Symmetries of the Julia sets of Newton’s method for multiple root

The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.     

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.     

Ostrowski’s fourth-order iterative method speedily solves cubic equations of state

Pressure-volume-temperature (P-V-T) data are required in simulating chemical plants because the latter usually involve production, separation, transportation, and storage of fluids. In the absence of actual experimental data, the pertinent mathematical model must rely on phase behaviour prediction by the so-called equations of state (EOS). When the plant model is a combination of differential and algebraic equations, simulation generally relies on numerical integration which proceeds in a piecewise fashion unless an approximate solution is needed at a single point. Needless to say, the constituent algebraic equations must be efficiently re-solved before each update of derivatives. Now, Ostrowski’s fourth-order iterative technique is a partial substitution variant of Newton’s popular second-order method. Although simple and powerful, this two-point variant has been utilised very little since its publication over forty years ago. After a brief introduction to cubic equations of state and their solution, this paper solves five of them. The results clearly demonstrate the superiority of Ostrowski’s method over Newton’s, Halley’s, and Chebyshev’s solvers.     

L-stable Simpson’s 3/8 rule and Burgers’ equation

In this paper we develop an unconditionally stable third order time integration formula for the diffusion equation with Neumann boundary condition. We use a suitable arithmetic average approximation and explicit backward Euler formula and then develop a third order L-stable Simpson’s 3/8 type formula. We also observe that the arithmetic average approximation is not unique. Then L-stable Simpson’s 3/8 type formula and Hopf-Cole transformation is used to solve Burger’s equation with Dirichlet boundary condition. It is also observed that this numerical method deals efficiently in case of inconsistencies in initial and boundary conditions.     

On the convergence of the secant method under the gamma condition

We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.      

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

On interpolation variants of Newton’s method for functions of several variables

A generalization of the variants of Newton’s method based on interpolation rules of quadrature is obtained, in order to solve systems of nonlinear equations. Under certain conditions, convergence order is proved to be 2d+1, where d is the order of the partial derivatives needed to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton’s classical method, whose convergence order is d+1 under the same conditions.     

A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich

From Kantorovich’s theory we establish a general semilocal convergence result for Newton’s method based fundamentally on a generalization required to the second derivative of the operator involved. As a consequence, we obtain a modification of the domain of starting points for Newton’s method and improve the a priori error estimates. Finally, we illustrate our study with an application to a special case of conservative problems.     

On Newton’s method for solving equations containing Fréchet-differentiable operators of order at least two

We provide sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation containing operators that are Fréchet-differentiable of order at least two, in a Banach space setting. Numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot [J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79(1997) 131-145; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985) 71-84].     

Sixteenth-order method for nonlinear equations

Modification of Newton’s method with higher-order convergence is presented. The modification of Newton’s method is based on King’s fourth-order method. The new method requires three-step per iteration. Analysis of convergence demonstrates that the order of convergence is 16. Some numerical examples illustrate that the algorithm is more efficient and performs better than classical Newton’s method and other methods.     

Numerical inversion of a general incomplete elliptic integral

We present a numerical method to invert a general incomplete elliptic integral with respect to its argument and/or amplitude. The method obtains a solution by bisection accelerated by half argument formulas and addition theorems to evaluate the incomplete elliptic integrals and Jacobian elliptic functions required in the process. If faster execution is desirable at the cost of complexity of the algorithm, the sequence of bisection is switched to allow an improvement by using Newton’s method, Halley’s method, or higher-order Schröder methods. In the improvement process, the elliptic integrals and functions are computed by using Maclaurin series expansion and addition theorems based on the values obtained at the end of the bisection. Also, the derivatives of the elliptic integrals and functions are recursively evaluated from their values. By adopting 0.2 as the critical value of the length of the solution interval to shift to the improvement process, we suppress the expected number of bisections to be as low as four on average. The typical number of applications of update formulas in the double precision environment is three for Newton’s method, and two for Halley’s method or higher-order Schröder methods. Whether the improvement process is added or not, our method requires none of the procedures to compute the incomplete elliptic integrals and Jacobian elliptic functions but only those to evaluate the complete elliptic integrals once at the beginning. As a result, it runs fairly quickly in general. For example, when using the improvement process, it is around 2–5 times faster than Newton’s method using Boyd’s starter (Boyd (2012) [25]) in inverting E(φ|m)$E\left(\phi \right|m)$, Legendre’s incomplete elliptic integral of the second kind.     

Third order iterative methods free from second derivative for nonlinear systems

In this work we present a family of predictor-corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.     

Weaker conditions for the convergence of Newton’s method

Newton’s method is often used for solving nonlinear equations. In this paper, we show that Newton’s method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Pták (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.     

Numerical analysis of a quadratic matrix equation

The quadratic matrix equation AX²+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional iteration methods based on Bernoulli’s method.Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions.We show that functional iteration applied to the quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem (²A + B + C)x = 0.     

Third and fourth order iterative methods free from second derivative for nonlinear systems

In this work, we develop a family of predictor-corrector methods free from second derivative for solving systems of nonlinear equations. In general, the obtained methods have order of convergence three but, in some particular cases the order is four. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Newton’s classical method and with other recently published methods.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

Local convergence of Newton’s method under majorant condition

A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications.     

A modified Newton’s method for best rank-one approximation to tensors

In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient.