> From a very slow simulation I got y= f(x): > x y > 5 0.8048174 > 6 0.8194384 > ... > 44 0.9706268 > 47 0.9724846 > ... > 48765 0.9999756 > 53765 0.9999776 > > For every x, I stop the simulation when the confidence interval for y > is less than 2,5*10^-6 (with 99% of confidence). > > I can't calculate all the x's (because the simulation is very slow), so > I need to interpolate; for example, I don't have y(45) or y(46). > > Using the Levenberg-Marquardt Least Squares Fitting, the best equation > I found gives an error that is too high (about 10^-4 for small x's). > > Then I thought to use a cubic spline, but I notice some "fluctuations" > on the tails. > > Should I use LM or spline? > > Thanks > Cristiano
For this type of problem it helps greatly if you have an approximate notion of what the function looks like. Then you fit either a multiplicative or additive correction to the approximate functional form.
You did not say what you were fitting to the data but most classes of functional fits get their results by oscillating between the observed values. That is what they are intended to do if you read enough explanations.
By the way, there are special variants on splines that are monotone. Assuming that you know that yu are fitting to a montone result. See first point!