Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions
Abstract
:1. Introduction
- Sharma’s Method (order three) [9]
- A New Fourth Order Method
- Motivation
- (P1)
- We consider an academic but motivational example that contradicts the above factors. We choose and the function H on the interval by if , and , if , where are parameters satisfying . It is straightforward to see that solves the equation . However, the third derivative of function is not continuous at . However, the results utilizing Taylor series which require the continuity of or even higher cannot guarantee convergence to it.
- (P2)
- There are no prior computable error estimates for . Therefore, the number of iterations required to achieve a specified error tolerance cannot be determined in advance.
- (P3)
- No computable neighborhood of does not contain another solution to (1).
- (P4)
- The choice of to assure convergence of these method to is very difficult or impossible, since the radius of convergence is not usually determined.
- (P5)
- The convergence of iterative methods is established only for the real line.
- (P6)
- For the majority of these techniques, the challenging semi-local examination of convergence is not mentioned in the earlier studies.
- Novelty
- The local convergence is assured using on the derivatives on the method, i.e.,
- A prior error estimates those which are computable, and are provided.Thus, the number of iterations required to achieve the tolerance is known in advance.
- A computable neighbourhood of is given that contains no other solution.
- The radius of convergence can be computed.
- The convergence is assured on S.
2. Convergence 1: Local Analysis
- (E1)
- There is a function that is continuous and non-decreasing (CD), such that the function has zeros in the interval . Let the smallest such zero be denoted as . Define .
- (E2)
- There exists a CD function such that for , defined byThe function has a , which is denoted by .
- (E3)
- Functions , are given as:Let and .Let , , and . Define the functions and by
- (E4)
- The function has a , which is denoted by . SetThen, we have for eachThe functions and relate to the derivative on the method (2).
- (E5)
- Let .The notation is used to denote an open real interval with center x and of radius . Moreover, the symbol is the closure of interval .
- (E6)
- For each , we have
- (E7)
- .
- (i)
- (ii)
- is the usual but not the most appropriate (necessarily) selection. However, in this case, is a simple solution of Equation (1). In Theorem 1, we do not assume is a simple solution. So, the method (2) is also useful to find solutions of multiplicity greater than the one provided that . The preceding notation, together with conditions –, is used in the local analysis of convergence for method (2). Related work on other methods for Banach space valued operators can be found in [1,7].
3. Convergence 2: Semi-Local Analysis
- (M1)
- There exists a CD function so that has a which is denoted by . Set .
- (M2)
- There exists a CD function .It is convenient to set and .Define the sequences and for , and each byThe following conditions assure the convergence of sequences and .
- (M3)
- There exists , such that for eachThis condition together with (28) imply and there exists such that . It is well known that is the unique least upper bound of the sequence . As in the local analysis, the functions and are related to .
- (M4)
- There exists and a non zero number L, such that for eachSet if we take in this condition and use the definition ofThus, , and the iterate exists. So, we can take
- (M5)
- For each
- (M6)
- .
- (i)
- The limit point can be exchanged by in condition .
- (ii)
- In the Proposition 2, assume and under all conditions –.
4. The Convergence Order of Method (2)
5. Numerical Experiments
- Local area convergence
- Semi-local area convergence
- Device Name: HP
- Edition: Windows 10 Enterprise
- Version: 22H2
- Installed RAM: 8.00 GB (7.89 GB usable)
- OS Build: 19045.2006
- Processor: Intel(R) Core(TM) i7-4790 CPU @ 3.60 GHz
- System type: 64-bit operating system, ×64-based processor
5.1. Examples for LAC
5.2. Examples for SLAC
Methods | n | CPU Timing | |||
---|---|---|---|---|---|
NM | 7 | 2.0000 | 0.0034916 | ||
SM | 5 | 3.0000 | 0.0106235 | ||
PM | 5 | 3.0000 | 0.0081417 | ||
OM | 4 | 4.0000 | 0.0152859 |
Methods | n | CPU Timing | |||
---|---|---|---|---|---|
NM | 9 | 2.0000 | 0.0004944 | ||
SM | 6 | 3.0000 | 0.0006272 | ||
PM | 7 | 3.0000 | 0.000563 | ||
OM | 5 | 4.0000 | 0.0008959 |
Methods | n | CPU Timing | |||
---|---|---|---|---|---|
NM | 7 | 2.0000 | 0.001264 | ||
SM | 5 | 3.0000 | 0.0032786 | ||
PM | 5 | 3.0000 | 0.0022388 | ||
OM | 4 | 4.0000 | 0.0049327 |
6. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Behl, R.; Argyros, I.K. Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions. Symmetry 2024, 16, 1179. https://doi.org/10.3390/sym16091179
Behl R, Argyros IK. Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions. Symmetry. 2024; 16(9):1179. https://doi.org/10.3390/sym16091179
Chicago/Turabian StyleBehl, Ramandeep, and Ioannis K. Argyros. 2024. "Convergence of a Family of Methods with Symmetric, Antisymmetric Parameters and Weight Functions" Symmetry 16, no. 9: 1179. https://doi.org/10.3390/sym16091179