Why are there problems when interpolating $f(x)=\arctan(10x)$?

Question

Given $f(x)=\arctan(10x)$, there would be a problem when we interpolate it by using Lagrange's method. This would have something to do with the derivatives of $f(x)$. I plotted some derivatives of $f$ but I did not come up with an answer. Can anybody tell me what goes wrong if we interpolate $arctan(10x)$ and why this comes from the derivatives of $f(x)$. thanks :-)

Oh.. second question: I also would like to know why this problem does not occur when using cubic Hermite interpolation... In addition, I want to know why this problem does not occur when using cubic spline interpolation ? I can not figure this out

Answer 1

If what you got is something like this:

where dashed lines are the interpolant.

Then this is Runge's phenomenon. What you said is a typical example of this phenomenon that using high degree polynomial to approximate a continuous function on evenly spaced sample points. This happens when we approximate some smooth function on a given interval, given $n$ evenly distributed sample points, and using a single globally defined polynomial.

If you use cubic Hermite spline, we would not have this phenomenon because of this interpolation's piecewise nature, it is piecewisely defined on each small interval. We have multiple locally defined polynomials, we only glue these different polynomials together using continuity (namely $f$ and $f'$ are continuous across the sample points).

Answer 2

If you do the interpolation intelligently (i.e. not at equally-spaced points) then you can avoid the Runge phenomenon, and you can approximate any reasonably smooth function very well with a polynomial on a bounded interval.

An excellent tool for constructing polynomial approximations is the Chebfun system. The Chebfun folks love to rant about how Lagrange interpolation has been criticized unfairly by people who don't know what they're talking about (including the authors of some textbooks on numerical analysis). The rants are fun to read.

If you want a polynomial approximation to be good on the entire real line, rather than a bounded interval, then you have a problem that's much worse than the Runge phenomenon. The value and the derivatives of any polynomial $p(x)$ will tend to $\pm\infty$ as $x \to \infty$ (unless it's linear). And for your function $\arctan(10x)$, the function value tends to $\pi/2$ and the derivative approaches zero as $x \to \infty$, so there is a dramatic mismatch that's difficult to handle. This is the problem you referred to in your question, I suppose.

The way forward is to use splines or rational approximations.