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In this paper we study the dynamical behavior of the $$(\alpha ,c)$$ -family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two parameters; so, as we show throughout this paper, its dynamics is really interesting, but complicated. In fact, we have found in the real $$(\alpha ,c)$$ -plane a line in which the corresponding elements of the family have a lower number of free critical points. As this number is directly related with the quantity of basins of attraction, it is probable to find more stable behavior between the elements of the family in this region. [ABSTRACT FROM AUTHOR]

Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation. [ABSTRACT FROM AUTHOR]

This paper presents a new iterative Newton-like method for solving nonlinear equations which is firstly compared with very recent results and we show that it reaches the solution in a lower number of iterations and with a lower total number of function evaluations, considering a variable length of the floating point arithmetic with multi-precision. Then, considering a fix length of the floating point arithmetic with multi-precision, we have compared it with other methods and we have obtained similar results. [ABSTRACT FROM AUTHOR]

The paper establishes the exact error energy compression ratio on each step of the two-step method, which shows a better convergence rate compared with the one-step method with optimal parameter. [ABSTRACT FROM AUTHOR]

The paper deals with a steady state version of a nonlocal nonlinear parabolic problem defined on a bounded polygonal domain. The nonlocal term involved in the strong formulation essentially increases the complexity of the problem and the necessary total computational work. The nonlinear weak formulation of the problem is reduced to a linear one suitable for applications of Newtonian type iterative methods. A discrete problem is obtained by the FEM. A fast and stable iterative method with ninth-order of convergence is applied for solving the discrete problem. The iterative algorithm is described by a pseudo-code. The method is computer implemented and the approximate solutions are presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of computing the $${\mathcal {H}}_\infty $$ -norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the $${\mathcal {H}}_\infty $$ -norm and the structured complex stability radius of a corresponding matrix pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that, we use a new fast iterative scheme which is based on certain rank-1 perturbations of a matrix pencil. Finally, we analyze the performance of our algorithm by using real-world examples. In particular we compare our method with different other algorithms including a recently and independently derived method from Guglielmi, Gürbüzbalaban and Overton. [ABSTRACT FROM AUTHOR]

In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methods. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the order of convergence and the efficiency of four families of iterative methods using frozen divided differences. The first two families correspond to a generalization of the secant method and the implementation made by Schmidt and Schwetlick. The other two frozen schemes consist of a generalization of Kurchatov method and an improvement of this method applying the technique used by Schmidt and Schwetlick previously. An approximation of the local convergence order is generated by the examples, and it numerically confirms that the order of the methods is well deduced. Moreover, the computational efficiency indexes of the four algorithms are presented and computed in order to compare their efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new parallel sparse iterative method (PPSIA) for computing the stationary distribution of large-scale Markov chains. The PPSIA method is based on Markov chain state isolation and aggregation techniques. The parallel method conserves as much as possible the benefits of aggregation, and Gauss-Seidel effects contained in the sequential algorithm (SIA) using a pipelined technique. Both SIA and PPSIA exploit sparse matrix representation in order to solve large-scale Markov chains. Some Markov chains have been tested to compare the performance of SIA, PPSIA algorithms with other techniques such as the power method, and the generalized minimal residual GMRES method. In all the tested models, PPSIA outperforms the other methods and shows a super-linear speed-up. [ABSTRACT FROM AUTHOR]

We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing iterative methods using the embedding approach for calculating the solution. The main aim of this paper is to design a numerical procedure for improving these algorithms. Some numerical examples are illustrated to show the applicability of the methods and to show the efficiency of the proposed algorithms. Here we report the numerical results of large-scaled fuzzy problems. [ABSTRACT FROM AUTHOR]

The paper presents iterative methods for calculating the neutron flux distribution in nonlinear problems of nuclear reactor criticality. Algorithms for solving equations for variations in the neutron flux are considered. Convergence of the iterative processes is studied for two nonlinear problems in which macroscopic interaction cross sections are functionals of the spatial neutron distribution. In the first problem, the neutron flux distribution depends on the water coolant density, and in the second one, it depends on the fuel temperature. Simple relationships connecting the vapor content and the temperature with the neutron flux are used. [ABSTRACT FROM AUTHOR]

