Abstract
Solving the root of algebraic and transcendental nonlinear equation f' (x) = 0 is a classical problem which has many interesting applications in computational mathematics and various branches of science and engineering. This paper examines the quadratic convergence iterative algorithms for solving a single root nonlinear equation which depends on the Taylor’s series and backward difference method. It is shown that the proposed iterative algorithms converge quadratically. In order to justify the results and graphs of quadratic convergence iterative algorithms, C++/MATLAB and EXCELL are used. The efficiency of the proposed iterative algorithms in comparison with Newton Raphson method and Steffensen method is illustrated via examples. Newton Raphson method fails if f' (x) = 0, whereas Steffensen method fails if the initial guess is not close enough to the actual solution. Furthermore, there are several other numerical methods which contain drawbacks and possess large number of evolution; however, the developed iterated algorithms are good in these conditions. It is found out that the quadratic convergence iterative algorithms are good achievement in the field of research for computing a single root of nonlinear equations.
Highlights
The estimation of a single root of nonlinear equations leads to wide range of applications in the field of numerical analysis, which emerges in applied sciences and engineering, for instance: distance, rate, time problems, population change and trajectory of a ball [1][2][3]
This study aims to work on these types of conditions and ascertain that the proposed iterated algorithms are decent attainment in these conditions in comparison with Newton Raphson method and Steffensen method for estimating a root of nonlinear equations
This section discusses the following nonlinear equations to demonstrate the efficiency of proposed Algorithm 3.1 and Algorithm 3.2 by equating with the Newton Raphson method xn+1
Summary
The estimation of a single root of nonlinear equations leads to wide range of applications in the field of numerical analysis, which emerges in applied sciences and engineering, for instance: distance, rate, time problems, population change and trajectory of a ball [1][2][3]. The Taylor series is an important and basic concept: h2 f (x) = f (xo) + hf (xo) + 2 f (xo) + . By using Taylor series, one of the most important technique Newton Raphson’s method has been developed [6][7][8] which is given as follows. To estimate the roots of Equation 1.1, several approaches have been established, see for instance [4][5].
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