Interpolation with Method->Spline

*To*: mathgroup at smc.vnet.net*Subject*: [mg98000] Interpolation with Method->Spline*From*: John_V <jvillar.john at gmail.com>*Date*: Fri, 27 Mar 2009 05:37:52 -0500 (EST)

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