From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Newsgroups: sci.math.num-analysis Subject: Re: trying to find a good interpolation formula Date: 31 Aug 1997 14:37:33 -0400 In article , bin hu wrote: >Hi > >Recently I met a problem which need to find a interpolation formula that >will not increase wildly outside the interpolation range. We know that >Lagrange interpolation formula will give very big value out side the >interpolation range. That's bad news, of course. The even worse news is: There are "innocent-looking" functions whose Lagrange interpolating polynomials, while doing what they are supposed to do at the equidistant interpolation nodes (match the values of the function), they explode in magnitude (diverge to infinity as you refine the grid) between nodes. The classical example is due to Runge (of Runge-Kutta fame): f(x) = 1/(1+25*x^2) for x in [-1, 1] >Does anyone know some formula, which can be written >into an analytical form, have such good property? The reasonable answers must be (sort of) evasive: it depends on the purpose of the fitting. If the expected form of the interpolant is a rational function rather than a polynomial, you can prescribe the degrees of the numerator and denominator to control the growth beyond the interpolation range. The obvious risk: you can obtain discontinuities (poles) in unexpected places. If you expect periodic behavior (and you know the length of the period), try trigonometric interpolation. Boundedness is guaranteed all over R, but the fit may not reflect the nature of the data. The only source I can recall off-hand is Berezin-Zhidkov: Numerical Methods. If you settle for linear behavior beyond the interval spanned by the x-values in the (x,y) table, try natural cubic splines. If your data are experimental, laden with errors, try least-squares fit of low degree. Other criteria of fit are also appropriate (uniform fit or first-power fit), depending on the nature of the errors. In general, go back to the nature of the data. If they are, for example, levels of spontaneous radiation in time, then looking for polynomial fit would be a declaration of incompetence; exponential fit (of low dimension) corresponds more closely to the physics of the problem. In population growth studies, logistic fit is desirable (although reported to be badly numerically conditioned). (Isn't uiuc strong in numerical analysis? Seek local information, too. It may require a physical exercise called walking...:-) Good luck, ZVK (Slavek).