A Mathematical Basis for an Error Analysis of Differential Analyzers

Abstract

1.1. Purpose and Literature. The purpose of this paper is to present a mathematical basis for a general error analysis for the solution of systems of ordinary differential equations by machine methods. Specifically, we shall concern ourselves with the effect of errors on the machme solutions. We show that the study of these effects for general non-linear systems can be referred to the solution of linear systems of ordinary differential equations, but we do this without "linearizing" or simplifying the given system.Since we permit perturbations in the solution as given by the machine, our discussion IS apphcable to both continuous computers and digital machines using step by step methods. However, stability discussions for such a digital process are most conveniently given in terms of difference equations rather than differential equations and are not given here.There , but an intensive analysis of this work is not appropriate here One of our major objectives is to avoid the "linearization" which appears in these and In this sense our results can be regarded as supplementing thIS work.The effect of errors on solutions obtained by means of continuous computers has been studied in the case in which the given problem involves a system of linear equations WIth C01stant coefficients, nota.bly by Raymond [34] and Macnee [27, 28].Our dIscussion is based to a certain extent on well known theories for the dependence of systems of ordmary differential equations on parameters. However, it was necessary to extend this theory m order to properly consider those errors which affect the order of the system. Order variations in systems of equations have been considered from other points of view by Coddington and LeVIn-

Sensitivity Analysis in Optimal Systems Based on the Maximum Principle

“…As can be seen from the above definition, the main task in the sensitivity analysis is to find the partial derivatives as given in Equation 3-namely, the sensitivity coefficients. A method of finding the sensitivity coefficients in the problem of solving a system of differential equations by differential analyzers has been developed by Miller and Murray (1953) and used in the sensitivity analysis of control systems (Chang, 1961;Tomovic, 1964). z(T) = f (10) where dfí/'dx is a row vector with components bH/dxt, (i = 1,2, .…”

“…or in short form Jw = cxw(T), where Jw is a row vector with components, bJfbwh i = 1, 2, The Jacobian xa can be obtained from the method proposed by Miller and Murray (1953). This method is based upon the differentiation of Equation 5with respect to w by assuming that the functions in the equation are differentiable and the change of w does not change the order of the differential equation.…”

The Use of Parameter Influence Coefficients in Computer Analysis of Dynamic Systems

“…If the system is described in terms of a set of differential equations where a,, a,, ---, an are system parameters of particular interest, then any information obtainable from the computer which serves to enhance one's knowledge of the system response as function of these parameters will be valuable. Let the solution of the differential equations (1) obtained for a prescribed set of parameters and initial conditions be expressed as Then the partial derivatives which will be referred to as parameter influence coefficients, will be of considerable interest. They can be used to predict system performance in the neighborhood of the known solution Xi by first-order approximation and to describe system sensitivity to certain design changes.…”