       Problem with spline function object in Mathematica 6

• To: mathgroup at smc.vnet.net
• Subject: [mg95663] Problem with spline function object in Mathematica 6
• From: borisov at sas.upenn.edu
• Date: Sat, 24 Jan 2009 06:19:12 -0500 (EST)

```Hello,

Here is a description of what I am trying to do. I have a tabulated
monotonically increasing function (imagine mass contained in concentric balls
with increasing radius). I need to obtain an interpolating function which
preserves the monotonicity. I also need the derivative of the original function
- the density (and it needs to be positive everywhere). For that I use spline as
polynomial interpolations do not satisfy these requirements. I was able to
obtain the derivative of the spline object via a method described before on
MathGroup. As I need the density as a function of distance and the spline
function object is a parametric object in the end I need the inverse functions
which give me the parameter in terms of the distance. The current problem I
have is that the spline object is only correctly evaluated if the argument is a
number. If it is a symbol it automatically evaluates to the same symbol. I am
providing an arbitrary table for the original function.

Here is my current code:
SplineTable={{0,0},{1,1},{2,8},{3,27},{4,64}};
Spl=SplineFit[SplineTable,Cubic];
derivSpline=ReplacePart[Spl,Map[Append[Rest[#]*{1,2,3},0]&,Spl[],{2}],4];
dMdr[x_]:=Last[derivSpline[x]]/First[derivSpline[x]];
drdt[x_] := First[derivSpline[x]];
sol3 = NDSolve[{param==0,(param'[x]-(1/drdt[param[x]]))==0},param,{x,0,4}]

So again the problem I have is that drdt[param[x]] evaluates to param[x]
automatically (and immediately gives a division by 0 error in NDSolve) instead
of attempting to evaluate param[x] to a number first.
The same happens with dMdr[x]. My goal in the end will be to evaluate
dMdr[param[x]] correctly when supplied with x.

I would also appreciate to know if there is an easier way to get the derivative
in terms of distance from the original table while satisfying the physical
requirements.
Thanks in advance: Alexander Borisov.

```

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