This work is devoted to the study of integration with respect to binomial measures. We develop interpolation quadrature rules and study their properties. Applying a local error estimate based on null rules, we test two automatic integrators with local quadrature rules that generalize the five points Newton Cotes formula.

Two polynomial expansions of the time-evolution superoperator to directly integrate Markovian Liouville-von Neumann (LvN) equations for quantum open systems, namely the Newton interpolation and the Faber approximation, are presented and critically compared. Details on the numerical implementation including error control, and on the performance of either method are given. In a rst physical appli...

In this paper we present a modified BFGS algorithm for unconstrained optimization. The BFGS algorithm updates an approximate Hessian which satisfies the most recent quasi-Newton equation. The quasi-Newton condition can be interpreted as the interpolation condition that the gradient value of the local quadratic model matches that of the objective function at the previous iterate. Our modified al...

The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...

The Linux [1] random number generator (LRNG) aims to produce random numbers with all the limitations due to a deterministic machine. Two recent analysis exist for this generator [2,5]. These analysis provide strong cryptographic details about LRNG. However both fail to give a mathematical explanation of the entropy estimator embedded. In this paper we propose an interpretation using Newton poly...

In this paper we propose a learning based iterative numerical method for computing a rootξ of a non-linear equation of the form f(x) = 0 in the interval [a,b]. Lag based method through Hermite interpolation modeled root discovery approach is developed and demonstrated in this paper. The new method has been tested for a series of functions considered by several researchers. The numerical experim...

We derive a simple error estimate for equally spaced, polynomial interpolation of power series that does not require the uniform bounds on derivatives of the Cauchy remainder. The key steps are expressing Newton coefficients in terms of Stirling numbers S(i, j) of the second kind and applying the concavity of lnS(i, j).

Given convex scattered data in R3 we consider the constrained interpolation problem of finding a smooth, minimal Lp-norm (1 < p < ∞) interpolation network that is convex along the edges of an associated triangulation. In previous work the problem has been reduced to the solution of a nonlinear system of equations. In this paper we formulate and analyse a Newton-type algorithm for solving the co...