respectively It is clear that pl)-rf method has achieved excellent performance where the algorithm of pl)-rf is terminated earlier by using the same convergence criterion In fact, we have shown that the R-order of convergence of pl)-rf method is greater than does p-rf method or O R pl) RF ),ξ) >O R p RF, ξ) 7 Acknowledgement We are indebted to Universiti ebangsaan Malaysia for funding this research under the grant UM-GUP-0-59 References [] A Gilat, MATLAB: An introduction with applications, obuken, NJ:Wiley, 008 [] E ansen et al, Interval forms of Newton s method, Computing, 0 978), [3] F Costabile, MI Gualtieri, SS Capizzano, An iterative method for the computation of the solution of nonlinear equations, Calcolo, ), 7-34 [4] G Alefeld and J erzberger, Introduction to Interval Computation, Academic Press, New York, 983

15 An improved parameter regula falsi method 36 [5] I Gargantini, Further applications circular arithmetic: Shoroeder-like algorithms with error bounds for finding zeroes of polynomial, SIAM J Numer Anal, 5 978), [6] J ou and Y Li, An improvement of the Jarrat method, Applied Mathematics and Computation,89 007), 86-8 [7] Wang & Feng, New predictor corrector methods of second-order for solving nonlinear equations, International Journal of Computer Mathematics, Vol88, 0), [8] NA Abd Rahmin, M Monsi, MA assan, F Ismail, A modification of inclusion of a zero of a function using interval method, Malaysian Journal of Mathematical Sciences, 3 009), 67-8 [9] NA Bakar, M Monsi, MA assan, WJ Leong, On The Modification of The p-rf Method, Applied Mathematical Sciences, 5 0), [0] P Parida, & D Gupta, A cubically convergent iteration method for multiple roots of fx) = 0, International Journal of Computer Mathematics, Vol87,3 00), [] SA oda Ibrahim, An improved exponential regula falsi methods with cubic convergence for solving nonlinear equations, J Appl Math & Informatics, 8 00), [] SM Rump, INTLAB - INTerval LABoratory:In Tibor Csendes, Developments in Reliable Computing, luwer Academic Publishers, Dordrecht, 999), [3] Wu and D Fu, New high-order convergence iteration methods without employing derivatives for solving nonlineat equations, Computers & Mathematics with Applications, 4 00), Received: September, 0