Keynote lecture: a one node fluctuation suppressing univariate integration via deviation removals between power means and mean powers
Authors: Metin Demiralp
Venue: Proceedings of the 10th WSEAS international conference on Mathematical and computational methods in science and engineering (2008)

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. An n-node Gauss quadrature uses a linear combination of the function's values at the nodes with positive combination coefficients which are called weight parameters. The nodal values and the weight parameters are universal. That is, they are not depending on the function although they may vary with the interval and the weight function of the integration. According to Fluctuationlessness Theorem these entities are chosen in such a way that the deviation of the (1, 1) element of the n dimensional matrix representation of, say, j-th power of the independent variable from the j-th power of the (1, 1) element of the independent variable's matrix representation is set equal to zero. The basis set should be an orthogonal polynomial set due the construction philosophy of Gauss quadrature. The nodes are the eigenvalues of the matrix representation of the independent variable in the space spanned by first n orthonormal polynomials and the weight parameters are the squares of the first elements of the corresponding eigenvectors. The construction of the nodal values and the weight parameters annihilates the first 2n-1 deviations mentioned above leaving remaining infinite number of deviations nonzero. The theory of fluctuation free integration matches Gauss quadrature when its basis set is chosen as the basis set of the Gauss quadrature. However, there is no limitation on the basis set of the fluctuation free integration except the orthonormality and the spanning of the whole Hilbert space where integration is performed. This means more flexibility in this case. The problem is to use this flexibility efficiently. If this is done then it is possible to get results better in precision than Gauss quadrature. According to mean value theorem of integration it is always possible to find a unique function value which is the image of an interior point of the domain interval under the function, such that, the integral of this function over its domain is equal to this value times the integration interval length as long as the function under consideration is continuous over integration interval. This implies that it is possible to construct an appropriate one node quadrature to get, at least approximately, this point. Gauss quadrature somehow does this. However, integration variable transformation may change the integral in such a way that a better value can be produced for this new integral. The presentation focuses on these aspects of the issue.