and at B.) No high precision is required here, the standard line search with a couple of quadratic fits should suffice. (See ) Let . Each amplitude is greater than or equal to E. The Theorem of de La Vallée Poussin and its proof also apply to with as the new lower bound for the best error possible with polynomials of degree n. Moreover, comes in handy as an obvious upper bound for that best possible error. Step 4: With and as lower and upper bound for the best possible approximation error, one has a reliable stopping criterion: repeat the steps until is sufficiently small or no longer decreases. These bounds indicate the progress. Variants Some modifications of the algorithm are present on the literature. These include: Replacing more than one sample point with the locations of nearby maximum absolute differences. Replacing all of the sample points with in a single iteration with the locations of all, alternating sign, maximum differences. Using the relative error to measure the difference between the approximation and the function, especially if the approximation will be used to compute the function on a computer which uses floating point arithmetic; Including zero-error point constraints. The Fraser-Hart variant, used to determine the best rational Chebyshev approximation. See also Approximation theory References External links Minimax Approximations and the Remez Algorithm, background chapter in the Boost Math Tools documentation, with link to an implementation in C++ Intro to DSP Polynomials Approximation theory Numerical analysis