We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a useroriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach al...

Abstract In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of error these Gauss–Kronrod formulae analytic functions are obtained, using techniques contour integration that were introduced by Gautschi and Varga (cf. SIAM J. Numer. Anal. 20 , 1170–1186 1983). Some illustrative numerical examples which show both accuracy sharpness...

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation second kind. To end, Lagrange interpolation polynomials together with Gauss-Legendre quadrature rule are used to transform source problem system algebraic equations. Afterwards, resulting can be solved by Newton method. The basic idea choose points same as integration. This...

The need to compute expressions of the form uf(A)v, where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and the...

In this paper we examine the computation of the potential generated by space-time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the cur...

Abstract. The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small prob...

The nodes xi and weights wi are uniquely determined by the choice of the domain D and the weighting kernel ψ(x). In fact, one may go as far as to say that the choice of the domain and the kernel defines a quadrature. In particular, the location of the nodes xi are given by the roots of the polynomial of order m in the sequence of orthonormal polynomials {πj} generated by the metric 〈πj|πk〉 := ∫...

1 Abstract Computing energy transfer between objects is the most expensive operation in radiosity systems. This energy transfer operation, known as the irradiance operator, is an integral that, in general, must be calculated numerically. We wish to increase the speed of this computation without severely compromising delity and perform a study of numerical integration techniques, Quadrature Meth...

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...