I agree re differentiation being numerically unstable, but I don't know the answer. The video does show a number of examples, but I don't recall this particular issue being addressed. Many of the examples used in the video were combinations of well-known functions -- e.g., sin(x), cos(x), whatever. I don't know if Chebfun calculated all of the approximating functions numerically or symbolically (e.g., using symbolic algebra and known properties of sin, cos, etc.) Trefethen has published widely, so I suspect that most of these issues are covered in his papers: http://people.maths.ox.ac.uk/trefethen/publication/publication.html At 01:22 PM 8/25/2012, Fred lunnon wrote:

Some issues are raised by the slides, one of which lists differentiation as a built-in functional. Maybe somebody else can clarify these for me?

Differentiation is numerically unstable; therefore the original function will have to be recomputed at increasing precision in order to maintain the prescribed error in higher derivatives. As far as I can see, simply increasing the number n of interpolating points is insufficient to remedy the problem, whatever approximations (Taylor, Chebychev or whatever) are employed.

Furthermore allowing n to vary dynamically seems to imply that the user must express his function f as an infinite Chebychev series (or maybe combination of built-in functions). If for example he possesses only a functional equation for f, it's unclear how this can be achieved within the framework supplied.

Fred Lunnon

On 8/25/12, Henry Baker <hbaker1@pipeline.com> wrote:

...

If you're not aware of the "Chebfun" package developed by Trefethen, et al, at Oxford U., you will be pleasantly surprised.

Basically, Chebfun approximates smooth curves over a finite interval to within machine precision. Chebfun also supports piecewise smooth curves by separately Chebbing each piece.

Since different smooth curves require different numbers of terms to converge, some of these sums can have 1000 or more terms.

As pointed out in the slides & video below, the whole point of this is to provide the flavor of symbolic computation with the efficiency and ease of numeric computation. For example, plotting, derivatives, integrals, root-finding are all relatively easy in this framework.

The major success of Chebfun's is in ODE's, both linear & non-linear.

It's a pity that Chebfun only works on Matlab; perhaps some student could be motivated to do this in Maxima.

After watching the video below, I started to wonder why we bother teaching Taylor series at all.

Here's a 1-hour tutorial about Chebfun (850 MBytes):

http://downloads.sms.cam.ac.uk/1160831/1160835.mp4

Here are the slides for this talk (1 MByte):

http://people.maths.ox.ac.uk/trefethen/chebfun.pdf

Some history of Cheb approximations

http://people.maths.ox.ac.uk/trefethen/cftalk.pdf