A quadratically convergent method for computing simple singular roots and its application to determining simple bifurcation points

We present an efficient method for computing roots of mappings on ? ⁿ in the case where the Jacobian has the rankn?1 at the root. For the accurate determination of such a rootx*∈? ⁿ an auxiliary system ofn equations inn+1 variables is constructed which possesses (x ^*, 1) as a turning point. This turning point can be computed by direct methods. We use an adapted method which requires only the solution of (n+1)-dimensional systems of linear equations and the evaluation of one Jacobian and 5 function values per step. This techniques is successfully applied to compute simple bifurcation points by means of a suitable system of nonlinear equations which has the properties mentioned above.     

Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter

Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.     

On the new solutions for a fully fuzzy linear system

In this paper, we propose a new method to present a fuzzy trapezoidal solution, namely “suitable solution”, for a fully fuzzy linear system (FFLS) based on solving two fully interval linear systems (FILSs) that are 1-cut and 0-cut of the related FILS. After some manipulations, two FILSs are transformed to 2n crisp linear equations and 4n crisp linear nonequations and n crisp nonlinear equations. Then, we propose a nonlinear programming problem (NLP) to computing simultaneous (synchronic) equations and nonequations. Moreover, we define two other new solutions namely, “fuzzy surrounding solution” and “fuzzy peripheral solution” for an FFLS. It is shown that the fuzzy surrounding solution is placed in a tolerable fuzzy solution set and the fuzzy peripheral solution is placed in a controllable fuzzy solution set. Finally, some numerical examples are given to illustrate the ability of the proposed methods.     

Computing simple bifurcation points using a minimally extended system of nonlinear equations

The present paper deals with the computation of simple bifurcation points of nonlinear parameter dependent equations. At first, a minimally extended system of nonlinear equations is constructed by addition of one parameter and two equations. This augmented system has an isolated solution which yields to the simple bifurcation point directly. Using the structural properties of this auxiliary system an adapted Newton-like method is developed not requiring evaluations of second derivatives. Finally, the results of some computer experiments show the efficiency of theR-quadratically convergent method.     

Locating,characterizing and computing the stationary points of a function

A method for the localization, characterization and computation of the stationary points of a continuously differentiable real-valued function ofn variables is presented. It is based on the combinatorial topology concept of the degree of a mapping associated with an oriented polyhedron. The method consists of two principal steps: (i) localization (and computation if required) of a stationary point in ann-dimensional polyhedron; (ii) characterization of a stationary point as a minimum, maximum or saddle point. The method requires only the signs of gradient values to be correct and it can be successfully applied to problems with imprecise values.     

The Symm-Wilkinson method for improving an approximate eigenvalue and its associated eigenvector

Here is discussed the Symm-Wilkinson method (called a relaxed algorithm in [4]) for improving an approximate simple eigenvalue of ann×n matrix and a corresponding approximate eigenvector which were obtained by some method. It is shown that their method is a Newton-like method applied to a system of nonlinear equations so that the process converges linearly under the usual assumptions. The Symm-Wilkinson method needs more multiplications than the standard Newton-like method applied to the same equations byn?1 at each step. Therefore, there does not seem to be any great advantage in using the former in place of the latter.     

On the computational complexity of the solution of linear systems with moduli

A problem of solvability for the system of equations of the formAx=D|x|+δ is investigated. This problem is proved to beNP-complete even in the case when the number of equations is equal to the number of variables, the matrixA is nonsingular,A≥D≥0,δ≥0, and it is initially known that the system has a finite (possibly zero) number of solutions. For an arbitrary system ofm equations ofn variables, under additional conditions that the matrixD is nonnegative and its rank is one, a polynomial-time algorithm (of the orderO((max{m, n})³)) has been found which allows to determine whether the system is solvable or not and to find one of such solutions in the case of solvability.     

Statistical control of control-affine nonlinear systems with nonquadratic cost functions: HJB and verification theorems

In statistical control, the cost function is viewed as a random variable and one optimizes the distribution of the cost function through the cost cumulants. We consider a statistical control problem for a control-affine nonlinear system with a nonquadratic cost function. Using the Dynkin formula, the Hamilton-Jacobi-Bellman equation for the nth cost moment case is derived as a necessary condition for optimality and corresponding sufficient conditions are also derived. Utilizing the nth moment results, the higher order cost cumulant Hamilton-Jacobi-Bellman equations are derived. In particular, we derive HJB equations for the second, third, and fourth cost cumulants. Even though moments and cumulants are similar mathematically, in control engineering higher order cumulant control shows a greater promise in contrast to cost moment control. We present the solution for a control-affine nonlinear system using the derived Hamilton-Jacobi-Bellman equation, which we solve numerically using a neural network method.     

On the maximization of the horizontal range and the brachistochrone with an accelerating force and viscous friction

The problem of maximizing the horizontal coordinate of a point moving in the vertical plane driven by gravity, viscous friction, which is proportional to the nth degree of the velocity, and an accelerating force is considered, along with the related brachistochrone problem. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. The qualitative analysis of the trajectories of this system is performed, and their characteristic features that allow us to elaborate the results obtained in other studies are revealed. The optimality of the found extremals is discussed.     

