Numerical study on some iterative methods for solving nonlinear equations by using Scilab programming - Aboamemah, Ahmed Hadi Mohammed (2019)
This research aimed to investigate the most efficient iterative method in solving scalarnonlinear equations. There are three iterative methods that are used to solve the nonlinearscalar equations that are Bisection, Secant and Newton Raphsons methods. Thesethree iterative methods have different order of convergence. Bisection method is linearlyconvergence while Secant method is super linear and Newton-Raphson methodhas quadratic convergence. It is well known that the method that has a higher orderof convergence, will perform much faster than others. Seven nonlinear scalar equationsare considered based on the combinations of two or three functions and are solvedby the Bisection, Secant and Newton-Raphson methods using Scilab programming language.The tolerance used is 1010 and the performances of these methods are based onnumber of function evaluation, number of iterations, and computational or CPU time.Based on the numerical results of the seven nonlinear equations, it is observed thatNewton-Raphson method is still the most efficient method but not for all the equations.Bisection method has fixed performances on all the nonlinear equations however, themethod failed to converge for the imaginary root. On the other hand, the performanceof Secant method is almost similar to Newton-Raphson method except for the nonlinearEquations (4.4), and (4.5) on the interval [1.3,2] and [0,1] respectively. In conclusion,Newton-Raphson method remains the best but not for all nonlinear equations sincethere are realistic circumstances that makes Newton-Raphson converges either sloweror identical to Secant method. It is also proven that Secant method can perform fasterthan Newton-Raphson method depending on the form of the curve functions that correspondsto the approximate values. As implications, more than three combinations ofthe functions can be investigated and also the research can be extended to system ofnonlinear equations.