Higher-order derivative-free iterative methods for solving nonlinear equations and their basins of attraction. Her current research is focused on dynamical systems and numerical analysis, with an emphasis on iterative methods for solving nonlinear equations and systems, and the dynamical study of the rational functions involved in these processes.
Preface to ”Iterative Methods for Solving Nonlinear Equations and Systems”
- Introduction
- Preliminaries
- Convergence Analysis of Iterative Methods
- Numerical Results
- Conclusions
A modified iterative method based on the third derivative approximant In fact, we let zk=xk−ff(x(xk) . k). Therefore, we obtained some modified two-step iterative methods without higher derivatives of the function.
Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations
- An Iterative Method
- Convergence Analysis
- Numerical Computations
- Ending Comments
This is one of the pioneering and fundamental memory-based methods for solving nonlinear equations. Recall that using a complex initial approximation, one can find the complex roots of the nonlinear equations using (9).
Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their
Numerical Examples
For comparison, we consider the following iterative methods for solving nonlinear equations: ψ4thSB, a method proposed by Sharma et al. The proposed 16thYM method converges in a smaller number of iterations with the smallest error in all tested functions.
Applications to Some Real World Problems 1. Projectile Motion Problem
Figure 1 shows the intersection of the path function, the enveloping parabola and the linear impact function for this application. Here λ is the wavelength of the radiation, it is the absolute temperature of the black body, the Boltzmann's constant, the Planck's constant and the speed of light.
Corresponding Conjugacy Maps for Quadratic Polynomials
Basins of Attraction
To summarize the results, we have compared the average number of iterations and the total number of functional evaluations (TNFE) for each polynomial and each method in Table 11. The method with the fewest number of functional evaluations per points is 8thSA2 closely followed by 4thYM.
Concluding Remarks and Future Work
A family of high-order multipoint iterative methods based on power averaging for solving nonlinear equations.Africa Mathematics. Comparison of two techniques for the development of higher-order two-point iterative methods for solving quadratic equations. SeMA J.2018, 1–22.
Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations
Development of Methods and Their Convergence Analysis
Based on the Potra–Pták method ( 3 ), we propose the following two-step method using a weight function, whose iterative expression is . In view of Theorem2, the following is the sixth-order method yn =xn−ff(x(xnn)),.
Numerical Examples
For some special parameter values, one of the methods marked with M84 is given by. From Tables 1–3 , we observe that the proposed methods are compatible with other existing methods (and sometimes perform better than other methods) of the appropriate order.
Applications
Under uniform flow conditions, the water flow in an open channel is given by Manning's equation. For a rectangular channel with the width and the water depth in the channel, it is known that. Now if it is necessary to determine the water depth in the channel for a certain amount of water, (27) can be rearranged as.
Because of its limited use in engineering, an alternative equation of state for gases is the given van der Waals equation [32–35].
Dynamical Analysis
In this sense, dynamical planes depict basins of attraction of points of attraction. The other fixed point operators have six reflected fixed points in addition to the roots of the corresponding polynomials: zF4,5=±i√55inzF6−9=±i. In order to visualize the basins of attraction of fixed points, several values of γ were chosen to implement the dynamic planes of the M41 method.
In order to analyze the stability of the presented methods, their dynamic behavior has been studied.
Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins
- Some Known Derivative-Free Methods
- Test Problems
- Basins of Attraction
- Concluding Remarks
11] presented memoryless derivative-free iterative methods with convergence orders of eight and sixteen for solving nonlinear equations. An optimal three-point eighth-order memoryless iterative method for solving nonlinear equations with its dynamics. Jpn. Some derivative-free iterative methods and their basins of attraction for nonlinear equations.Discret.
Higher-order derivative-free families of Chebyshev-Halley type methods with or without memory for solving non-linear equations.
A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics
Underlying Purely Imaginary Extraneous Fixed Points
Methods and Special Cases
In the last substep of scheme (1), xn+1=O(e16) can be obtained based on the Kung-Traub conjecture. Relations were sought among all free parameters of Jf(s,u,v), giving us a simple governing equation for extraneous fixed points in the proposed family of methods (1). 25) To get a simple form for Jf(s,u,v), we had to carefully examine how it is related to Kf(s,u). We seek relations between the free parameters that give purely imaginary extraneous fixed points in the proposed family of methods when f(z) = (z2−1)mis is used.
In fact, the degree of Φ(t) will be reduced by destroying the relevant coefficients containing free parameters to make all its roots negative.
The Dynamics behind the Extraneous Fixed Points
Then the external fixed points for case 2 discussed earlier all turn out to be repugnant. Although not described in detail here due to space constraints, external fixed points for Case 1, through a proof similar to that shown in Proposition 2, were shown to be indifferent in [19]. If f(z) = p(z) is a generic polynomial other than (z2−1)m, theoretical analysis of the relevant dynamics may not be feasible due to the greatly increased algebraic complexity.
Basins of attraction for different polynomials are illustrated in Section 5 to observe the complicated dynamics behind fixed points or external fixed points.
Results and Discussion on Numerical and Dynamical Aspects
In the current experiments, W3G7 has slightly better convergence for f5 and slightly worse convergence for all other test functions than the other listed methods. According to the definition of the asymptotic error constantη(ci,Qf,Kf,Jf) =limn→∞|Rf(xn)−α|/. Due to the high order, we take a smaller square and use 601 × 601 initial points, uniformly distributed in the domain.
It seems that most of the methods left have a larger pool for root−i, ie. the boundary does not exactly match the real line.
Conclusions
On the convergence of a family of fourth-order optimal methods and its dynamics.Appl. A class of sixth-order multiple-zero two-point finders of modified double Newton type and their dynamics.Appl. A sixth-order family of modified Newton-like three-point multiroot finders and dynamics behind their external fixed points.Appl.
