Let K be a field. Generalizing the binomial coefficient polynomials ( X n ) , the Newton interpolation basis polynomials relative to an infinite sequence a = (ai)i=0 of distinct elements of K are defined by [X n ] a = ∏n−1 i=0 X−ai an−ai ∈ K[X ]. There is a complete and natural set of relations for these polynomials, namely, [X m ] a [X n ] a = ∑m+n l=max(m,n)[m,n, l]a [X l ] a , where the coef...

Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in Rn. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or “interpolated”. The sequence of univariate calculations is more efficient than multivariat...

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x 1 ,. .. , x r be closed points in general position in projective space P n , then the linear subspace V of H 0 (P n , O(d)) (the space of homogeneous polynomials of degree d on P n) formed by those polynomials which are singular at each x i , ...

We show how to combine incidence matrices, which admit Hermite-Birkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the co...

This paper deals with the summation problem of power series of the form Sb a(f ;x) = ∑ a≤k≤b f(k)x k, where 0 ≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for Sb a(f ;x), by the ...

In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of n...

We employ the generalized factorials to define a Stirling-type pair {s n, k;α,β, r , S n, k;α,β, r } which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The g...

We consider families of sparse Laurent polynomials f1, . . . , fn with a finite set of common zeroes Zf in the torus T n = (C − {0}) . The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Zf . We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the fi when the Newton polytopes of the fi ...

In this paper, we construct a piecewise interpolation method of approximate solution the transport equation based on Newton polynomial two variables. We transform to algebraic form with numerical coefficients; leads us sequence iterations, which improves accuracy approximation. The is implemented in software and experiments are performed. possibility generalizations systems partial differential...

We study the local topological structure of generic multijet preimages of algebraic varieties and prove their stratifiability with additional quantitative estimates. The result is a necessary technical tool for analyzing bifurcations of periodic orbits from heteroclinic polygons of vector fields in space. The proof is based on interpretation of Newton interpolation formula as a resolution of si...