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.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

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.     

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 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.     

The convergence of the perturbed Newton method and its application for ill-conditioned problems

Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples.     

Majorizing sequences for Newton’s method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.     

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.     

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].     

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.     

Split Newton iterative algorithm and its application

Inspired by some implicit-explicit linear multistep schemes and additive Runge-Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction-diffusion equations, such as Burger’s-Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.     

Convergence behavior of Gauss–Newton’s method and extensions of the Smale point estimate theory

The notions of Lipschitz conditions with L$L$ average are introduced to the study of convergence analysis of Gauss–Newton’s method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss–Newton’s method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, γ$\gamma$-conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.     

Steffensen type methods for solving nonlinear equations

In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.     

Improving order and efficiency: Composition with a modified Newton’s method

In this paper a zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to compose a given iterative method with a modified Newton’s method that introduces just one evaluation of the function. To carry out this procedure some classical methods with different orders of convergence are used to obtain new methods that can be generalized in Banach spaces.     

Efficient methods for the inclusion of polynomial zeros

Using a suitable zero-relation and the inclusion isotonicity property, new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived. These methods produce disks in the complex plane that contain the polynomial zeros in each iteration, providing in this manner an information about upper error bounds of approximations. Starting from the basic method of the fourth order, two accelerated methods with Newton’s and Halley’s corrections, having the order of convergence five and six respectively, are constructed. This increase of the convergence rate is obtained without any additional operations, which means that the methods with corrections are very efficient. The convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions, which is of practical importance. Two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods.     

Improved local convergence of Newton’s method under weak majorant condition

We provide a local convergence analysis for Newton’s method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before [14]. Special cases and numerical examples are also provided in this study.     

Newton’s method and its use in optimization

Newton’s method is a basic tool in numerical analysis and numerous applications, including operations research and data mining. We survey the history of the method, its main ideas, convergence results, modifications, its global behavior. We focus on applications of the method for various classes of optimization problems, such as unconstrained minimization, equality constrained problems, convex programming and interior point methods. Some extensions (non-smooth problems, continuous analog, Smale’s results, etc.) are discussed briefly, while some others (e.g., versions of the method to achieve global convergence) are addressed in more details.     

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.     

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.     

A singular value decomposition algorithm based on solving hyperplane constrained nonlinear systems

A new algorithm for singular value decomposition (SVD) is presented through relating SVD problem to nonlinear systems whose solutions are constrained on hyperplanes. The hyperplane constrained nonlinear systems are solved with the help of Newton’s iterative method. It is proved that our SVD algorithm has the quadratic convergence substantially and all singular pairs are computable. These facts are also confirmed by some numerical examples.