Re: Re: Interpolation with Method->Spline

*To*: mathgroup at smc.vnet.net*Subject*: [mg98093] Re: [mg98024] Re: Interpolation with Method->Spline*From*: John_V <jvillar.john at gmail.com>*Date*: Mon, 30 Mar 2009 04:41:49 -0500 (EST)*References*: <gqia9h$oo8$1@smc.vnet.net> <200903281042.FAA04440@smc.vnet.net>

Setting all first derivatives is not a requirement for *ordinary *interpolation. That is, if you use the example data that I gave, but leave out the Method->Spline, you get a polynomial interpolation with 0 derivative at the first endpoint, just as you'd expect. It is only with Spline that it seems not to work. It wouldn't make sense to require all derivatives to be specified. (This would render the interpolation less useful since we often don't know what the derivatives ought to be, and in such cases we'd like to leave the interpolation free to optimize other properties.) The problem of fitting a spline through points in an interval is ambiguous to the extent of two boundary conditions, one at each end of the interval. A frequent choice is to arbitrarily require the curvature at the endpoints to be 0--but another common choice is to specify some other derivative there, which is what I'd like to do. On Sat, Mar 28, 2009 at 6:42 AM, <rommel.ua at gmail.com> wrote: > > You have to set the first derivative value for ALL points in your > series. In other case Mathematica will ignor the third number for > first point. > I add random numbers as third element for your points (and reevaluate > interpolation each time after addition) and the interpolated data > curve was changed only when I set all points with derivative value. > All data must be with derivative or all without it. > John_V had written I upgraded to Mathematica 7.01 today so I could try the new Method->Spline option of the Interpolation function. I've experimented with it, and it appears to give the natural spline (second derivative = 0 at the data endpoints). I would like to specify the *first *derivative at the endpoints. It is possible to do this for splines outside of Mathematica (e.g., Numerical Recipes describes it), and Mathematica permits it for ordinary polynomial interpolation, so I tried using the same syntax. Here's an example: In[53]:= exampData = {{{0.}, 0., 0.}, {{0.5}, 0.00042}, {{1.}, 0.0013}, {{1.5}, 0.00614}, {{2.}, 0.026}, {{2.2}, 0.0622}, {{2.4}, 0.153}, {{2.6}, 0.188}} Out[53]= {{{0.}, 0., 0.}, {{0.5}, 0.00042}, {{1.}, 0.0013}, {{1.5}, 0.00614}, {{2.}, 0.026}, {{2.2}, 0.0622}, {{2.4}, 0.153}, {{2.6}, 0.188}} Notice that the first data input has an extra 0 after the function value to specify a 0 first derivative at x=0. This works for polynomial interpolation. However, for spline interpolation: In[54]:= f = Interpolation[exampData, Method -> "Spline"] Out[54]= InterpolatingFunction[] If you plot this (e.g., Plot[f[x], {x, 0, 1}] ) you'll see that f clearly has a negative slope at x=0. In[56]:= f[0.1] Out[56]= -0.0000687322 Unfortunately for me, f represents a physical quantity for which negative values are impossible (and cause mischief in later calculations). This was why I was trying to specify the 0 derivative at x=0: in hopes of making the spline behave itself near the endpoint so I could enjoy its other nice properties (minimum curvature) elsewhere. This is a new and not-yet-well-documented feature, so maybe there's a syntax or a workaround to do what I want. Does anyone know of one? John

**References**:**Re: Interpolation with Method->Spline***From:*rommel.ua@gmail.com