A new correction of Newton ’ s procedure to achieve convergence of higher order

An easy correction to the standard Newton procedure for estimating the root of a univariate function is explained and dissolution. For a given number of functions and derivative appraisements, The correction procedure converges quicker, By the convergence of the procedure is 1 2 2.4   While the convergence of standard Newton procedure is equal to 2 . Numerical examples illustrate the quicker convergence attained with this correction of Newton’s method. This corrected Newton–Raphson procedure is partly easy and stable.


Introduction
For solving nonlinear equations numerical methods are used.We consider the equation ( ) 0 fx , suppose ( ) 0 f   ; For obtaining the root  , we consider an initial guessed 0 x then we have the following sequence generated: 12 , ,....., ,.....

n x x x
This sequence is generated based on the initial guessed 0 x and the aim is to converge to the root  .The calculations will converge when the difference computed between each of the two consecutive sequence elements decreases.Newton's procedure has been one of the numerical procedures to solve the nonlinear equation ( ) 0 fx , which is a long term research because of its simplicity and speeding up the convergence.
In general, the Newton method is an iterative generated sequence of estimations that converge to the root, and the order of convergence is equal to 2. http://www.ispacs.com/journals/cacsa/2015/cacsa-00045/International Scientific Publications and Consulting Services -Newton's procedure for designating a root of a nonlinear equation   0 fx has long been desirable for its easiness and rapid rate of convergence.Using alone the function and its first derivative, Newton's procedure repetitively produces a sequence of estimations to a naïve root.While several of the procedures with higher convergence order are too long known [1], these have the bad feature of needing higher order derivatives.
Recently, several authors who had been brought in [2][3][4][5][6] have concluded a multistep, predictor-corrector procedure that obtain higher convergence order but although they need the function and the first derivative of the function; all these procedures need more function or derivative appraisals in each iteration than Newton's procedure.However these additional calculations are compensated by a better rate of convergence.A predictor-corrector procedure that was designed is presented in ref [7], which obtained a well initial approximation of the root for beginning the iteration and there was also a good approximation of the first derivative of the function at the point.This additional derivative data, composed with the seeing that if () fx The primary iteration to detect 1 x is the standard Newton-Raphson design.While to detect 2 x the function's derivative is not specified at 1 x .In exchange, the available approximation of the derivative is once again used to detect a medium value of 1 x  , and the derivative is appraised at * 11 1 2 xx    We have named the step to detect the medium value, 1 x  , the ''predictor step'', while the step to detect the subsequent value of x , 2 x ,(using the derivative specified at 1 2 xx    ) has been named ''the corrector'' step.
We propose a predictor-corrector method of the following * 00 0 () () Wherein 0 f  showed the approximation of the derivative at 0 x .The initial predictor step is only a Newton step based on the approximate derivative, whenever the corrector step is motivated by the implied relation (1.1).The next iterations of the [7] procedure is alike, besides that in the predictor step the approximated derivative is superseded by the derivative calculated in the prior iteration, that is (beginning at k = 1), A useful feature is the re-use in Equation (1.4) of the derivative computed in the prior iteration that causes the rule to be exclusively effective.each predictor-corrector step requires just one function and one derivative evaluation.In this article, the [7] procedure is developed to a more total context and it is illustrated that the rule converges at a rate of 1 2 2.41  at a cost of one function and its first derivative evaluation http://www.ispacs.com/journals/cacsa/2015/cacsa-00045/International Scientific Publications and Consulting Services Each iteration, causing this rule has been more effectively compared with the standard Newton-Raphson rule for the same cost.

The corrected Newton-Raphson procedure
Again the iteration is determinated, our general iteration is obtained in the forms (1.4) and (1.5) above.As mentioned, this is a predictor-corrector rule in which the predictor step is based on the derivative computed in the prior iteration, and the corrector step is motivated by the implied Equation (1.1).We need two start values, 0 x and * 0 x .To begin the iteration we have an initial approximate 0 x of the root.Now to estimate * 0 x use the standard procedure obtains that * 00 0 () () For values We now explain in more details the main and important properties of our corrected Newton-Raphson procedure.The value 2 x is computed from 1 x using the function value at 1 x and the value of * 11 1 () 2 x x     .Now the key features is that this value of the derivative is again used in the subsequent predictor step to Calculate * 3 x .This property of the derivative is used again meaning that the evaluations of the starred values of x in Eq. (2.8) basically come for free, which then provides the more suitable value of the derivative to be used in the corrector step Equation (2.9).

