An Algorithm Derivative-Free to Improve the Steffensen-Type Methods

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.

A Modification of Newton–Raphson Power Flow for Using in LV Distribution System

The Newton–Raphson (NR) method is still frequently applied for computing load flow (LF) due to its precision and quadratic convergence properties. To compute LF in a low voltage distribution system (LVDS) with unbalanced topologies, each branch model in the LVDS can be simplified by defining the neutral and ground voltages as zero and then using Kron’s reduction to transform into a 3 × 3 branch matrix, but this decreases accuracy. Therefore, this paper proposes a modified branch model that is also reduced into a 3 × 3 matrix but is derived from the impedances of the phase-A, -B, -C, neutral, and ground conductors together with the grounding resistances, thereby increasing the accuracy. Moreover, this paper proposes improved LF equations for unbalanced LVDS with both PQ and PV nodes. The improved LF equations are based on the polar-form power injection approach. The simulation results show the effectiveness of the modified branch model and the improved LF equations.

Consistent Tangent Stiffness of a Three-Dimensional Wheel–Rail Interaction Element

Consistent tangent stiffness plays a crucial role in delivering a quadratic rate of convergence when using Newton’s method in solving nonlinear equations of motion. In this paper, consistent tangent stiffness is derived for a three-dimensional (3D) wheel–rail interaction element (WRI element for short) originally developed by the authors and co-workers. The algorithm has been implemented in finite element (FE) software framework (OpenSees in this paper) and proven to be effective. Application examples of wheelset and light rail vehicle are provided to validate the consistent tangent stiffness. The quadratic convergence rate is verified. The speeds of calculation are compared between the use of consistent tangent stiffness and the tangent by perturbation method. The results demonstrate the improved computational efficiency of WRI element when consistent tangent stiffness is used.

A generalized multivariable Newton method

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

The Use of Solvable Directed Graphs in a Jacobi-like Algorithm

In this paper, we introduce a Jacobi-like algorithm (we call D-NJLA) to reduce a real nonsymmetric n × n matrix to a real upper triangular form by the help of solvable directed graphs. This method uses only real arithmetic and a sequence of orthogonal similarity transformations and achieves ultimate quadratic convergence. A theoretical analysis is constructed and some experimental results are given.

Newton's method for the eigenvalue problem of a symmetric matrix

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.