A Variational Approach for Finding Initial Approximation in Numerical Solution of an Equation

Abstract

An attempt to estimate the initial approximation in numerical solution of equation is introduced. In this paper, the aim is to find strong starting approximation from favorable initial approximation in the field of both real and complex equations. We have established that the strong initial approximation convergent to the root of the equation under iteration.

Introduction: Each numerical method needs a good starting approximation to diminish the frequency of iteration and to converge the root of a function.  Generally intermediate value theorem (IVT) is widely used to estimate an interval containing the root of a real function and as it is well known that the theorem depends on the alternative sign of a function. When the range of a function is non-negative or non-positive in its natural domain, IVT does not suggest any interval containing root.

Sometimes the linear interpolation method is also recommending to estimate the initial approximation of a root, but here we need prior information about the interval containing root; that is, the linear interpolation technique is depends on IVT. In some cases, graph of the corresponding functions are also used to estimate initial approximation as an alternative. But tracing of graph is not always an easy task without using mathematical tool. Supposed that  is a real valued function and it can be expressed as other two real valued function such as  (say), then their intersection point can be used as an initial approximation of the function . But here difficulty is that, the solution of the expression  may not be calculate easily.

The content of this paper is organized as follows: we briefly review the literature associated with initial approximation in Section-2. The main results on proposed technique are studied in both real and complex field with examples in Section-3 and Section-4 respectively. The paper end with a conclusion in Section-5.

Objectives: The prime objective of this paper is proposing a technique to estimate a better initial approximation to a root of a function. In this technique our main focus is to overcome the aforementioned limitations and make it user friendly.

Methods: The Wiersma basin attraction root point Theorem, Schrder Fixed Point Theorem are used. The concept of variational inenequality problem approach is used to find the root interval of the basin attraction root point of a function.

Results: The basin of attraction of the equations of real function and complex function are established. The results are established with the concrete examples.

Conclusions: In this piece of work, we suggested a new strategy to estimate the real and complex initial approximation for both real and complex equations respectively. Any standard iteration by function will definitely convergent to the root by choosing these initial approximation and it will also diminish the frequency of iteration.