ANALYSIS OF CONVERGENCE AND STABILITY OF ITERATIVE SCHEMES FOR NONLINEAR EQUATIONS

Abstract

Stability and convergence of iterative methods of nonlinear equations is studied. Many sciences and engineering offer nonlinear equations and solutions which are typically vulnerable to numerical methods because of solutions that cannot be easily determined. The paper seeks to analyze some of the iterative methods which include; fixed point iteration, secant method, Newton Raphson process and higher order two step methods which are usually in use.

They are compared and contrasted systematically with one another in terms of their convergence speed, accuracy as well as dependability in a variety of initial conditions. The acquired results have shown that besides the fact that, classical methods are easy to carry out and to carry out, they also are convergent more slowly and erratic. The greater the number of iteration steps the smaller it is and the rate of convergence of the functions is also very high especially in nonlinear functions; complex functions.

The importance of first approximations in the effectiveness of the iterative processes is also laid down in the study. It is noted that methods have superior starting values near the true root, and they may be unsteady or erratic with very inappropriate starting values. When the study of stability is also made, the approaches of higher order are found to be less susceptible to the small scale perturbation, and can be more widely applicable to practice.

Overall, the paper provides a comprehensive comparison of iterative schemes and a need to select the methods that are to be used based on the type of problems. The results make difference in the areas of numerical analysis as they give information as to how the performance of the solution to the nonlinear equation can be enhanced in terms of performance and accuracy