The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the Gauss-Kronrod scheme. An essential progress was made only recently, based on new bounds and as-ymptotic properties for the S...

In this work the asymptotic behavior of the partial sums of the divergent asymptotic moment series 2% \ MiA'> where \i.l are the moments of the weight functions w{x) = x"e~ , a > 1 , and w(x) = x"Em(x), a > 1 , m + a > 0, on the interval [0, oo), is analyzed. Expressions for the converging factors are derived. These converging factors form the basis of some very accurate numerical quadrature fo...

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximant...

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...

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?