How can I accurately interpolate a C1 function with infinite second derivative?

[bruno67]

I have to interpolate a function between a small number of points n (say, 3-5), at which I know both the value of the function and its first derivative. Normally, this would be a good candidate for polynomial interpolation.

The only problem is that at the first point the function is only once differentiable (its second derivative is infinite), while it is perfectly smooth in the rest of the interval. How can I get an estimate of the interpolation error? The usual estimate found in textbooks requires the function to have at least n+1 continuous derivatives in the closed interval in which one is interpolating.

Thanks.

 

[AlephZero]

If you know the second derivative is infinite at one end of the range, you should use that fact to choose the form of your interpotation function, otherwise you will be introducing systematic errors and reducing the order of accuracy.

Presumably you know the general form of the function near the end point, so you should include it in your interpolation, for eample

y = ax^3/2 + b + cx + dx² + ...

or
y = sqrt(x)(ax + bx² + cx³ + ...)
or whatever.

The square roots are just an example of a function that gives an infinite second derivative. Use the function that fits the physics of your situation. That may be a "special function" like a Bessel function etc, not a polynomial.