From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: Differential equation Date: 5 Nov 2000 15:14:48 -0500 Newsgroups: sci.math Summary: [missing] In article , Arthur Meunier wrote: >"Herman Rubin" wrote in message >news:8u19ns$2pti@odds.stat.purdue.edu... >>In article , >>Arthur Meunier wrote: >>>I know there is a way to compute an approximative solution to a differential >>>equation wih initial conditions, but I'd like to know if there is a way to >>>control the error, the precision.... thank you in advance >>>( and sorry for my hesitating English ) >> There certainly is; one can use interval arithmetic. >In fact this is exactly what I'm looking for, I'd like to compute with >interval arithmetic, unfortunately, on an interval there is no way I can >know the maximum/minimum og the derivative, even if I know they exist ! >..... :-( So if you have an idea >> It is a little more complicated than it looks, because the >> bounds of [f(x+a) - f(x)]/a have to be computed by putting >> the bounds into the recursion step. As I said, it is a little more complicated than it looks. Suppose that one has the equation y' = f(x,y). In general, a unique solution requires that f is locally Lipshitz in y. If f is continuous and satisfies, this, then |f(x,y) - f(x_0,y_0)| <= eps + C|y - y_0| for x close enough to x_0; in evaluating the bounds, the continuity in x and locally bounded difference quotient in y are enough to enable the intervals to be obtained. If f is locally integrable in x, one can do better, as it is only the difference quotient in y which leads to uncertainty. For example, in integrating y' = y, starting with y_0 >0, one has immediately that in an interval of length h, we have y_0 < y' < y_0 + h*m, where m is the maximum of y in the interval. This gives m < y_0/(1 - h). Interval arithmetic tends to be slow and overly conservative. Higher order methods are likely to be far more efficient. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558