J.Basic.Appl.Sci.Res., 8(3)9-13,2018 ISSN 2090-4304
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Umair Khalid Qureshi
Department of Basic Science & Related Studies, Mehran University of Engineering and technology, Pakistan
Received: December 23, 2017 Accepted: March 1, 2018
ABSTRACT
This studyhas been developed a Modified Iterative Method for estimating a single root of nonlinear equations and analyzed. The proposed Modified Technique is a Modification of Regula-Falsi Method and Newton Raphson Method, and it is cubic order of convergence. The Modified Cubic Convergence Method faster than cubic Methods such as variant of Newton Raphson Method and Halley Method. The comparison in table-1 for different test functions demonstrate the faster convergence of proposed method. EXCEL and C++ are implemented for the results and graphical representations. KEYWORD: Non-linear equations, cubic methods, order of convergence, Absolute percentage error, accuracy
Finding roots of nonlinear equations efficientlyhas widespread applications in numerous branches of pure science and applied science, it is deliberated in general framework of the nonlinear equations [2], such as nonlinear equations
0
For estimating a root of non-linear equations0, utmost researchers and scientists took interest and lots of variants of accelerated methods had been introduced. Such as most commonly used bracketing techniques includes bisection method and regula-falsi method [1]. These methods are useful bracketing techniques which require two initial guesses. Both techniques are linear convergence order, while in some cases regula-falsi technique struggles due to sluggish convergence. Besides, Newton Raphson Method are fast converging numerical techniques but are not reliable because keeping a kind of pitfall [3].
–
`Where n=0,1,2, … However, it is most valuable and modest numerical techniques. In recent years, in literature several modifications had been done byusing these techniques for solving nonlinear equations [4-6].Furthermore, modification in Newton Raphson Method to increasing order of convergence and computational efficacy a Variant of Newton Raphson Technique had been proposed byusing Quadrature rule [9], such as
–
`
2
``
Similar investigation, combined the Bisection, Regula-Falsi and Newton Raphson techniques are given some techniques for solving non-linear equations with better accuracy sight as well as iteration perspective [7-8]. Correspondingly, in this paper a Modified cubic iterated method has been suggested. The proposed method is assortment of regula-falsi method and Newton Raphson Method. The Modified cubic method has been compared
Corresponding Author: Umair Khalid Qureshi, Department of Basic Science & Related Studies, Mehran University of Engineering and technology, Pakistan. Email:khalidumair531@gmail.com
9
Qureshi, 2018
with cubic method in reference [9, 10]. The Modified Method is fast converging and more efficient to approaching the root.
The new developed iterative method is based on Regula-FalsiMethod and Newton Raphson Method, such as
Or1
Where,
–2
`Byusing 21,we get
–`
–
`
or
–`
–
`
Finally, we obtain
``–
`
Hence this is a proposed method.
The following statement will be shows that the developed Method is keeping cubic convergence.
Proof:
Using the relationain Taylor series, thereforefrom Taylor series we estimate
``
,`–with using this condition cand ignoring higher order term for easy to
``
solve, such as
```12Byusing 2,we get `
`12
1121121c
Thus,
`cc`1
–c`1
`Byusing ,3,we get
J. Basic. Appl. Sci. Res., 8(3)9-13, 2018
``
``12`12c`1[112]
1c11
[112]1
c1[12]1
1
[122]1132113232332332
Hence this proves that the established iterative method is cubic order of convergence.
This segment the established method is practical on few examples of nonlinear equations and tested developed method with the variant ofNewton Raphson Method [11]
–
`2``
and HalleyMethod [12]
2`2```
Since numerical result in table-1, it has been detected that the cubic iterative method is reducing the number of iterations which is less than the number of iteration of cubic methods and likewise accuracy side. Mathematical package such as C++ and EXCEL have used to justify the proposed Iterative Method. From the results and comparison of proposed cubic iterative method with the cubic methods that the proposed cubic iterative method is well execution than prevailing cubic methods.
Table-1
FUNCTIONS | METHODS | ITERATIONS | X | A E% |
---|---|---|---|---|
Sinx-x+1 | Halley Method | 2 | 4.33922e-5 | |
X=2 | Variant Newton Method | 3 | 1.93456 | 1.19209e-7 |
New Method | 2 | 3.01600e-5 | ||
2x-lnx-7 | Halley Method | 3 | 3.10421e-4 | |
X=4 | Variant Newton Method | 2 | 4.21991 | 1.19209e-5 |
New Method | 2 | 2.86102e-6 | ||
x 3 -9x+1 | Halley Method | 2 | 1.53050e-4 | |
X=0 | Variant Newton Method | 4 | 0.111264 | 1.49012e-7 |
New Method | 2 | 4.24683e-7 | ||
Cosx-x3 X=1 | Halley Method Variant Newton Method New Method | 2 6 2 | 0.865474 | -31.28573e 1.78814e-7 1.76102e-3 |
xxe –2 | Halley Method | 5 | -85.96046e | |
X=3 | Variant Newton Method | 10 | 0.852605 | 5.96046e-8 |
New Method | 5 | 2.98023e-8 |
Qureshi, 2018
In this paper, a Modified Iterated Method has been designed to find the root of nonlinear equations. The Modified Iterated Method has a cubic order of convergence, and it is derived from Regula-Falsi Method and Newton Raphson Method. Throughout the study, we can be concluded that the Modified Cubic Method is good execution forthe comparison of cubic Methods such as variant of Newton Raphson Method and Halley Method. A new developed method is fast converging to approaching the root. Henceforth the proposed method is superior andperforming well for solving non-linear equations.
REFERENCES
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