This article describes a fast direct solver (i.e., not iterative) for partial hierarchically semiseparable systems. This solver requires a storage of O(N logN) and has a computational complexity of O(N logN) arithmetic operations. The numerical benchmarks presented illustrate the method in the context of interpolation using radial basis functions. The key ingredients behind this fast solver are...

Three division algorithms are presented for univariate Bernstein polynomials: an algorithm for finding the quotient and remainder of two univariate polynomials, an algorithm for calculating the GCD of an arbitrary collection of univariate polynomials, and an algorithm for computing a μ-basis for the syzygy module of an arbitrary collection of univariate polynomials. Division algorithms for mult...

In this paper, we present a numerical method for solving Hallen’s integral equation based on radial basis functions (RBFs). This method will represent the solution of Hallen’s integral equation by interpolating the radial basis functions based on Legendre-Gauss-Lobatto(LGL) nodes and weights. The numerical results show that the proposed method for Hallen’s integral equation is very accurate and...

Determination of the basis of the space of all root functionals of a system of polynomial equations and of the basis of its ideal by the operation of the extension of bounded root functionals (Presented by Corresponding Member of the NAS of Ukraine A. A. Letichevsky) The notion of a root functional of a system of polynomials or an ideal of polynomials is a generalization of the notion of a root...

In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.

In my PhD thesis 1965 and the subsequent publication 1970 in aequationes mathematicae, I introduced the notion of Gröbner bases and proved a charaterization theorem on which an algorithm for constructing Gröbner bases can be based. The characterization theorem states that a set F of multivariate polynomials (over a field) is a Gröbner bases iff all S-polynomials of F reduce to 0 w.r.t. F. The a...

Two models were recently proposed to explore the robust hardness of Gröbner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gröbner basis for the ideal generated by the remaining polynomials. For the q-Fractional Gröbner Basis Problem the algorithm is allowed to ignore a...

Division algorithms for univariate polynomials represented with respect to Lagrange and Bernstein basis are developed. These algorithms are obtained by abstracting from the classical polynomial division algorithm for polynomials represented with respect to the usual power basis. It is shown that these algorithms are quadratic in the degrees of their inputs, as in the power basis case.

in this paper we demonstrate the existence of a set of polynomials pi , 1 i  n , which arepositive semi-definite on an interval [a , b] and satisfy, partially, the conditions of polynomials found in thelagrange interpolation process. in other words, if a  a1  an  b is a given finite sequence of realnumbers, then pi (a j )  ij (ij is the kronecker delta symbol ) ; moreover, the sum of ...