Effective Numerical Methods for Non-Linear Equations

Abstract

According to the recent studies related to the multiplicative numerical methods, more accurate numerical results can be obtained in many applications. The process of determining roots of non-linear functions arise from many applications of various fields such as image and audio processing, mathematical analysis, fuzzy systems, fluid mechanics etc. Especially a nonlinear equation which involves exponential and logarithmic functions can be effectively approximated by using multiplicative methods. On the other hand, corresponding ordinary numerical methods work better for the given non-linear equation consists only polynomial function. Therefore, the idea of joined numerical methods consisting both ordinary and multiplicative methods should be considered for optimal solution. The derivation of Halley method provides a sufficient formation of joint root-finding methods to derive more accurate root finding methods. Prior to the derivation of these joint methods, multiplicative and Volterra Halley methods are derived based on respective Taylor theorems. Moreover, numerical results of the paper show that ordinary Halley method may not converge on an initial value. In this case, a multiplicative method converges for the same initial value.