Re: How can I evaluate interval initial value problem solver? > firstly, thank you for your consideration! > as a newcomer, I have so many problems. > 1) to a single interval solver, how to compute the EXCESS width?? >  y\\'(t)=f(y,t)  the initial condition is y0=[a,b] ,for t in [0,tend] > the following is my view, choose enough points in y0=[a,b] randomly as initial > value ,then from a point NOT interval solver,we can get y(tend) according to > the initial value in y0. > if the number of points is large enough,we can get an interval to contain all > the y(tend) with the minimum width. > then we can view this interval as the exact solution interval and from this > and the [y(tend)] from interval solver,we can get the EXCESS width >  > is it right? No.  There are two flaws, both fundamental to understanding how interval people look at the world. #1.  You have no guarantee of the correctness of the approximate solutions. Hence, the true set of solutions could look very different. #2.  Even if the approximate solutions were perfect, you have no guarantee that the Monte Carlo simulation you suggest explores the true extremes.  It is possible for there to be narrow spikes in the true solution set that happen nor to be exercised by your sampling. The method you outlined, while not guaranteed, can often give insight. Alternatives: 1. Exact solution, if it is available, but you already knew that. 2. If the problem has a special form, e.g., 1 dimensional, linear, etc., there may be special techniques you can use. 3.  For some problems, you can use differential inequalities to obtain inner enclosures, or sets which are guaranteed to be inside the true solution set. 4.  If the dimension is not too high, you can subdivide the original initial condition and integrate on sub-boxes. > 2)I still can not understand of Taylor Series Method well. > I have read "Validated solutions of initial value problems for ordinary > differential equations" ,a very helpful paper to me. > Alough with it, I still cannot program this method rightly in Matlab. There are MANY subtle implementation details.  It is NOT as easy as the authors try to make it sound in papers. > I wanna ask: > which c/c++ interval solver package can be use under Windows? Get Ned Nedialkov's VNODE package.  You can download it from http://www.cas.mcmaster.ca/~nedialk/.  I strongly recommend you also read his thesis and his more recent papers. You might be interested in an interval tutorial I gave in August, http://www.eng.mu.edu/corlissg/05DTUworkshop/Latex/.  I have attached my most recent Survey of ODEs talk. Dr. George F. Corliss Electrical and Computer Engineering Marquette University PO Box 1881  1515 W. Wisconsin Ave. Milwaukee WI 53201-1881 USA 414-288-6599; Fax: 288-5579; Dept. 288-6280 Office: Haggerty Engineering 296 George.Corliss@Marquette.edu >    > ----- Original Message ----- > From: George Corliss  > To: ,  > Cc:  > Subject: Re: How can I evaluate interval initial value problem solver? >  >> for example >> y\\\\\\'=f(y,t)  the initial condition is y0=[a,b] ,t=0 to tend >>  >> is the width of interval y(tend) an index to evaluate the solver?? > I\\'m not sure exactly what you are asking.  Are you asking about how to > compare the quality of two interval solvers?  If so, comparing the width of > the interval y(tend) is one measure.  A more comprehensive measure would be > some norm of width of the interval y(t) for t in [0, tend].  You could also > compare the widths of the defects y\\'(t) - f(t,y). >  > If you do any of those, you should be careful of concluding too much.  You > will only have compared the tightness performance of the two methods on that > problem.  The tightness performance on another problem might be quite > different. >  > If what you want is an evaluation of the quality of a single interval > solver, you probably want some measure of the EXCESS width, although > authorities differ on the most appropriate way to express excess width, > especially in dimensions greater than one. >  > Dr. George F. Corliss > Electrical and Computer Engineering > Marquette University > PO Box 1881  > 1515 W. Wisconsin Ave. > Milwaukee WI 53201-1881 USA > 414-288-6599; Fax: 288-5579; Dept. 288-6280 > Office: Haggerty Engineering 296 > George.Corliss@Marquette.edu >  >  >  >