A New Algorithm For Best Average Rational Approximation

In the mathematics, rational approximation is an important problem. It cloud solve not converge and instability in polynomial approximation or inter-polation approximation and so on, so it has the value of research.Rational average approximation should be better than polynomial approx-imation in a wide rang of function approximation, but its study is few up to now. For example, it had high precision of the Taylor series expansion points, but some distance points were not good, the same to rational pade approxi-mation. Lagrange interpolation approximation is used to applying, however, when the function of curvature changes large approximation, accuracy may be very poor. Mr Wang write a book of rational interpolation approximation, it meat the demanding solvability conditions, it was very difficult to guarantee the accuracy of the whole.We have analyzed the singularity in the approximation and further pro-posed a new algorithm:regularization+Newton, by which better results can be obtained. Through several approximation algorithms, its results showed that the nature of the function was not good, the effect was still good. Sec-ondly, we have analyzed weighting best average rational approximation, its de-fect was that the error of some distance points was reduced, so that the error of the whole was reduced.Finally,we have analyzed specified boundary value of best average rational approximation,its defect was that the error of some distance points was zero. The conclusion is that they were verified by our numerical experiments.