On the convergence of an iterative method for bounding the inverses of an interval matrix

An iterative method for improving an initial inclusion for the inverses of an interval matrix is [roposed. For this we prove a convergence theorem with sufficient conditions on the width of the interval matrix in question and of the initial inclusion so that our method yields an improvement. Numerical examples are discussed.     

A symmetric iterative interval method for systems of nonlinear equations

An iterative method for nonlinear systems of equations is presented that is based on the idea of symmetric methods known from linear systems. Due to the use of interval arithmetic the convergence to a solution can be proved under relatively weak conditions provided an initial inclusion of that solution is known. The concept of symmetry leads to a reduction of computation time compared to some well-known methods.     

Accelerating Krawczyk-like interval algorithms for the solution of nonlinear systems of equations by using second derivatives

Recently the properties of Krawczyk-like iterative interval methods for the solution of systems of nonlinear equations have been discussed in several papers (e.g. [2], [3], [5], [6]). These methods converge to a solution under relatively weak conditions provided an initial inclusion vector is known. In the present paper we describe a method that improves the convergence speed for an important class of problems by using second partial derivatives. This method is particularly interesting for large systems with a Jacobi matrix whose off-diagonal coefficients are all constant.     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37–55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.     

Mean square numerical solution of random differential equations: Facts and possibilities

This paper deals with the construction of numerical solutions of random initial value differential problems. The random Euler method is presented and the conditions for the mean square convergence are established. Numerical examples show that random Euler method gives good results even if the sufficient convergence conditions are not satisfied.     

非线性迭代学习控制问题的延拓修正牛顿法

对于非线性迭代学习控制问题,提出基于延拓法和修正Newton法的具有全局收敛性的迭代学习控制新方法.由于一般的Newton型迭代学习控制律都是局部收敛的,在实际应用中有很大局限性.为拓宽收敛范围,该方法将延拓法引入迭代学习控制问题,提出基于同伦延拓的新的Newton型迭代学习控制律,使得初始控制可以较为任意的选择.新的迭代学习控制算法将求解过程分成N个子问题,每个子问题由换列修正Newton法利用简单的递推公式解出.本文给出算法收敛的充分条件,证明了算法的全局收敛性.该算法对于非线性系统迭代学习控制具有全局收敛和计算简单的优点.     

On the guaranteed convergence of a cubically convergent Weierstrass-like root-finding method

Initial conditions that provide guaranteed and fast convergence of the Weierstrass-like cubically convergent iterative method for the simultaneous determination of all simple zeros of a polynomial are considered. It is proved that this method is convergent under suitable conditions stated in the spirit of Smale's point estimation theory. The proposed convergence conditions are computationally verifiable since they depend only on initial approximations and the degree of a given polynomial, which is of practical importance.     

The improved Farmer–Loizou method for finding polynomial zeros

An improvement of the Farmer–Loizou method for the simultaneous determination of simple roots of algebraic polynomials is proposed. Using suitable corrections of Newton's type, the convergence of the basic method is increased from 4 to 5 without any additional calculations. In this manner, a higher computational efficiency of the improved method is achieved. We prove a local convergence of the presented method under initial conditions which depend on a geometry of zeros and their initial approximations. Numerical examples are given to demonstrate the convergence behaviour of the proposed method and related methods.     

Visualizing steepest descent

The convergence of the method of steepest descent can be studied with computer graphics. The number of iterations required for convergence and the basin of attraction determined by the algorithm are both sensitive to initial conditions. The convergence in valleys is studied and beautiful, chaotic, images of the convergence are given.     

Estimation of pulse transfer function parameters by quasilinearization

An iterative algorithm for the identification of single-input single-output linear stationary discrete systems is developed using the method of quasilinearization. The resulting procedure is similar to mode 1 of the method of Steiglitz and McBride but has the advantage of the quadratic convergence property of quasilineariza-tion. It is shown that this algorithm becomes mode 1 if the measured plant output is used in the calculations in place of the model output. Consequently, the two methods are extremely compatible and it is a simple matter to combine them in a single program, which generates its own initial estimates has the wide range of convergence of mode 1, and possesses the quadratic convergence property of quasilineariza-tion for final convergence to a solution. The method also permits the estimation of plant initial conditions in those cases where they must be considered. Results of a few numerical applications are discussed.     

