Algorithm 414: Chebyshev approximation of continuous functions by a Chebyshev system of functions

The second algorithm of Remez can be used to compute the minimax approximation to a function, ƒ( x ), by a linear combination of functions, { Q i ( x )} n 0 , which form a Chebyshev system. The only restriction on the function to be approximated is that it be continuous on a finite interval [ a , b ]. An Algol 60 procedure is given, which will accomplish the approximation. This implementation of the second algorithm of Remez is quite general in that the continuity of ƒ( x ) is all that is required whereas previous implementations have required differentiability, that the end points of the interval be “critical points,” and that the number of “critical points” be exactly n + 2. Discussion of the method used and of its numerical properties is given as well as some computational examples of the use of the algorithm. The use of orthogonal polynomials (which change at each iteration) as the Chebyshev system is also discussed.