New Simpson type method for solving nonlinear equations

Abstract: Finding root of a nonlinear equation is one of the most important problems in the real world, which arises in the applied sciences and engineering. The researchers developed many numerical methods for estimating roots of nonlinear equations. The this paper, we proposed a new Simpson type method with the help of Simpson 1/3rd rule. It has been proved that the convergence order of the proposed method is two. Some numerical examples are solved to validate the proposed method by using C++/MATLAB and EXCEL. The performance of proposed method is better than the existing ones.


Introduction
Q uadrature is a historical term which means the process of determining area and is used to computed the value of definite integrals by Newton cotes formula. Indeed, Newton cotes formula is also known as quadrature rule. Newton cotes formula is developed by Isaac Newton and Roger Cotes. Five basic commonly used Quadrature rules for numerical integration are Trapezoidal rule, Simpson 1/3 rule, Simpson 3/8 rule, Boole's rule and Weddle's rule.
Many scientist took interest and developed numerous techniques for numerical integration with the help of mid-point, arithmetic mean, geometric mean, harmonic mean and heroinic mean [1][2][3]. Now a days, estimating a root of nonlinear equations are one of the most important topics and many numerical techniques have been developed so far [4][5][6]. The famous iterative methods for finding roots of nonlinear equations are; the Newton's method; the Steffensen method; the Trapezoidal method; , and the Quadrature method; The aim of this paper is to develop a new numerical method called 'Simpson type method' by using quadrature rule for solving nonlinear equations. The proposed Simpson type method is based on Simpson 1/3rd rule which converges quadratically. The proposed Simpson type method is compared with Newton Raphson method, Steffensen method, Quadrature method and Trapezoidal method [8-10].

Simpson algorithm
This segment developed a Simpson type method for solving nonlinear equations with the help of Simpson 1/3rd rule. The Simpson 1/3rd rule is Taking derivative, we have The placement of certain integration, we get where n = 2 and h = x−x 0 2 . Substitute the value of h in (1), we have where x o is an initial guess, which is near to root x. Now using f (x) = 0, (2) becomes The basic numerical technique [7] is or Substitute (4) and (5) in (3), we get In general Equation (6) is the Simpson type method for solving nonlinear equations.

Convergence analysis
In the following theorem, we prove that the proposed Simpson type method converges quadratically.
Theorem 1. Let f : D ⊆ R → R be a sufficiently differentiable function and a ∈ D be a root of it in the open interval D. If x 0 is sufficiently close to a, then the proposed Simpson type method has second-order convergence and satisfies the following error equation: Proof. Let a be the root of the function f . By using Taylor series we have, we have implies By taking derivative of (9), we get Using (7) and (8) in (10), we get Taking derivative of (12), we get Using (7) and (8) in (13), we get By solving (8), (9) and (14), we get Now using (7)-(15) in (6), we get = e n f (a) + ce 2 n (1 + 7 f (a)).

Numerical results
Now, we solve some examples numerically to compare the performance of our proposed Simpson type method by using C++ and EXCEL. We compared our Simpson type method (P S Method) with Newton Raphson method (NR Method), Steffensen method (S Method), Trapezoidal method (T Method), and Quadrature method (Q Method).

Example 1. Consider the nonlinear equation;
which represents the mass of the jumper. By taking x 0 = 1 as initially guess, the comparison of NR Method, S Method, T Method, Q Method and P S Method is given in Table 1 and error analysis if given in Table 2. It can be observed from Table 1 that P S Method converges faster to the root of the Equation (18) than NR Method, S Method, T Method and Q Method. The graphical comparison of error analysis is shown in Figure 1.
which represents the diameter of the pipe. By taking x 0 = 1 as initially guess, the comparison of NR Method, S Method, T Method, Q Method and P S Method is given in Table 3 and error analysis if given in Table 4. It can be observed from Table 3 that P S Method converges faster to the root of the Equation (19) than NR Method, S Method, T Method and Q Method. The graphical comparison of error analysis is shown in Figure 2.  2.77556e-016 Example 3. Consider the nonlinear equation; which represents the volume of the gas depends. By taking x 0 = 1 as initially guess, the comparison of NR Method, S Method, T Method, Q Method and P S Method is given in Table 5 and error analysis if given in Table  6. It can be observed from Table 5 that P S Method converges faster to the root of the Equation (20) than NR Method, S Method, T Method and Q Method. The graphical comparison of error analysis is shown in Figure  3.
which represents the anti-symmetric buckling of a beam. By taking x 0 = 1 as initially guess, the comparison of NR Method, S Method, T Method, Q Method and P S Method is given in Table 5 and error analysis if given in Table 8. It can be observed from Table 7 that P S Method converges faster to the root of the Equation (21) than NR Method, S Method, T Method and Q Method. The graphical comparison of error analysis is shown in Figure 4.

Conclusion
We proposed a new numerical method for solving nonlinear equations which is derived from Simpson 1/3rd rule. It has been proved that the proposed method converges quadratically. We solved different nonlinear equations numerically by using Newton Raphson method, Steffensen method, quadrature method, trapezoidal method and our proposed method. The numerical results show that the proposed method is more accurate and faster than Newton Raphson method, Steffensen method, Quadrature method and Trapezoidal method.