Variable-step Method Research Articles

Pattern Recognition Using Neural Networks. Comparison to the Nearest Neighbour Rule

The purpose of this paper is to compare two pattern recognition methods : Neural Networks and the Nearest Neighbour (NN) rule. As we shall see, a neural network approach corresponds to an approximation of the decision function.First of all, we present the neural network approach and the properties of the error function (which must be minimized) as a function of the network parameters, the a priori probabilities and the conditional probability density parameters. Considerations about the convergence of a classical steepest descent algorithm are presented, which lead to variable step method application. Since the learning “quality” is related to the number of cells in the network, we introduce a criterion for determining the optimum number of cells. Examples corresponding to discrimination beetwen two clusters in R2 are shown and the results are compared to those obtained by the NN rule.Both methods will be applied to automatic analysis of human sleep. Our results will be compared to the visual classification achieved by an expert.

A new variable step method for the numerical integration of the one-dimensional Schrödinger equation

The numerical integration of the one-dimensional Schrodinger equation is considered. A new variable step method is presented within the “Canonical Functions” scheme, characterized namely by: (i) the replacement of the Schrodinger equation by the “equivalent” integral equation; (ii) the representation of the wavefunction by a superposition of simple integrals; (iii) the definition of “Taylor Series like” difference equations. At each interval, an estimate of the step-size h is determined in order to obtain the desired local accuracy. For the practical case of a given potential function with polynomial interpolations, the step-size h is directly given by a simple formula depending on: (i) the potential function in the interval; (ii) the “initial values” of the wavefunction at the origin of the interval. This variable step method is applied to the Morse potential function used for the recent Raptis and Cash method ( Comput. Phys. Commun. 36, 113 (1985)). For this application, the present method presents some advantages over the previous one.

A variable step method for the numerical integration of the one-dimensional Schrödinger equation

Most numerical methods which have been proposed for the approximate integration of the one-dimensional Schrödinger equation use a fixed step length of integration. Such an approach can of course result in gross inefficiency since the small step length which must normally be used in the initial part of the range of integration to obtain the desired accuracy must then be used throughout the integration. In this paper we consider the method of embedding, which is widely used with explicit Runge-Kutta methods for the solution of first order initial value problems, for use with the special formulae used to integrate the Schrödinger equation. By adopting this technique we have available at each step an estimate of the local truncation error and this estimate can be used to automatically control the step length of integration. Also considered is the problem of estimating the global truncation error at the end of the range of integration. The power of the approaches considered is illustrated by means of some numerical examples.

A variable interval variable step method for the solution of linear second order coupled differential equations

A new method for the numerical solution of the linear second order coupled differential equations of quantum scattering theory is presented. A formal framework is set up which makes clear the interrelationships between many currently used integration methods. Our variable interval variable step method is designed to reduce the total number of matrix (N3) operations required for solution at a given accuracy. The method is tested on several problems and is shown to be uniformly and rapidly convergent, stable, and significantly faster than previous methods.

The logarithmic derivative-variable interval variable step hybrid method for the solution of coupled linear second-order differential equations

The logarithimic derivative method and variable interval variable step method for the solution of coupled linear second-order differential equations are joined to give a hybrid method which is more efficient than either method alone.

Second Derivative Multistep Methods for Stiff Ordinary Differential Equations

The difficulty associated with the numerical solution of stiff ordinary differential equations is considered and the stability requirements of methods suitable for stiff equations are described. A class of second derivative formulas is developed and the stability of these formulas is investigated. These k-step $(k + 2)$nd order formulas are shown to be suitable for stiff equations for $k \leqq 7$. These formulas have been implemented in a variable order, variable-step method and some numerical results are presented.