Newton-Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function values are only available on a regular grid. Yet, these rules rapidly become unstable for high orders. In this paper we review two techniques to construct stable high-order quadrature rules using equidistant quadrature points. The stabil...

The Krylov subspace approximation techniques described by Gallopoulos and Saad 2] for the numerical solution of parabolic partial diierential equations are extended. By combining the weighted quadrature methods of Lawson and Swayne 6] with Krylov subspace approximations, three major improvements are made. First, problems with time-dependent sources or boundary conditions may be solved more eeci...

WENJUN LIU, YONG JIANG, AND ADNAN TUNA Abstract. By introducing a parameter, we give a unified generalization of some quadrature rules, which not only unify the recent results about error bounds for generalized mid-point, trapezoid and Simpson’s rules, but also give some new error bounds for other quadrature rules as special cases. Especially, two sharp error inequalities are derived when n is ...

Abstract. This paper presents a modification of Krylov subspace spectral (KSS) methods, which build on the work of Golub, Meurant and others, pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to variable-coefficient time-dependent PDEs. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, our modification us...

In this paper a combination of discontinuous, piecewise linear, finite elements with implicitexplicit time stepping is considered for convection-reaction equations. Combined with low order quadrature rules, this leads to convenient schemes. We shall consider the effect of such low order quadrature rules on accuracy and stability for one-dimensional problems. 2000 Mathematics Subject Classificat...

Inequalities are obtained for quadrature rules in terms of upper and lower bounds of the first derivative of the integrand. Bounds of Ostrowski type quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case. Applications to numerical integration are demonstrated.

Quadrature rules on semi-infinite and infinite intervals are considered involving weight functions of the Laguerre and Hermite type. It is shown that such quadrature rules cannot have equal coefficients and real nodes unless the algebraic degree of accuracy is severely limited.

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...

Recently Laurie presented a fast algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of Gauss-Kronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.