Although numerous writers have stated that the class of single-step (“Runge-Kutta”—like) methods of numerical integration of ordinary differential equations are stable under calculational or round-off error, no one has given formal equations for the bounds on the propagated error to indicate this stability. Rutishauser [1] justifies the stability by noting that there is only one solution to the approximating difference equation, and Hildebrand [2] calculates a propagated error bound for the simplest (Euler) case. However, the latter bound does not indicate the stability for even that case. It is the purpose of this paper to investigate this “stability” of the Kutta fourth order procedure for integration of the ordinary differential equation dy / dx = ƒ( x, y ), (1) where ƒ( x, y ) possesses a continuous first-order partial derivative with respect to y throughout a region D in which the integration is to take place. (By alteration of the proof below, this condition can be replaced by a Lipschitz condition.) Since the Kutta process is the most complicated of such single-step procedures, it should be apparent that similar error bounds can be derived for the various other single-step methods of same or lower order (and in particular the Gill variant method, probably most often used in machine integration because of the storage savings). It is plausible that such error bounds can also be extended to the stable (extrapolation) multi-step methods, such as the Adams method, and to systems of ordinary differential equations. If the variational equation d η/ dx = ∂ƒ( x, y )/∂ y η (2) for the above ordinary differential equation has ƒ v ( x, y ) < 0 throughout a region D the ordinary differential equation is termed stable in D , and for small enough variations in the initial conditions the absolute value of the propagated error decreases with growing x . Todd [3], Milne [4], Rutishauser [1], and others have shown that numerous multi-step numerical integration techniques are unstable in that even when the differential equation is stable, the difference equation will introduce spurious solutions for any step-size h . For the Kutta fourth order process, as seen below, this is not the case; for the stable differential equation the propagated error in the difference approximation remains bounded for small enough (but not too small!) step-size h ; and for a given value of x , bounds on the propagated error decrease to a minimum given as a function of the round-off error as the step-size is decreased. Similar statements can be proved (but are not proved here) for other single-step processes. For no round-off (an “infinite word-length machine”) the process converges as h goes to zero. In addition, an algorithm for determining the step-size as a function of the partial derivative is given below so as to keep the propagated error within a given bound. The classical Kutta procedure [5] gives the value of y at the ( i + 1)th step in terms of the value at the i th step and the step-size h as follows: y i +1 = y i + h /6( k 1 + 2 k 2 + 2 k 3 + k 4 ) + O ( h 5 ) k 1 = ƒ( x i , y i ) k 2 = ƒ( x i + h /2, y i + k 1 h /2) k 3 = ƒ( x i + h /2, y i + k 2 h /2) k 4 = ƒ( x i + h, y i + k 3 h ) (3) When the O ( h 5 ) term is neglected, the value of y i +1 , here called y t i +1 , is only an approximation. An error bound for the truncation error at each step is given by Bieberbach [6, 7] and Lotkin [8]. This guarantees that for h small enough the truncation error at a single step, starting at y i , will be bounded by | y i +1 - y t i +1 | ≦ C i +1 h 5 , (4) where C i +1 ≧ 0 is a function only of i , the function ƒ( x, y ), and its partial derivatives of the first three orders; and y t i is the true solution to the equations (3) truncated so that the O (

I too, like Dr. Householder, want to thank the

article Free Access Share on Prevention of Propagation of Machine Errors in Long Problems Authors: J. H. Brown University of Michigan, Ann Arbor, Mich. University of Michigan, Ann Arbor, Mich.View Profile , John W. Carr University of Michigan, Ann Arbor, Mich. University of Michigan, Ann Arbor, Mich.View Profile , Boyd Larrowe University of Michigan, Ann Arbor, Mich. University of Michigan, Ann Arbor, Mich.View Profile , J. R. McReynolds University of Michigan, Ann Arbor, Mich. University of Michigan, Ann Arbor, Mich.View Profile Authors Info & Claims Journal of the ACMVolume 3Issue 4Oct. 1956 pp 348–354https://doi.org/10.1145/320843.320853Published:01 October 1956Publication History 3citation312DownloadsMetricsTotal Citations3Total Downloads312Last 12 Months9Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

article Free Access Share on On the Demonstration of High-Speed Digital Computers Author: John W. Carr The Ramo-Wooldridge Corporation and the University of Michigan The Ramo-Wooldridge Corporation and the University of MichiganView Profile , Editor: Walter F. Bauer The Ramo-Wooldridge Corporation and the University of Michigan The Ramo-Wooldridge Corporation and the University of MichiganView Profile Authors Info & Claims Journal of the ACMVolume 1Issue 401 October 1954pp 177–182https://doi.org/10.1145/320783.320787Published:01 October 1954Publication History 1citation520DownloadsMetricsTotal Citations1Total Downloads520Last 12 Months13Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF