This paper addresses the problem of improving the unknown syndrome representations to develop algebraic decoding of the (17,9,5) and (23,12,7) binary quadratic residue codes up to true minimum distance, respectively. The proposed unknown syndrome representations are expressed as binary polynomials in terms of the single known syndrome, which is different from the known syndrome in [Chang-Lee, A...

We present fast evaluating and differentiating algorithms for the Hermite interpolating polynomials with the knots of multiplicity 2, which are generated dynamically in a field , , K K by the recurrent formula of the form 1 0 1,2,.., 1; i i x x i n x . As in the case of Lagrange-Newton interpolating algorithms, the running time of these algorithms is C n O n base operations from the field K, wh...

In the paper [8], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [−1, 1], and derived a compact form of the corresponding Lagrange interpolation formula. In [1] we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows ...

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonome...

where n is even, h = (b − a)/n, xi = a + ih, and ξ ∈ (a, b). We usually derive (1) using Lagrange polynomials or making the formula exact for f(x) = 1, x, x. A standard exercise for a numerical analysis class is to use the composite Simpson’s rule to approximate an integral with n equal to successive powers of two, and verify that (as long as the fourth derivative of f is well behaved) the erro...

School of Science, Dalian Nationalities University Dalian 116600, Liaoning, P.R. China Abstract In this work, we mainly review two methods for improving the uniform convergence of the known Lagrange interpolation polynomial put forward by Bernstein, namely, the method of weighted mean for basic functions of interpolation and the method of lowering some interpolation conditions. Moreover, we pre...

Due to the elimination property held by the lexicographic monomial order, the corresponding Gröbner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolatio...

Abstract This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on Scott–Vogelius pair Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials degree k , pressure ( – 1) without continuity constraints. A Lagrange multiplier that with respect to partition is introduced enforce conditions mitigate lack pressure-rob...

In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then exploit these extra degrees of freedom in the case of type A to give the first proof of Kadell’s orthogonality ...

In this paper, an approach based on statistical spline model (SSM) and collocation method is proposed to solve Volterra-Fredholm integral equations. The set of collocation nodes is chosen so that the points yield minimal error in the nodal polynomials. Under some standard assumptions, we establish the convergence property of this approach. Numerical results on some problems are given to describ...