The authors are best approximating functions \(f\in C[\alpha, \beta]\) by functions of a \(m\)-parameter smooth subfamily \(M\) of \(C[\alpha, \beta]\), which satisfies the local and the global Haar condition. Under proper assumptions, the corresponding exchange algorithm will generate references, where the nonlinear error equation can be solved, if the error is sufficiently small, while the deviation in the references is not increasing and converging to the minimal deviation. If \(f\in C^2[\alpha, \beta]\) and if the approximating family is also smooth of the second-order, then the convergence is quadratic.