Investigate the Relationship Between the type of Non-Linear Equation and the Initial Estimation on the Results of the Roots of the Non-Linear Equation using Open Numerical Methods with an Internal Stop Condition

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This research focuses on highlighting the impact of initial guessing on finding the roots of a nonlinear equation using open numerical methods with an internal stopping condition. Three examples were taken to understand the relationship between the initial guess and the results of the roots of the nonlinear equation using open numerical methods with an internal stopping condition. Precise values of the roots were calculated using the direct mathematical method. Subsequently, the roots of the nonlinear equation were calculated using open numerical methods, which are: Newton-Raphson method, secant method, and fixed-point method. The relative percentage error for each method was calculated for the three examples, in order to study and compare the results. The findings indicated that the type of equation directly controls the possibility or impossibility of finding the initial guess that meets the convergence condition for the three methods, and thus, it controls the results of the roots of the nonlinear equation using the internal stopping condition. Keywords: Fixed Point Method, Secant Method, Newton Raphson Method.