Constraint preconditioners have proved very efficient for the solution of ill-conditioned finite element (FE) coupled consolidation problems in a sequential computing environment. Their implementation on parallel computers, however, is not straightforward because of their inherent sequentiality. The present paper describes a novel parallel inexact constraint preconditioner (ParICP) for the efficient solution of linear algebraic systems arising from the FE discretization of the coupled poro-elasticity equations. The ParICP implementation is based on the use of the block factorized sparse approximate inverse incomplete Cholesky preconditioner, which is a very recent and effective development for the parallel preconditioning of symmetric positive definite matrices. The ParICP performance is experimented with in real 3D coupled consolidation problems, proving a scalable and efficient implementation of the constraint preconditioning for high-performance computing. ParICP appears to be a very robust algorithm for solving ill-conditioned large-size coupled models in a parallel computing environment. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variational inequality with a variational inequality constraint over a set of fixed points of a nonexpansive mapping called triple hierarchical variational inequality. We propose two iterative methods, one is implicit and another one is explicit, to compute the approximate solutions of our problem. We present an example of our problem. The convergence analysis of the sequences generated by the proposed methods is also studied. [ABSTRACT FROM AUTHOR]

The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized Navier-Stokes equations (Oseen's problem). We utilize the so-called augmented Lagrangian framework, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen's problem, are of two-by-two block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the block-preconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an element-by-element fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block. [ABSTRACT FROM AUTHOR]

Magnetic field synthesis problems have been discussed at length in scientific literature but they still remain as a topic of research in electrical engineering, physics and medical applications. In these disciplines, there is often a need to design an electromagnetic arrangement which can generate a magnetic field of required distribution. Such an arrangement can also work as an active shield. The aim of an active shield is to generate a specified magnetic field which counteracts the external magnetic fields in a protected region. The idea of active shielding is to construct a suitable arrangement of coils, excited with currents that generate an opposite magnetic field sufficient to cancel out the unwanted external fields. The opposite field must have the same frequency and amplitude as the external field. If the incident field presents a wide bandwidth, the final aim is to generate an opposite field in the same frequency range or at least in a range as large as possible. Two independent methods of magnetic fields synthesis, i.e. iteratively regularized Gauss-Newton method and Genetic Algorithm coupled with Bezier curves-based method, are discussed and compared in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we use the auxiliary principle technique to suggest and analyze an implicit iterative method for solving bifunction variational inequalities. We also study the convergence criteria of this new method under pseudomonotonicity condition. [ABSTRACT FROM AUTHOR]

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT( κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541-559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings. [ABSTRACT FROM AUTHOR]

In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f( x)=0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence { x n} generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods. [ABSTRACT FROM AUTHOR]

Many optimization problems reduce to the solution of a system of linear inequalities (SLI). Some solution methods use relaxed, averaged projections. Others invoke surrogate constraints (typically stemming from aggregation). This paper proposes a blend of these two approaches. A novelty comes from introducing as surrogate constraint a halfspace defined by differences of algorithmic iterates. The first iteration is identical to surrogate constraints methods. In next iterations, for a given approximation x¯, besides the violated constraints in x¯, we also take into consideration the surrogate inequality, which we have obtained in the previous iteration. The motivation for this research comes from the recent work of Scolnik et al. (Appl. Numer. Math. 41, 499—513, 2002), who studied some projection methods for a system of linear equations. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed. [ABSTRACT FROM AUTHOR]

Time Discontinuous Galerkin methods require the factorization of a matrix larger than that exploited in standard implicit schemes. Therefore, they lend themselves to implementations based on predictor-multicorrector solution algorithms. In this paper, various convergent and computationally efficient iterative methods implemented in the unknown displacements for determining the solution of non linear systems are proposed. The iterative solutions presented here differ from those implemented in the unknown velocities in that they are computationally superior. The results of numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements which are designed to evaluate the efficacy of these iterative methods with non-linear systems, show a low-computational expense when compared to earlier iterative schemes. [ABSTRACT FROM AUTHOR]