When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as to match the most important poles of the integrand. We describe two methods for generating such quadrature ...

Many authors have investigated numerical methods for solving stochastic differential equations and their boundary value problems. The situation with numerical stochastic integration, as a subject of its own, has so far been different. It has not been studied with the same vigor as of investigating ordinary numerical integration. Consequently, a timely question appears to arise here on whether u...

In this correspondence, the authors develop a novel method based on spherical radial cubature and Gauss–Laguerre quadrature rule for non-linear state estimation problems. The proposed filter, referred as cubature quadrature Kalman filter (CQKF) would be able to overcome inherent disadvantages associated with the earlier reported cubature Kalman filter (CKF). The theory and formulation of CQKF h...

We construct a formula for numerical integration of functions over the unit ball in R that uses n Radon projections of these functions and is exact for all algebraic polynomials in R of degree 2n&1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadratur...

We introduce a finite class of weighted quadrature rules with the weight function |x|−2a exp −1/x2 on −∞,∞ as ∫−∞|x|−2a exp −1/x2 f x dx ∑n i 1 wif xi Rn f , where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, andRn f is the error value. We show that the above formula is valid only for the finite values of n. In...