Convergence analysis
Eearlier we knew that the distance ratio of any two consecutive Sequence elements Convergent root equal to the rate of convergence.Considering and frequentatived replacement and Taylor expansion equation (2.9).demonstrations that to leading order Investigatinge the sequence of powers over pairs of serial steps demonstrates the estimation converges quicker than quadratic.By accuracy in these expressions, it is most simply seen that the powers of 0 The ratio R of the successive numbers in this series is seen quickly, estimating a constant number of 2.41 r  .The Taylor sequences computations above illustrate that the numbers in the series (3.13) prove the claim [8].Therefore for solving ( ) 0 fx using the corrected Newton-Raphson procedure, we obtain the convergence rate

r 
. While the convergence rate of the standard Newton procedure is quadratic.

Comparison between ''cubic' and ''secant'' proced
The proposed way of comparing these numerical methods is to define the rate of convergence per function or first derivative evaluation, the so-called ''efficiencies'' of the numerical methods.Numerical methods have a cubic convergence property introduced in Refs.[2,5,3,11].These numerical methods in each iteration need three appraisements of either the function or its first derivative.Based on this, foundation, the cubic convergence procedure has an efficiency of However the secant procedure has the disadvantage that is, very close to the root, while the procedure not always able to avoid this unwanted behavior.

Numerical examples
Computational tests for the reported family of procedures were done with the CLN multi accuracy library [13], using a 64-digit floating point calculation.In Table 2 we compare the efficiencies of the two procedures; Newton's procedure, the corrected rules of [2,5], and our corrected Newton-Raphson procedure using the three test functions.http://www.ispacs.com/journals/cacsa/2015/cacsa-00045/International Scientific Publications and Consulting Services 22 In addition considering [2,5] the adopted stopping criteria for the computer programs were

 
In Table 2 it is seen that our corrected Newton-Raphson procedure universally achieved at the definitely repeated solution with less function and derivative evaluations than the standard Newton-Raphson procedure and the cubic procedures.In fact the cubic procedures are universally not better to the standard Newton procedure, maybe becausein each iteration of the requirement to do three function evaluations, so that the cubic procedures often requirement to complete such a full iteration whiles the standard Newton-Raphson procedure may attain the favorabled precision after its two function or derivative evaluations of its first iterations Table 1: Numerical comparisons of the methods of Newton, [5] and the present paper.The number of iterations N is shown in column and the number of function or derivative evaluations f N required to complate the iteration.We implemented a more wide sequences of tests than is shown in Table 2, using all the eight test functions and their multiple starting values of x explained in [2], making a total of 16 test cases.Here we report on the comparative proficiency of the procedure of this paper compared with that of the other procedures explained above.In each of these 16 test cases, our procedure outperformed the standard Newton procedure, and in both cases, the standard Newton procedure did not converge to a solution within 50 iterations (100 function or derivative evaluations).our procedure also outperformed the procedure of [3] in every case and outperformed the [2] procedure in 12 of the 16 cases.Our rating of performance here is to note the number of function or derivative evaluations required to decrease the error of the initial conjecture by 30 orders of value.The evaluation of the derivative in our procedure at the mid-point amongst the maximum recent solution and a predictor value of the solution (that is not unlike Newton's method) is the proof why our procedure converges to a solution more often than does the standard Newton procedure.In summary, the procedure presented in this paper outperformed Newton's procedure and all of the cubic procedures.The relatively robust feature of the corrected Newton procedure presented is demonstrated by the fact that in 2 of the 16 cases, Newton's procedure did not converge.

Conclusion
The corrected Newton-Raphson procedure introduced in this paper described a rate of convergence over Newton's Procedure increased without additional cost.In fact the corrected procedure in terms of total function evaluations is found to offer a better efficiency than other so-called cubic convergence procedures.It is the re-use of the formerly calculated values of the derivative that offer our corrected procedure its numerical efficiency contrasted with the standard Newton-Raphson procedure.The corrected Newton-Raphson procedure presented in this paper is rather simple and is stable.In the corrected Newton-Raphson procedure presented in this paper a stopping criteria can be applied after every evaluation of either the function or its first derivative.This property is an additional superior in terms of total calculation efficiency compared to procedures that need several function or derivative evaluations in order to perform a full iteration.The efficiency index of our corrected Newton procedure, is 10% more than that of Newton's procedure itself (1.5538/1.4142≈ 1.099).The beginning values should be more attentively selected to guarantee convergence to the root because the rather little addition in performance provided by the higher order methods should be equivalented against the less stable nature of these higher order schemes.

.
While the corrected Newton-Raphson procedure of the present paper has is greater than the efficiency of our corrected Newton-Raphson procedure.