Efficient method for computing nodes and weights of Gaussian-Zernike quadrature

Abstract

Properties of the underlying Zernike-circle polynomials of the Gaussian-Zernike quadrature are investigated, and an extremely efficient method for computing the nodes and weights is devised. The method allows direct computation of the nodes and weights for any given order, therefore, can be embedded in application programs to calculate the Gaussian-Zernike quadrature formula on the fly. Another advantage of the present method is that it is easy to implement. This new method facilitates application of the Gaussian-Zernike quadrature to large antennas, for which the advantages resulted from reduced number of sampling points become remarkable.