Re: Correction for: Interpolating function with two important characteristics

In article <4356396c$0$29558$4fafbaef@xxxxxxxxxxxxxxxxxxx>,
mike@xxxxxxxxxxxxxxxxxxx wrote:

> 1) Passes through the points.
> 2) At least the SECOND-derivative must be continuos, thus the
> first-derivative must be "smooth", with no abrupt, sudden changes
> in slope.
[...]
> Is it even mathematically possible? Should I give up? Or what
> should I Google for? :P

Yes, it is possible to do what you want. I do not have the
software, but I once wrote a 4th order spline that has continuous
function, first derivatives, and second derivatives at the node
points. If I remember correctly, you end up with a tridiagonal
linear equation to solve, the same as for the typical cubic spline
(which has only continuous function and first derivatives at the
nodes). Also, you have to assume more endpoint conditions to get a
well-defined set of equations. In my case, I assumed that the third
derivatives at the endpoints were zero.

If you can't find software for this anywhere, then you will have to
write it yourself. Work through the cubic spline case (which is
explained in textbooks, Numerical Recipes, etc.), and generalize the
same approach using the higher-order polynomial and the additional
continuity conditions. When I did it that way, it was
straightforward.

$.02 -Ron Shepard
.

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