An Approach to Strengthen Some Particular Numerical Iterative Methods

Abstract

In this paper our main endeavor is to strengthen the usual routine root finding strategy of a numerical iterative method such as the Bisection method the Regula Falsi method the Newton Raphson method etc of a given equation fx 0 To achieve this goal we first frame this usual routine root finding strategy as a theorem and referred to the weak common root finding strategy of a numerical iterative method WCRFSNIM defined on the real axis We point out certain limitations and weaknesses of the WCRFSNIM theorem such as a numerical iterative method lies in this theorem unable to provide i an even number of real roots in the open interval of the intermediate value theorem IMVT defined on the real axis of a given equations fx 0 ii an even number of complex roots in the open disk of the intermediate value theorem IMVT defined on the complex plane of a given equation fz 0 and iii finalizing a proper choice of an initial approximation PCIA from a neighborhood point of such an odd or even number of real and complex roots Further we develop a comprehensive plan to extend the WCRFSNIM theorem in to the strong theorems so as the above limitations and weaknesses i iii of the WCRFSNIM theorem are overcome by these our developed extended strong theorems These extended theorems we call the strengthened common root finding strategy of a numerical iterative method SCRFSNIM defined on the Real Axis and next on the Complex Plane To demonstrate the performance and effectiveness of the SCRFSNIM theorem defined on the Real Axis and next the SCRFSNIM theorem defined on the Complex Plane certain typical application oriented examples are illustrated Mathematics Subject Classification 65B99 65G40.