Development of a New Multi-step Iteration Scheme for Solving Non-Linear Models with Complex Polynomiography (original) (raw)
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Journal of Nonlinear Sciences and Applications, 2016
In this paper, we describe the new Husehölder's method free from second derivatives for solving nonlinear equations. The new Husehölder's method has convergence of order five and efficiency index 5 1 3 ≈ 1.70998, which converges faster than the Newton's method, the Halley's method and the Husehölder's method. The comparison table demonstrate the faster convergence of our method. Polynomiography via the new Husehölder's method is also presented.
Numerical Methods With Engineering Applications and Their Visual Analysis via Polynomiography
IEEE Access, 2021
Polynomiography is a fusion of Mathematics and Art, which as a software results in a new form of abstract art. Rendered images are through algorithmic visualization of solving a polynomial equation via iteration schemes. Images are beautiful and diverse, yet unique. In short, polynomiography allows us to draw unique and complex-patterned images of polynomials which be re-colored in different ways through different iteration schemes. In the modern age, polynomiography covers a variety of applications in different fields of art and science. The aim of this paper is to present polynomiography using newly constructed root-finding algorithms for the solution of non-linear equations. The constructed algorithms are two-step predictor-corrector methods. For reducing computational cost and making the algorithm more effective, we approximate the second derivative via interpolation technique. These methods have been derived by employing Househölder's method, interpolation technique and Taylor's series expansion. The convergence criterion of the newly developed algorithms has been discussed and proved their sixth-order convergence which is higher than many existing algorithms. To analyze the accuracy, validity and applicability of the proposed methods, several arbitrary and engineering problems have been tested and the obtained numerical results certify the better efficiency of the suggested methods against the other well-known iteration schemes given in the literature. Finally, we present polynomiography through the constructed iteration schemes and give a detailed comparison with the other iteration schemes which reflects the convergence properties and graphical aspects of the constructed algorithms.
Convergence rate for the hybrid iterative technique to explore the real root of nonlinear problems
Mehran University Research Journal of Engineering and Technology
This study explored the convergence rate of the hybrid numerical iterative technique (HNIT) for the solution of nonlinear problems (NLPs) of one variable ( f (x) = 0) . It is sightseen that convergence rate is two for the HNIT. By the HNIT, several algebraic and transcendental NLPs of one variable have been illustrated as an approximate real root for efficient performance. In many instances, HNIT is more vigorous and attractive than well-known conventional iterative techniques (CITs). The computational tool MATLAB has been used for the solution of iterative techniques.
NUMERICAL HYBRID ITERATIVE TECHNIQUE FOR SOLVING NONLINEAR EQUATIONS IN ONE VARIABLE
JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES, 2021
In recent years, some improvements have been suggested in the literature that has been a better performance or nearly equal to existing numerical iterative techniques (NIT). The efforts of this study are to constitute a Numerical Hybrid Iterative Technique (NHIT) for estimating the real root of nonlinear equations in one variable (NLEOV) that accelerates convergence. The goal of the development of the NHIT for the solution of an NLEOV assumed various efforts to combine the different methods. The proposed NHIT is developed by combining the Taylor Series method (TSM) and Newton Raphson's iterative method (NRIM). MATLAB and Excel software has been used for the computational purpose. The developed algorithm has been tested on variant NLEOV problems and found the convergence is better than bracketing iterative method (BIM), which does not observe any pitfall and is almost equivalent to NRIM.
A Novel Multistep Iterative Technique for Models in Medical Sciences with Complex Dynamics
Computational and Mathematical Methods in Medicine
This paper proposes a three-step iterative technique for solving nonlinear equations from medical science. We designed the proposed technique by blending the well-known Newton’s method with an existing two-step technique. The method needs only five evaluations per iteration: three for the given function and two for its first derivatives. As a result, the novel approach converges faster than many existing techniques. We investigated several models of applied medical science in both scalar and vector versions, including population growth, blood rheology, and neurophysiology. Finally, some complex-valued polynomials are shown as polynomiographs to visualize the convergence zones.
Second Order Methods for Solving Non-Linear Equations
Iterative methods for solving non-linear system of equations are developed in this paper. Particularly, the methods are, in all cases, based on the Taylor series of the function f(x) around a generic point x near a root r. The substitution of the Taylor series up to the second order term into the homogeneous equation f(x)=0 is the basis of the methods. Two forms of the second order methods are studied: Richmond's method and Newton's Method. These forms depend on how the local cuadratic problem is solved for each iteration. In the first case, an internal iterative process of the type of a fixed point method is used. In the second case, the internal iterative process is of the type of the Newton-Raphson method. For these methods, three different relaxation factors may be tuned in order to increase the efficiency. The first one is the main relaxation factor. The second one is the relaxation factor for the internal iterative process. The third one is a factor of modulation to select the steepest descent direction, which varies from the Newton-Raphson method to the pure second order method. A well-accepted characteristic of second order methods is that they may also be used to find a local maximum or minimum of the evaluated functions. This feature is taken into account in the algorithm. Finally, the convergence and the stability are studied and evaluated for the second order methods and a significant improvement is noted from the analysis and from the fractal representation of some examples.
Computer Oriented Numerical Scheme for Solving Engineering Problems
Computer Systems Science & Engineering, 2022
In this study, we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously. Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation. Some non-linear equations are taken from physics, chemistry and engineering to present the performance and efficiency of the newly constructed method. Some real world applications are taken from fluid mechanics, i.e., fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations. These non-linear equations are then solved using newly developed simultaneous iterative schemes. Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance, using very less number of function evaluations in each step. Local convergence order of single root finding method is computed using CAS-Maple. Local computational order of convergence, CPU-time, absolute residuals errors are calculated to elaborate the efficiency, robustness and authentication of the iterative simultaneous method in its domain.
Efficient iterative scheme for solving non-linear equations with engineering applications
Applied Mathematics in Science and Engineering, 2022
A family of three-step optimal eighth-order iterative algorithm is developed in this paper in order to find single roots of nonlinear equations using the weight function technique. The newly proposed iterative methods of eight order convergence need three function evaluations and one first derivative evaluation that satisfies the Kung-Traub optimality conjecture in terms of computational cost per iteration (i.e.2 n−1). Furthermore, using the primary theorem that establishes the convergence order, the theoretical convergence properties of our schemes are thoroughly investigated. On several engineering applications, the performance and efficiency of our optimal iteration algorithms are examined to those of existing competitors. The new iterative schemes are more efficient than the existing methods in the literature, as illustrated by the basins of attraction, dynamical planes, efficiency, log of residual, and numerical test examples.
Numerical scheme for estimating all roots of non-linear equations with applications
AIMS Mathematics, 2023
The roots of non-linear equations are a major challenge in many scientific and professional fields. This problem has been approached in a number of ways, including use of the sequential Newton's method and the traditional Weierstrass simultaneous iterative scheme. To approximate all of the roots of a given nonlinear equation, sequential iterative algorithms must use a deflation strategy because rounding errors can produce inaccurate results. This study aims to develop an efficient numerical simultaneous scheme for approximating all nonlinear equations' roots of convergence order 12. The numerical outcomes of the considered engineering problems show that, in terms of accuracy, validations, error, computational CPU time, and residual error, recently developed simultaneous methods perform better than existing methods in the literature.