Any ideas on how GeneralMiniMaxApproximation is implemented?

Question

GeneralMiniMaxApproximation is used to construct minimax approximations of parametrically defined functions. I am curious about how GeneralMiniMaxApproximation is implemented (just a general overview). Any ideas? Or is this a Mathematica trade secret?!

The document states that MiniMaxApproximation uses Remez's second algorithm, but does this algorithm extend to parametrically defined functions? I can't seem to locate any references.

Answer 1

The references mentioned in the other answer do not really pertain too much to the problem of minimax approximation; they are more concerned with the other functions in FunctionApproximations` that deal with numerical differential equation solving; that is, the content that was once in the old NumericalMath`packages Butcher` and OrderStar`.

Indeed, a look at the contents of the old NumericalMath`Approximations` package, where GeneralMiniMaxApproximation[] once was, yields the following refs:

Carl-Erik Fröberg, Numerical Mathematics: Theory and Computer Applications, Benjamin/Cummings, 1985, pp. 250-266

A. Ralston & P. Rabinowitz, A First Course in Numerical Analysis (2nd. ed.), McGraw-Hill, New York, 1978

I do not possess a copy of the first ref, but I have a copy of the second one; the discussion of Remez's second algorithm is here. The papers in that book's bibliography are also to be pursued.

Answer 2

You can check here for Function Approximations Package References

References

• Hairer, E. and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1991.
• Iserles, A. and S. P. Nørsett. Order Stars. Chapman and Hall, 1991.
• Lambert, J. D. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons, 1991.
• Sofroniou, M. "Order Stars and Linear Stability Theory." Journal of Symbolic Computation 21, no. 1 (1996): 101-131.

and if you check the Mathematica\8.0\AddOns\Packages\FunctionApproximations\Approximations.m you can browse the code. The fourth method for approximation is what you need. It's been stated that GeneralMiniMaxApproximation--the same as MiniMaxApproximation, except that the function may be specified parametrically, and the error being minimized need not be the relative error.