Keynote lecture II: who is afraid of integration: no fluctuation approximation for any arbitrary precision
Authors: Metin Demiralp
Venue: Proceedings of the 2nd conference on European computing conference (2008)

This lecture is focused on the no fluctuation approximation for univariate integration. This approximation technique has been developed quite recently by the author's group and its development is continuing towards several directions. The basic idea behind this novel approach is not to approximate a function but its matrix representation when it is considered as a multiplication operator. As is well-known almost all of the numerical integration methods based on quadratures where the integral is approximated by a linear combination of the integrand function's values at interior points (sometimes end points of the integration interval can be taken into consideration depending on the considered method's nature) with positive combination coefficients. ∫abdxW(x)f(x) ≈ Σi=1nwif(xi) where a < b are real paremeters which can be finite or infinite in magnitude and W(x) stands for a given weight function which is positive everywhere except (in certain cases) some number of interior points of the interval. The right hand side entities, wi and xis are positive coefficients which are called weight factors, and, the interval's interior points (certain cases may involve end points also) which are called nodes respectively. The integrand of the integral is generally approximated by either polynomials, splines, or sometimes orthogonal functions to get (1). Depending on the idea behind the approximations the weight factors and nodes take different set of values and this affects the quality and stability of the approximation. Although we do not intend to give these topics in all details, we will emphasize on important issues about the approximation quality mostly. The main emphasis will be given on the no fluctuation approximation which approximates not the function f(x) but an algebraic operator whose action on its operand is just the multiplication by the function f(x) for a specified x independent variable value. This operator is linear and is defined on a Hilbert space of square integrable functions. It can be noticed without any difficulty that the matrix representation of the operator which multiplies its operand by f(x) over an orthonormal of basis set for the Hilbert space under consideration is equal to the image of the matrix representation for the operator whose action on its operand is just to multiply by independent variable x under the function f. This equivalence does not hold for the matrix representations over the subspaces of Hilbert space under consideration. However, as the author of this talk had proven this equivalence does approximately hold and the approximation quality increases as the dimension of the subspace under consideration grows up to infinity. If we consider the basis functions which are in fact natural number powers of the independent variable, that is, 1, x, x2,..., then we can construct a polynomial basis function set by orthonormalizing this set in the order of ascending natural number powers. If this orthonormal polynomials are denoted by ui(x), i = 1, 2,... where i - 1 is the degree of the polynomial, and, we denote the matrix representation of f(x) by M (f') where f' stands for the operator whose action on its operand is the multiplication by f(x) then we can write Mij(f) ≡ ∫ab dxW(x)ui(x)f(x)uj(x), i,j = 1,2,... If W(x) is normalized to make its integral equal to 1 then u1(x) becomes 1 everywhere in the interval of the integration. This enables us to write ∫ab dxW(x)f(x)=e1TM(f)e1 where e1 stands for the first standard unit vector whose all elements except the first one which is 1 vanish in infinite dimensional cartesian space. If the infinite matrixM (f') is replaced by its left uppermost n×n block where n is a positive integer then we obtain the n-th order no fluctuation approximation. The quality discussions on this focus will also be comprehensively included in the lecture.