Numerical Solving of Nonlinear Equation Using Iterative Algorithms

In the given algorithm, we propose a development to the evaluations of Newton's numerical


Introduction
Newton's technique that approximates the zeros of a nonlinear equation including one variable by the value of the function and it is first derivative.This method is standard and very common, best known and most widely method for getting the roots of the functions from free second derivative.We suggested and analyzed some iterative techniques free from second derivatives of the function for solving nonlinear equation of a single diode of a solar cell based on it's equivalent circuit.There are many ways improved on the advancement the convergence of Newton's method, in order to attain lesser iterations than it .
The suggested algorithm TSM2 requires 5 evaluations of the function while the other technique (DM) needs 4 evaluation of the function.The following steps are investigate the procedure of this work: section two, three and four investigating the modelling and the root finding of TSM2 and DM algorithms respectively while; section five and six indicate the numerical problems, discussion and conclusion results respectively.
The current I pv and power P pv of the cell is given by The final equation from the circuit is given by

Three Step Method (TSM2M)
The following steps introduce this method Step 1: let the starting value x 0 Step 2: calculate algorithm 1: Newton Raphson Method (NRM) Step 3: determine algorithm 2: Two Step Method (TM) Step 4: compute algorithm 3: Three Step Method (TSM): given by the following formulas

Dekker's Algorithm (DM)
This method obtain when we combine the Bisection and Secant Methods achieved by Dekker in 1969.
Step 1: The first one called linear interpolation secant method using the following formula Step 2: the second one can be obtained by bisection method where: a n : the "contrapoint" this means that f(x n ) and f(b k ) have opposite signs, so the interval [a n , b n ] consist of the solution.
For the two algorithms, the tolerance is If|f(

Results and Discussion
Two numerical iterations is suggested to introduce the performance of the Three Step Method (TSM2) represented in Eq. 3 acquired in the present paper in order to solve non-linear equation with the initial value x 0 = 1 and we compare it with Dekker's Algorithm (DM) represented in Eq. 4 with two initial values x 0 and x 1 .For convergence criteria, the distance between two consecutive iterates is based on Eq. 5, less than 10-9.Five examples in Eq. 2 are used for numerical testing with the R values from 1-5 ohm, represents (load resistance) of the circuit.All determinations are carried out with the algorithm precision introduced in Tables and Figures 1 to 5 and the number of function evaluations needed are extracted from the Eq. 2. The numerical examples and the approximate solutions produced by two techniques for solving Eq. 2.
The following Tables and Figs.indicate that TSM algorithm needs 5 iterations while DM technique need 4 iterations to reach to the convergence which proves that DM is faster than TSM.

Conclusion
We have observed that DM has sixth computations provided first derivatives of the function exist.the most important analyze of the Dekker's algorithm is that unlike the other three step methods because it is not required to determine the second derivative of the function just need first derivative of it to carry out the evaluations.Determined results [Table 1, 2, 3, 4 and 5] reveal the accuracy and efficient and absolute error of the proposed method compared by the other ones.In addition for the computed results [Tables] that the total number of function computations needed less than that of the standard one.