We consider the problem of strong approximations of the solution of Itô stochastic functional differential equations involving a distributed delay term. The mean-square consistency of a class of schemes, the Θ-Maruyama methods, is analysed, using an appropriate Itô-formula. In particular, we investigate the consequences of the choice of a quadrature formula. Numerical examples illustrate the th...

In this paper in the space L (m) 2 (0, 1) the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed N and for any m ≥ 2. It is ...

We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier-Tchebycheff coefficients given by Michhelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a func...

Abstract. Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel’s contour integral. For example, Temme evaluates this integral based on steepest-decent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by ...

Let f : [−1, 1] → R be continuously differentiable. We consider the question of approximating f (1) from given data of the form (tj , f(tj)) M j=1 where the points tj are in the interval [−1, 1]. It is well known that the question is ill–posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order q...

The solution of an integral equation arising in a covariance factorization problem is obtained by a Newton-Raphson iteration that is almost always globally convergent. Interpretations of the iterates are given, and the result is shown to specialize to known algorithms when the covariance is stationary with a rational Fourier transform.

We study optimal quadrature formulas for convex functions in several variables. In particular, we answer the following two questions: Are adaptive methods better than nonadaptive ones? And: Are randomized (or Monte Carlo) methods better than deterministic methods?