We investigated an interpolation algorithm for computing the MoorePenrose inverse of a given polynomial matrix, based on the LeverrierFaddeev method. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier-Faddeev algorithm is given as the improvement of the interpolation algorithm. Algorithms are implemented in the symbolic programming language MATHEMATICA, ...

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using ...

We apply multivariate Lagrange interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore genera...

Let T be a Jordan curve in the complex plane, and let Í) be the compact set bounded by T. Let / denote a function analytic on O. We consider the approximation of / on fî by a polynomial p of degree less than n that interpolates / in n points on T. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation ...

An algorithm is described for extracting a polynomial matrix factor featuring any subset of the zeros of a given non-singular polynomial matrix. It is assumed that the zeros to be extracted are given as input data. Complex or repeated zeros are allowed. The algorithm is based on interpolation and relies upon numerically reliable subroutines only. It makes use of a procedure that computes the ge...

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [15], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more effective estimation of degrees of polynomial matrices generated in Leverrier-Faddev method. Algorithms are implemented and tested i...

Let Ln [f ] denote the Lagrange interpolation polynomial to a function f at the zeros of a polynomial Pn with distinct real zeros. We show that f − Ln [f ] = −PnHe [ H [f ] Pn ] , where H denotes the Hilbert transform, and He is an extension of it. We use this to prove convergence of Lagrange interpolation for certain functions analytic in (−1, 1) that are not assumed analytic in any ellipse wi...

The problem of choosing " good " nodes on a given compact set is a central one in multivariate polynomial interpolation. Besides unisolvence, which is by no means an easy problem, for practical purposes one needs slow growth of the Lebesgue constant and computational efficiency. In this talk, we present new sets of nodes recently studied for polynomial interpolation on the square that are asymp...

If the SRC is performed between arbitrary sampling rates, then the SRC factor can be a ratio of two very large integers or even an irrational number. An efficient way to reduce the implementation complexity of a SRC system in those cases is to use polynomial-based interpolation filters with the impulse response ha(t) having the following properties. ha(t) is nonzero for an interval 0 ≤ t < NT w...

A corrected interpolating polynomial is derived. Error inequalities of Ostrowski type for the corrected interpolating polynomial are established. Some similar inequalities are also obtained.