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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.

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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.

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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 is the same as MiniMaxApproximation, except that the function may be specified parametrically, and the error being minimized need not be the relative error.

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