V.B. Kumar Vatti^{1}, Ramadevi Sri^{2} and M.S.Kumar Mylapalli^{2}
^{1}Dept. of Engineering Mathematics, Andhra University, Visakhapatnam, India
^{2}Dept. of Mathematics, Gitam University, Visakhapatnam, India,
Corresponding author Email: drvattivbk@yahoo.co.in
DOI : http://dx.doi.org/10.13005/ojcst/10.04.19
ABSTRACT:In this paper, the Eighteenth Order Convergent Method (EOCM) developed by Vatti et.al is considered and this method is further studied without the presence of second derivative. It is shown that this method has same efficiency index as that of EOCM. Several numerical examples are given to illustrate the efficiency and performance of the new method.
AMS Subject Classification: 41A25, 65K05, 65H05.
KEYWORDS:Iterative method; Nonlinear equation; Newton’s method; Convergence analysis; Higher order convergence
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Introduction
It is well known that a wide class of problems, which arises in diverse disciplines of mathematical and engineering science can be studied by the nonlinear equation of the form
f (x) = 0 (1.1)
Where f : D ⊂ R → R is a scalar function on an open interval D and f (x) may be algebraic, transcendental or combined of both. The most widely used algorithm for solving (1.1) by the use of value of the function and its derivative is the well known quadratic convergent Newton’s method (NM) given by
starting with an initial guess x0 which is in the vicinity of the exact root x* The efficiency index of Newton’s method is 2√2 = 1.4142
The Extrapolated Newton’s method (ENM) suggested by V.B.Kumar, Vatti et.al [11] which is developed by extrapolating Newton’s method (1.2) introducing a parameter ‘αn’ given by
Which is same as Halley’s method having third order convergence which requires three functional evaluations. The efficiency index of this method is 3√3 = 1.4422
A three step Predictor-corrector Newton’s Halley method (PCNH) suggested by Mohammed and Hafiz [10], is given by:
For a given x0 compute xn+1 by using
This method has eighteenth order convergence and its efficiency index is 8√18 = 1.4352
The three step Eighteenth Order Extrapolated Newton’s method(EOCM) developed by Vatti et.al [12] is given by:
For a given x_{0}, compute x_{n+1} by the iterative schemes
This method has eighteenth order convergence and its efficiency index is 8√18 = 1.4352
In section 2, we consider the Second derivative free Eighteenth Order Extrapolated Newton’s method and discuss the convergence criteria of this method in section 3. Few numerical examples are considered to show the superiority of this method in the concluding section.
Second Derivative Free Eighteenth Order Convergent Method (Seocm):
Considering the Eighteenth Order Extrapolated Newton’s method(EOCM) (1.8) with (1.9) and (1.10), and expanding f (w_{n}) about x_{n} we obtain.
and rewriting equations (2.2), (2.3) in (1.8), we thus have the following algorithm:
Algorithm 2.1: For a given x_{0}, compute x_{n+1} by the iterative schemes
Where p_{n} and p_{n} are as given in (2.2) and (2.3).
This algorithm can be called as Second derivative free Eighteenth Order Extrapolated Newton’s method (SEOCM) and it requires eight functional evaluations.
Convergence Criteria
Theorem 3.1.
Let x* ∈ D be a single zero of a sufficiently differentiable function f : D ⊂ R → R for an open interval D and let x_{0} be in the vicinity of x* then the algorithm (2.1) has eighteenth order convergence.
Numerical Examples
We consider the same examples considered by Mohammed and Hafiz [10] and V.B.Kumar, Vatti et.al [13] and compared EOCM with NM and PCNH methods. The computations are carried out by using mpmath-PYTHON software programming and comparison of number of iterations for these methods are obtained such that |x_{n+1}- x_{n}| <10^{-201} and |f(x_{n+1})|<10^{-201.}
It is evident from these tabulated values that SEOCM is superior to the methods (1.2) and (1.7) considering the number of iterations and accuracy and the rate at which SEOCM converged and the convergence rate is almost same as that of the method (1.8). Of these methods, SEOCM is free from second derivatives.
Table 1: Comparison of different methods |
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