10th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Bucharest, Romania, 7 - 09 November 2008, pp.15-16
This work attempts to develop a one node quadrature which gives better results than standard Gauss Quadrature by using just one node. This is based on a coordinate transformation in such a way that the resulting integral's weight becomes the multiplication of the original weight by a nonnegative function which is in fact the square of a finite linear combination of a given basis set for the Hilbert space where the integration is performed. The basis set can be chosen as a set of polynomials which are orthonormals under the weight and over the interval of the integration. In that case certain drastic simplifications appear in the matrix representations of integration variable powers as it occurs in the Gauss quadrature. However it is not mandatory to use these polynomials. Depending on the nature of the integrand various sets of functions with more complicated structures can be taken into consideration as basis set. It is just a matter of expertise. Presentation keeps the basis set general without specifying its structure in the conceptual part and uses orthogonal polynomials in illustrative implementations.