scholarly journals Inverse Family of Numerical Methods for Approximating All Simple and Roots with Multiplicity of Nonlinear Polynomial Equations with Engineering Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Shams Forruque Ahmed ◽  
Nazir Ahmad Mir ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Lower Bound ◽  
Order Of Convergence ◽  
Nonlinear Models ◽  
Computational Time ◽  
Computer Algebra Systems ◽  
Multiple Roots ◽  
Polynomial Equations ◽  
Simultaneous Methods ◽  
Engineering Sciences

A new inverse family of the iterative method is interrogated in the present article for simultaneously estimating all distinct and multiple roots of nonlinear polynomial equations. Convergence analysis proves that the order of convergence of the newly constructed family of methods is two. The computer algebra systems CAS-Mathematica is used to determine the lower bound of convergence order, which justifies the local convergence of the newly developed method. Some nonlinear models from physics, chemistry, and engineering sciences are considered to demonstrate the performance and efficiency of the newly constructed family of inverse simultaneous methods in comparison to classical methods in the literature. The computational time in seconds and residual error graph of the inverse simultaneous methods are also presented to elaborate their convergence behavior.

scholarly journals Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Babar Ahmad ◽  
Nazir Ahmad Mir
Keyword(s):  
Lower Bound ◽  
Nonlinear Equation ◽  
Iterative Method ◽  
Local Convergence ◽  
Nonlinear Models ◽  
Numerical Test ◽  
Convergence Order ◽  
Iterative Technique ◽  
Simultaneous Methods ◽  
Engineering Sciences

We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

scholarly journals LCSS-Based Algorithm for Computing Multivariate Data Set Similarity: A Case Study of Real-Time WSN Data

Sensors ◽  
2019 ◽  
Vol 19 (1) ◽  
pp. 166 ◽  
Author(s):  
Rahim Khan ◽  
Ihsan Ali ◽  
Saleh M. Altowaijri ◽  
Muhammad Zakarya ◽  
Atiq Ur Rahman ◽  
...  
Keyword(s):  
Dynamic Programming ◽  
Dna Analysis ◽  
Multivariate Data ◽  
Longest Common Subsequence ◽  
Sensor Data ◽  
Computational Time ◽  
Data Sets ◽  
Data Set ◽  
Classical Dynamic ◽  
Engineering Sciences

Multivariate data sets are common in various application areas, such as wireless sensor networks (WSNs) and DNA analysis. A robust mechanism is required to compute their similarity indexes regardless of the environment and problem domain. This study describes the usefulness of a non-metric-based approach (i.e., longest common subsequence) in computing similarity indexes. Several non-metric-based algorithms are available in the literature, the most robust and reliable one is the dynamic programming-based technique. However, dynamic programming-based techniques are considered inefficient, particularly in the context of multivariate data sets. Furthermore, the classical approaches are not powerful enough in scenarios with multivariate data sets, sensor data or when the similarity indexes are extremely high or low. To address this issue, we propose an efficient algorithm to measure the similarity indexes of multivariate data sets using a non-metric-based methodology. The proposed algorithm performs exceptionally well on numerous multivariate data sets compared with the classical dynamic programming-based algorithms. The performance of the algorithms is evaluated on the basis of several benchmark data sets and a dynamic multivariate data set, which is obtained from a WSN deployed in the Ghulam Ishaq Khan (GIK) Institute of Engineering Sciences and Technology. Our evaluation suggests that the proposed algorithm can be approximately 39.9% more efficient than its counterparts for various data sets in terms of computational time.

Microcomputer-Assisted Mathematics: Polynomial Equations Revisited

1986 ◽  
Vol 79 (9) ◽  
pp. 732-737
Author(s):  
Jillian C. F. Sullivan
Keyword(s):  
High School ◽  
High School Students ◽  
Numerical Methods ◽  
Polynomial Equations ◽  
School Students ◽  
The Real ◽  
Real Solutions ◽  
Quadratic Equations ◽  
Computational Difficulty

Although solving polynomial equations is important in mathematics, most high school students can solve only linear and quadratic equations. This is because the methods for solving cubic and quartic equations are difficult, and no general methods of solution are available for equations of degree higher than four. However, numerical methods can be used to approximate the real solutions of polynomial equations of any degree. Because they involve a great deal of computation they have not traditionally been taught in the schools. Now that most students have access to calculators and computers, this computational difficulty is easily overcome.

Efficient Family of Sixth-Order Methods for Nonlinear Models with Its Dynamics

2019 ◽  
Vol 16 (05) ◽  
pp. 1840008
Author(s):  
Ramandeep Behl ◽  
Prashanth Maroju ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Complex Dynamics ◽  
Nonlinear Models ◽  
Computational Cost ◽  
Sixth Order ◽  
System Of Nonlinear Equations ◽  
Van Der Pol ◽  
Numerical Work ◽  
The Family

In this study, we design a new efficient family of sixth-order iterative methods for solving scalar as well as system of nonlinear equations. The main beauty of the proposed family is that we have to calculate only one inverse of the Jacobian matrix in the case of nonlinear system which reduces the computational cost. The convergence properties are fully investigated along with two main theorems describing their order of convergence. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. From this study, we intend to have more information about these methods in order to detect those with best stability properties. In addition, we also presented a numerical work which confirms the order of convergence of the proposed family is well deduced for scalar, as well as system of nonlinear equations. Further, we have also shown the implementation of the proposed techniques on real world problems like Van der Pol equation, Hammerstein integral equation, etc.

scholarly journals A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Multiple Root ◽  
Multiple Roots ◽  
Asymptotic Error ◽  
Error Constant ◽  
Asymptotic Error Constant

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

scholarly journals Superradiant stability of five and six-dimensional extremal Reissner–Nordstrom black holes

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
Jia-Hui Huang ◽  
Tian-Tian Cao ◽  
Mu-Zi Zhang
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Numerical Methods ◽  
Effective Potential ◽  
Scalar Perturbation ◽  
Polynomial Equations ◽  
Potential Well ◽  
Massive Scalar ◽  
Higher Dimensional ◽  
The Stability

AbstractWe revisit the superradiant stability of five and six-dimensional extremal Reissner–Nordstrom black holes under charged massive scalar perturbation with a new analytical method. In each case, it is analytically proved that the effective potential experienced by the scalar perturbation has only one maximum outside the black hole horizon and no potential well exists for the superradiance modes. So the five and six-dimensional extremal Reissner–Nordstrom black holes are superradiantly stable. The new method we developed is based on the Descartes’ rule of signs for the polynomial equations. Our result provides a complementary support of previous studies on the stability of higher dimensional extremal Reissner–Nordstrom black holes based on numerical methods.

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