Narrow Results

We introduce FPSOLVE, an implementation of generic algorithms for solving fixpoint equations over semirings. We first illustrate the interest of generic solvers by means of a scenario. We then succinctly describe some of the algorithms implemented in the tool, and provide some implementation details.

We derive coupled on-line learning rules for the singular value decomposition (SVD) of a cross-covariance matrix. In coupled SVD rules, the singular value is estimated alongside the singular vectors, and the effective learning rates for the singular vector rules are influenced by the singular value estimates. In addition, we use a first-order approximation of Gram-Schmidt orthonormalization as decorrelation method for the estimation of multiple singular vectors and singular values. Experiments on synthetic data show that coupled learning rules converge faster than Hebbian learning rules and that the first-order approximation of Gram-Schmidt orthonormalization produces more precise estimates and better orthonormality than the standard deflation method.

A direct series solution for the Falkner–Skan equation is obtained by first transforming the problem using the Crocco–Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient $\alpha$. Then, $\alpha$ is determined by using the secant method or Newton’s method. The derivative needed for Newton’s method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner–Skan equation for several cases reported previously in the literature.

Superconducting radio frequency cavities meet the demanding performance requirements of modern accelerators and high-brilliance light sources. Their design requires a precise knowledge of their electromagnetic resonances. A numerical solution of Maxwell’s equations is required to compute the resonant eigenmodes, their frequencies and losses due to the complex cavity shape. The consideration of resonances damped by external losses leads to a nonlinear eigenvalue problem. Previous work showed that, using State-Space Concatenation to construct a reduced order model and Newton iteration to solve the arising eigenvalue problem, solutions can be obtained on workstation computers even for large-scale problems without extensive simplification of the structure itself. In this paper, we augment the solution workflow by Beyn’s contour integral algorithm to increase the number of found eigenmodes. Numerical experiments are presented for one academic and two real-life superconducting cavities and partially compared to measurements.

In this paper, an edge point, line point, or curve point is called a feature point. A new approach to extract junction points and to describe feature points is proposed here. It accepts as input data a binary image resulting from a feature detector without thinning. In the binary image, each black point is first classified based on the number of lines passing through it and on a local property that the classes of its neighboring points are almost the same. Next, an aggregation method is presented to group those classified points into several segments. The orientation of each segment is kept either clockwise or counterclockwise. Conic curves are then used to describe these segments. Finally, junction points including corner points, cross points, branch points, and inflection points are located. It is worth mentioning that the proposed method does not use any thinning process and curvature information. The effectiveness of the approach is also verified by one illustrative example and two experimental results.

We compare the iterative root-finding methods of Newton and Halley applied to cubic polynomials in the complex plane. Of specific interest are those "bad" polynomials for which a given numerical method contains an attracting cycle distinct from the roots. This implies the existence of an open set of initial guesses whose iterates do not converge to one of the roots (i.e. the numerical method fails). Searching for a set of bad parameter values leads to Mandelbrot-like sets and interesting figures in the parameter plane. We provide some analytic and geometric arguments to explain the contrasting parameter plane pictures. In particular, we show that there exists a sequence of parameter values λ_n for which the corresponding numerical method has a superattracting n cycle. The λ_n lie at the centers of a converging sequence of Mandelbrot-like sets.

This paper presents an iterative method for locating invariant tori of maps. Specifically, through the introduction of a shift function on the torus corresponding to the projection of the dynamics on the torus onto toral parameters, a discrete system of equations may be formulated whose solution approximates the spatial location of the torus. As invariant tori of continuous flows may be considered invariant under the application of the flow for a fixed time, or may correspond to invariant tori (of one less dimension) of suitably introduced Poincaré maps, the methodology applies to discrete and continuous dynamical systems alike. Moreover, the insensitivity of the method to the local stability characteristics of the invariant torus as well as to the precise nature of the flow on the torus imply, for example, that the method can be employed for the continuation of unstable invariant tori on which the dynamics are attracted to a periodic trajectory as long as the torus is sufficiently smooth. The proposed methodology as well as reduced formulations that result from a priori knowledge about the invariant torus are illustrated through some sample dynamical systems.

The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z² + c)e^Q(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.

We study numerically the $\alpha$- and $\omega$-limits of the Newton maps of quadratic polynomial transformations of the plane into itself. Our results confirm the conjectures posed in a recent work about the general dynamics of real Newton maps on the plane.

In this paper, a class of generalized secant methods was investigated. The convergence of these methods was discussed, and their striking fractal patterns are generated to represent the chaotic behavior of these methods in the complex plane.

We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics of Newton maps $N_f$ associated to real polynomial maps $f:{ℝ}^2\to {ℝ}^2$ with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map $N_f$ has no wandering domain, almost every point under $N_f$ is asymptotic to a fixed point and there is some non-empty open set of points whose $\alpha$-limit equals the set of non-regular points of the Julia set of $N_f$. The first two points were proved by B. Barna in the real one-dimensional case.

Statistical multiplexing in telecommunications packet networks allows more connections to be carried by a router or trunk of a given bandwidth than could be carried using time division multiplexing. We provide a simple model of statistical multiplexing which can be used to quickly estimate the effect of such changes as using a higher bandwidth router or trunk.

The Newton–Kantorovich theorem enjoys a special status, as it is both a fundamental result in Numerical Analysis, e.g., for providing an iterative method for computing the zeros of polynomials or of systems of nonlinear equations, and a fundamental result in Nonlinear Functional Analysis, e.g., for establishing that a nonlinear equation in an infinite-dimensional function space has a solution. Yet its detailed proof in full generality is not easy to locate in the literature.

The purpose of this article, which is partly expository in nature, is to carefully revisit this theorem, by means of a two-tier approach.

First, we give a detailed, and essentially self-contained, account of the classical proof of this theorem, which essentially relies on careful estimates based on the integral form of the mean value theorem for functions of class with values in a Banach space, and on the so-called majorant method. Our treatment also includes a careful discussion of the often overlooked uniqueness issue. An example of a nonlinear two-point boundary value problem is also given that illustrates the power of this theorem for establishing an existence theorem when other methods of nonlinear functional analysis cannot be used.

Second, we give a new version of this theorem, the assumptions of which involve only one constant instead of three constants in its classical version and the proof of which is substantially simpler as it altogether avoids the majorant method. For these reasons, this new version, which captures all the basic features of the classical version could be considered as a good alternative to the classical Newton–Kantorovich theorem.

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math.79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

We propose a new finite element surface fitting method which can handle very large databases. This method uses finite element discretization to find an approximation of a smooth function which minimizes a sum of data residuals and second derivatives under some constraints on data. The finite element discretization derives a large scale constrained quadratic program, which can be reformulated as a system of piecewise linear equations. We develop a preconditioned Newton method to solve the system efficiently. We apply this method to form surfaces over Aomori Region in Japan by geographic databases, such that every bridge became associated with environmental information.

In this paper, we present a finite element scheme with Crank–Nicolson method for solving nonlocal parabolic problems involving the Dirichlet energy. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

In this study, we design a new efficient family of sixth-order iterative methods for solving scalar as well as system of nonlinear equations. The main beauty of the proposed family is that we have to calculate only one inverse of the Jacobian matrix in the case of nonlinear system which reduces the computational cost. The convergence properties are fully investigated along with two main theorems describing their order of convergence. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. From this study, we intend to have more information about these methods in order to detect those with best stability properties. In addition, we also presented a numerical work which confirms the order of convergence of the proposed family is well deduced for scalar, as well as system of nonlinear equations. Further, we have also shown the implementation of the proposed techniques on real world problems like Van der Pol equation, Hammerstein integral equation, etc.

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

Our principle aim in this paper is to present a new reconstruction of classical Chebyshev–Halley schemes having optimal fourth and eighth-order of convergence for all parameters $\alpha$ unlike in the earlier studies. In addition, we analyze the local convergence of them by using hypotheses requiring the first-order derivative of the involved function $f$ and the Lipschitz conditions. In addition, we also formulate their theoretical radius of convergence. Several numerical examples originated from real life problems demonstrate that they are applicable to a broad range of scalar equations, where previous studies cannot be used. Finally, a dynamical study of them also demonstrates that bigger and more promising basins of attractions are obtained.