Averaging of three time scale singularly perturbed control systems

We investigate the asymptotic properties of singularly perturbed control systems with three time scales. We apply the averaging method to construct a limiting system for the slowest motion in the form of a differential inclusion. Sufficient conditions for the uniform convergence of the slowest trajectories are given.     

The levenberg-marquardt method for approximation of solutions of irregular operator equations

An ill-posed problem is considered in the form of a nonlinear operator equation with a discontinuous inverse operator. It is known that in investigating a high convergence of the methods of the type of Levenberg-Marquardt (LM) method, one is forced to impose very severe constraints on the problem operator. In the suggested article the LM method convergence is set up not for the initial problem, but for the Tikhonov-regularized equation. This makes it possible to construct a stable Fejer algorithm for approximation of the solution of the initial irregular problem at the conventional, comparatively nonburdensome conditions on the operator. The developed method is tested on the solution of an inverse problem of geophysics.     

A semi-analytical iterative technique for solving nonlinear problems

We present a semi-analytical iterative method for solving nonlinear differential equations. To demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial/boundary conditions. In each of the examples we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a high order of convergence.     

On initial conditions in iterative learning control

Initial conditions, or initial resetting conditions, play a fundamental role in all kinds of iterative learning control methods. In this note, we study five different initial conditions, disclose the inherent relationship between each initial condition and corresponding learning convergence (or boundedness) property. The iterative learning control method under consideration is based on Lyapunov theory, which is suitable for plants with time-varying parametric uncertainties and local Lipschitz nonlinearities.     

Application of dynamic relaxation to the large deflection elasto-plastic analysis of plates

The application of dynamic relaxation, a finite difference based iterative analysis, to the study of plates to date is reviewed. The extension of the method to include both geometrical and material non-linear effects in plates is then described in detail. Particular attention is paid to aspects of the iteration parameters which control convergence. The advantages of interlacing finite difference meshes is discussed and the mesh refinement necessary for the accurate analysis of plates in compression and in shear is considered. The usual elastic out-of-plane boundary conditions are generalised by the inclusion of terms applicable in the elasto-plastic range, and the role of in-plane boundary restraints is discussed in respect of plating in both bridge and ship structures. The scope of the method is demonstrated by examples of a long plate with patch loading normal to the undeflected plane of the plate and a panel of a box girder web. In both cases, the effects of initial out-of-plane geometric distortions and welding residual strains on behaviour up to and beyond collapse are considered.     

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods     

Iterative methods for systems of equations with interval coefficients and linear form

We introduce iterative methods for systems of equations with interval coefficients and linear form by suitable matrix splittings. When compared to the iterative methods for systems amenable to iteration introduced in [1], improved convergence and inclusion properties can be proved under suitable conditions. The method can also be used in the solution of specific nonlinear systems of equations by interval arithmetic methods.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic

Starting from separated rectangles in the complex plane which contain polynomial complex zeros, an iterative method of second order for the simultaneous inclusion of these zeros is formulated in rectangular arithmetic. The convergence and a condition for convergence are considered. Applying Gauss-Seidel approach to the proposed method, two accelerated interval methods are formulated. TheR-order of convergence of these methods is determined. An analysis of the convergence order is given in the presence of rounding errors. The presented methods are illustrated numerically in examples of polynomial equations.     

Contraction mapping techniques applied to the hybrid computer solution of parabolic PDE's

Using contraction mapping techniques a proof is given of the convergence of the discrete-space-continuous-time (DSCT) hybrid computer method of solution of the one-dimensional constant coefficients heat equation. Numerical results are given for a particular choice of initial temperature distribution and boundary conditions.