On polynomial and polynomial matrix interpolation

Abstract

The classical algorithms for computations with polynomials and polynomial matrices use elementary operations with their coefficients. The relative accuracy of such algorithms is relatively small and for polynomials of higher order and polynomial matrices of higher dimension the executing time grows very quickly. Another possibility is to use symbolic manipulation package but even this is applicable only for moderate problems. This paper improves a new method based on polynomial interpolation. Its principle is as follows [1]: firstly a sufficient number of interpolation points is chosen, then the interpolated object is evaluated in these points and finally it is recovered from both series of values. The choice of interpolation points is crucial to have a well-conditioned task. Typically, a random choice of real points leads to a badly conditioning for higher order of interpolated polynomial. However, a set of complex points regularly distributed on the unit circle in the complex plane gives a perfectly conditioned task. Moreover very efficient algorithm of fast Fourier transform can be used to recover the resulted polynomial or polynomial matrix. The efficiency is demonstrated on determination of inverse to polynomial matrix. © Springer-Verlag Berlin Heidelberg 2002.