An efficient solution algorithm for boundary element equations

Boundary element techniques result in the solution of a linear system of equations of the type HU = GQ + B, which can be transformed into a system of equations of the type AX = F. The coefficient matrix A requires the storage of a full matrix on the computer. This storage requirement, of the order of n^*n memory positions (n = number of equations), for a very large n is often considered negative for the boundary element method. Here, two algorithms are presented where the memory requirements to solve the system are only n^*(n - 1)/2 and n^*n/4 respectively. The algorithms do not necessitate any external storage devices nor do they increase the computational efforts.     

Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing

This paper proposes a method for finding solutions of arbitrarily nonlinear systems of functional equations through stochastic global optimization. The original problem (equation solving) is transformed into a global optimization one by synthesizing objective functions whose global minima, if they exist, are also solutions to the original system. The global minimization task is carried out by the stochastic method known as fuzzy adaptive simulated annealing, triggered from different starting points, aiming at finding as many solutions as possible. To demonstrate the efficiency of the proposed method, solutions for several examples of nonlinear systems are presented and compared with results obtained by other approaches. We consider systems composed of n   equations on Euclidean spaces ?n${?}^n$ (n variables: x₁, x₂, x₃, ? , x_n).     

Some techniques for solving linear equation systems with guarantee

Methods are described for solving a system of linear equations with error bounds. Rectangular and spherical intervals ofR ⁿ are used combined. The objective is to get guaranteed accuracy with a minimal effort of computing time.     

基于多维泰勒网的非线性时间序列预测方法及其应用

针对非线性时间序列, 提出一种基于多维泰勒网的时间序列预测方法. 其特点在于利用非线性时间序列的观测数据, 通过多维泰勒网得到?? 元一阶多项式差分方程组, 在无需待预测系统的任何先验知识和机理的情况下获得动力学特性描述, 实现对非线性时间序列的预测. 最后分别采用Lorenz 混沌时间序列, 以及某大型建筑在顶升施工安全性监测中的结构振动响应数据进行实证研究, 所得结果表明了该方法的有效性.

     

SolvingN+m nonlinear equations with onlym nonlinear variables

We derive a method for solvingN+m nonlinear algebraic equations inN+m unknownsy≠R ^m andz≠R ^N of the formA(y)z+b(y)=0, where the(N+m) × N matrixA(y) and vectorb(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null(A ^T (y)) the problem is reduced to solvingm nonlinear equations iny only. These equations are solved by Newton's method inm variables. Details of computational implementation and results are provided.     

Über Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten

In this paper an intervalanalytic generalization of the theorem ofPrager-Oettli is used to characterize the solution-set of an, n-system of linear equations with interval coefficients as union of convex polyhedra with special properties. Then it is shown how to deduce from the theorem ofPrager-Oettli a nearly optimaln-interval contained in the solution-set. On the other hand the problem of finding with reasonable expense sharpn-intervals containing the solution-set is solved only for special cases. Some results on this problem are discussed; a numerical example shows the importance of criteria, under which sharpn-intervals can be computed with reasonable effort.     

Efficient classes of Runge-Kutta methods for two-point boundary value problems

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.     

The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems

The method of multiple scales is implemented in Maple V Release 2 to generate a uniform asymptotic solutionO(ϵ^r) for a weakly nonlinear oscillator.In recent work, it has been shown that the method of multiple scales also transforms the differential equations into normal form, so the given algorithm can be used to simplify the equations describing the dynamics of a system near a fixed point.These results are equivalent to those obtained with the traditional method of normal forms which uses a near-identity coordinate transformation to get the system into the “simplest” form.A few Duffing type oscillators are analysed to illustrate the power of the procedure. The algorithm can be modified to take care of systems of ODEs, PDEs and other nonlinear cases.     

Über eine Methode zur Lösung eines linearen Gleichungssystems

In the paper a direct method for the solution of a system of linear equations with a square, regular matrix ofn-th order is given. The method solves this system in \(\frac{{3 - \sqrt 2 }}{6}n^3 + O(n^2 )\) multiplications. By the recursive application of this method the number of multiplications is decreasing to \(\frac{{n^3 }}{6} + O(n^2 )\) . The results of numerical experiments and their comparison with Gauß-elimination are also given.     

A relation between Newton and Gauss-Newton steps for singular nonlinear equations

The Gauss-Newton step belonging to an appropriately chosen bordered nonlinear system is analyzed. It is proved that the Gauss-Newton step calculated after a sequence of Newton steps is equal to the doubled Newton step within the accuracy ofO(‖x?x ^*‖²). The theoretical insight given by the proof can be exploited to derive a Gauss-Newton-like algorithm for the solution of singular equations.     

New exact solutions for modified nonlinear dispersive equations mK(m,n) in higher dimensions spaces

With the use of some proper transformations and symbolic computation, we present a general and unified method for investigating the general modified nonlinear dispersive equations mK(m,n) in higher dimensions spaces. The work formally shows how to construct the general solutions and some special exact-solutions for mK(m,n) equations in higher dimensional spatial domains. The general solutions not only contain the solutions by Wazwaz [Math. Comput. Simulation 59 (2002) 519] but also contain many new compact and noncompact solutions.