Abstract
We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.
Highlights
Given the data set for the variables x and yCitation: Uteshev, A.; Baravy, I.; x yKalinina, E
As a matter of fact, the referred results should be treated as the gist of the algorithm, which is nowadays known as the Berlekamp–Massey algorithm [19,20]; it was suggested for the decoding procedure in Bose–Chaudhury–Hocquenghem (BCH) codes and for finding the minimal polynomial of a linear recurrent sequence. The deployment of this Hankel polynomial formalism to the main problem of the paper is started in Section 4 with the treatment of the polynomial interpolation
One can utilize the last expression for the direct deduction of the validity of the Formula (53) as a solution to the polynomial interpolation problem
Summary
Numerator and denominator of the rational interpolant can be expressed as Hankel polynomials of appropriate orders with the generating sequences chosen in the form of suitable symmetric functions of the data set (1), namely k. As a matter of fact, the referred results should be treated as the gist of the algorithm, which is nowadays known as the Berlekamp–Massey algorithm [19,20]; it was suggested for the decoding procedure in Bose–Chaudhury–Hocquenghem (BCH) codes and for finding the minimal polynomial of a linear recurrent sequence The deployment of this Hankel polynomial formalism to the main problem of the paper is started in Section 4 with the treatment of the polynomial interpolation.
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