Basins of attraction for optimal eighth-order methods for finding simple roots of nonlinear equations. Appl.
An Efficient Family of Optimal Eighth-Order Multiple Root Finders
- Construction of the Family
- Some Special Cases of Weight Function
- Now, we suggest a mixture of rational and polynomial weight functions satisfying condition Equation (8) as follows
- Now, we suggest another rational and polynomial weight function satisfying Equation (8) as follows
- Numerical Experiments
The derivation of the proposed class is based on a univariate and trivariate weight function approach. The rest of the paper is organized as follows: Section 2 provides the construction of the new family of iterative methods and the convergence analysis to prove the eighth order of convergence. In section 4, the numerical performance and comparison of some special cases of the new family with the existing ones are given.
In the following result, we demonstrate that the order of convergence of the proposed family reaches optimal order eight.
A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots
Construction of the Higher Order Scheme
In Theorem 1, we illustrate that the constructed scheme reaches at least sixth-order convergence and for α=2 it goes to eighth-order without using any extra functional evaluation. It is interesting to observe that H(η,τ) plays a significant role in the construction of the proposed scheme (for details, please see Theorem 1). Then, the current scheme (4) achieves at least sixth-order convergence for each α, but for a particular value of α=2, it achieves the optimal eighth-order convergence.
The above expression (15) proves that our proposed scheme (4) achieves eighth-order convergence for α=2 using only four functional evaluations per full iteration.
Numerical Experiments
Finally, we show the initial guess and approximate zeros up to 25 significant digits in the corresponding example where an exact solution is not available. For a particular case study, the problem is given as follows: Suppose that a particular population contains 1,000,000 individuals initially, that 300,000 individuals immigrate to the community in the first year, and that 1,365,000 individuals are present at the end of a year. The determination of the volume V of the gas in terms of the remaining parameters requires the solution of a non-linear equation in V,.
The desired zero of the function f4 above is α=2 with a multiplicity of order 50 and for this problem we choose an initial guess x0=2.1.
Conclusions
A comparison of methods for accelerating the convergence of Newton's method for multiple polynomial roots.ACM Signum Newsl.
An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots
Development of the Scheme and Convergence Analysis
The following theorem demonstrates that the proposed scheme in equation (9) achieves the optimal eighth order of convergence without increasing the number of functional evaluations. We expand f(xn) and f(xn) with Taylor's series expansion aboutx=μusing Mathematica (computer-based algebra software) to obtain. To obtain eighth-order convergence, we are restricted to choosing the values of parameters given by:
The above error equation (equation (22)) confirms that the presented scheme in equation (9) achieves the optimal order of convergence eight by using only four functional evaluations (using .
Special Cases of Weight Functions
The polynomial form of the weight function satisfying the conditions in Equation (10) can be represented as
- Numerical Tests
Therefore, we conclude that the proposed family is comparable and robust among existing multi-root methods. In this paper, we present a new family of eighth-order optimal schemes for finding multiple roots of nonlinear equations. Finally, numerical and dynamic tests confirmed the theoretical results and showed that the three members of the new family SM-1, SM-2 and SM-3 outperform the existing multi-root methods.
An efficient family of optimal fourth-order iterative methods for finding multiple roots of nonlinear equations. Proc.
An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations
Algorithm: Motivation and Convergence Result
Throughout this article, we assume the following (G1) The mappingFis monotone, that is. G2) The mapping Fis Lipschitz continuous, i.e. there exists a positive constantL such that F(x)−F(y) ≤Lx−y, ∀x,y∈Rn. Assuming that assumptions (G1)–(G3) hold, then the sequences {xk}and {zk} generated by Algorithm1(CGD) are bounded. By the definition of zk, Equation (12), monotonicity of and the Cauchy-Schwatz inequality, we get. 16) The boundedness of the sequence {xk}, together with Equations (15) and (16), implies that the sequence {zk} is bounded.
Since {zk} is bounded, then for every ¯x∈Ψ, the sequence {zk−x}¯ is also bounded, that is, there exists a positive constantν>0 such that.
Suppose lim inf
The performance profile criteria considered are; number of iterations, CPU time (in seconds) and number of function evaluations. Figure 4 is a graph of the decrease in the residual norm against the number of iterations for problem 9 with x4 as the starting point. The CGD method is an extension of the well-known conjugate gradient method for unconstrained optimization CG-DESCENT [20] for solving regularized 1-norm problems.
Finally, we provide some numerical examples to show the effectiveness of the proposed method in terms of number of iterations, CPU time, and number of function evaluations compared with some related methods for solving convex constrained nonlinear monotonic equations and its application in image restoration problems.
Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications
Semi-Local Convergence Analysis
Then, we first present a semi-local convergence result relating the major sequence {tn} to Newton's method and hypotheses (A). Then, the sequence {zn} generated by Newton's method is well defined, remains in S(z0,t∗) and converges to a solution z∗∈S(z0,t∗) of the equation G(x) =0. Moreover, by repeating Newton's method, equations (23) and (25) and induction hypotheses, we obtain that.
Note that the convergence criterion is even weaker than the corresponding one for the modified Newton's method given in [11].
Application: Planck’s Radiation Law Problem We consider the following problem [15]
Viru: Ko investigación oipytyvõ parcialmente programa de apoyo investigación Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14, proyecto MTM2014-52016-C2-1-P Ministerio español de Ciencia ha Innovación rehegua. Ko artíkulo oñemyasãi Creative Commons Atribución (CCBY) Licencia rupive (http://creativecommons.org/licenses/by/4.0/).
A Third Order Newton-Like Method and Its Applications
