Coefficients in Quadrature Formulas

Quadrature formulas for Fourier coefficients

Anti-Gaussian quadrature formulas

An anti-Gaussian quadrature formula is an (n+ 1)-point formula of degree 2n− 1 which integrates polynomials of degree up to 2n+ 1 with an error equal in magnitude but of opposite sign to that of the n-point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show tha...

Stochastic Quadrature Formulas

A class of formulas for the numerical evaluation of multiple integrals is described, which combines features of the Monte-Carlo and the classical methods. For certain classes of functions—defined by smoothness conditions—these formulas provide the fastest possible rate of convergence to the integral. Asymptotic error estimates are derived, and a method is described for obtaining good a posterio...

On Birkhoff Quadrature Formulas

In an earlier work the author has obtained new quadrature formulas (see (1.3)) based on function values and second derivatives on the zeros of nn(i) as defined by (1.2). The proof given earlier was quite long. The object of this paper is to provide a proof of this quadrature formula which is extremely simple and indeed does not even require the use of fundamental polynomials of (0,2) interpolat...

OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W ( m , m − 1 ) 2 SPACE

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W (m,m−1) 2 [0, 1] space for calculating Fourier coefficients. Using S. L. Sobolev’s method we obtain new optimal quadrature formulas of such type for N + 1 ≥ m, where N + 1 is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the...