scholarly journals Efficient iterative methods for finding simultaneously all the multiple roots of polynomial equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Computational Efficiency ◽  
Numerical Test ◽  
Polynomial Equation ◽  
Multiple Roots ◽  
Simultaneous Methods ◽  
Numerical Performance ◽  
Very High ◽  
High Computational Efficiency

AbstractTwo new iterative methods for the simultaneous determination of all multiple as well as distinct roots of nonlinear polynomial equation are established, using two suitable corrections to achieve a very high computational efficiency as compared to the existing methods in the literature. Convergence analysis shows that the orders of convergence of the newly constructed simultaneous methods are 10 and 12. At the end, numerical test examples are given to check the efficiency and numerical performance of these simultaneous methods.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350078 ◽  
Author(s):  
XIAOFENG WANG ◽  
TIE ZHANG
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Type Method ◽  
New Methods ◽  
Hermite Interpolating Polynomial ◽  
With Memory ◽  
Theoretical Results ◽  
Very High ◽  
High Computational Efficiency

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

Efficient Methods for Numerical Modeling of Laminar Friction in Fluid Lines

2006 ◽  
Vol 128 (4) ◽  
pp. 829-834 ◽  
Author(s):  
D. Nigel Johnston
Keyword(s):  
Computational Efficiency ◽  
Method Of Characteristics ◽  
High Accuracy ◽  
Slight Reduction ◽  
Improved Method ◽  
Frequency Dependent ◽  
Transmission Line Method ◽  
Laminar Pipe Flow ◽  
Very High ◽  
High Computational Efficiency

An improved method for simulating frequency-dependent friction in laminar pipe flow using the method of characteristics is proposed. It has a higher computational efficiency than previous methods while retaining a high accuracy. By lumping the frequency-dependent friction at the ends of the pipeline, the computational efficiency can be improved further, at the expense of a slight reduction in accuracy. The technique is also applied to the transmission line method and found to give a significant improvement in accuracy over previous methods, while retaining a very high computational efficiency.

scholarly journals A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Naila Rafiq ◽  
Mudassir Shams ◽  
Nazir Ahmad Mir ◽  
Yaé Ulrich Gaba
Keyword(s):  
Computer Method ◽  
Polynomial Equations ◽  
Literature Analysis ◽  
Complex Polynomials ◽  
Highly Efficient ◽  
Engineering Problems ◽  
Numerical Performance ◽  
And Performance ◽  
Very High ◽  
High Computational Efficiency

A highly efficient two-step simultaneous iterative computer method is established here for solving polynomial equations. A suitable special type of correction helps us to achieve a very high computational efficiency as compared to the existing methods so far in the literature. Analysis of simultaneous scheme proves that its convergence order is 14. Residual graphs are also provided to demonstrate the efficiency and performance of the newly constructed simultaneous computer method in comparison with the methods given in the literature. In the end, some engineering problems and some higher degree complex polynomials are solved numerically to validate its numerical performance.

scholarly journals On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
System Of Nonlinear Equations ◽  
Single Root ◽  
Root Finding

AbstractIn this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

2020 ◽  
Author(s):  
Mudassir Shams ◽  
Nazir Mir ◽  
Naila Rafiq
Keyword(s):  
Simultaneous Determination ◽  
Iterative Methods ◽  
Linear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
Single Root ◽  
Non Linear ◽  
Non Linear Equations

We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals On an efficient method for the simultaneous approximation of polynomial multiple roots

2014 ◽  
Vol 8 (1) ◽  
pp. 73-94 ◽  
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic ◽  
Jovana Dzunic
Keyword(s):  
Convergence Analysis ◽  
Computational Efficiency ◽  
Positive Root ◽  
Single Step ◽  
Multiple Roots ◽  
Step Method ◽  
Computational Aspects ◽  
Parallel Mode ◽  
Fourth Order Method ◽  
High Computational Efficiency

An iterative method in parallel mode for the simultaneous determination of multiple roots of algebraic polynomials is stated together with its single-step variant. These methods are more efficient compared to all simultaneous methods based on fixed point relations. To attain very high computational efficiency, a suitable correction resulting from Li-Liao-Cheng?s two-point fourth order method of low computational complexity and Gauss-Seidel?s approach are applied. Considerable increase of the convergence rate is obtained applying only n additional polynomial evaluations per iteration, where n is the number of distinct roots. A special emphasis is given to the convergence analysis and computational efficiency of the proposed methods. The presented convergence analysis shows that the R-order of convergence of the proposed single-step method is at least 2 + ?v; where ?v?2 (4,6) is the unique positive root of the polynomial gv(t) = tn-4n-1 t-22n-1: The convergence order of the corresponding total-step method is six. Computational aspects and some numerical examples are given to demonstrate high computational efficiency and very fast convergence of the proposed methods.

Efficient Methods for Numerical Modelling of Laminar Friction in Fluid Lines

2004 ◽  
Author(s):  
D. Nigel Johnston
Keyword(s):  
Computational Efficiency ◽  
Method Of Characteristics ◽  
High Accuracy ◽  
Slight Reduction ◽  
Improved Method ◽  
Frequency Dependent ◽  
Transmission Line Method ◽  
Laminar Pipe Flow ◽  
Very High ◽  
High Computational Efficiency

An improved method for simulating frequency-dependent friction in laminar pipe flow using the Method of Characteristics (MOC) is proposed. It has a higher computational efficiency than previous methods whilst retaining a high accuracy. By lumping the frequency-dependent friction at the ends of the pipeline the computational efficiency can be improved further, at the expense of a slight reduction in accuracy. The technique is also applied to the Transmission Line Method (TLM) and found to give a significant improvement in accuracy over previous methods, whilst retaining a very high computational efficiency.

scholarly journals Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Naila Rafiq ◽  
Saima Akram ◽  
Nazir Ahmad Mir ◽  
Mudassir Shams
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Dynamical Behavior ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Single Root ◽  
Root Finding ◽  
The Stability

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

scholarly journals New Mono- and Biaccelerator Iterative Methods with Memory for Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
T. Lotfi ◽  
F. Soleymani ◽  
S. Shateyi ◽  
P. Assari ◽  
F. Khaksar Haghani
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Acceleration Of Convergence ◽  
Convergence Orders ◽  
With Memory ◽  
High Computational Efficiency

Acceleration of convergence is discussed for some families of iterative methods in order to solve scalar nonlinear equations. In fact, we construct mono- and biparametric methods with memory and study their orders. It is shown that the convergence orders 12 and 14 can be attained using only 4 functional evaluations, which provides high computational efficiency indices. Some illustrations will also be given to reverify the theoretical discussions.

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