Extended Fluctuationlessness Theorem and its Application to Numerical Integration via Taylor Series ERCAN GÜRV ˙ IT ∗ Marmara University Mathematics Department Göztepe Campus ˙ Istanbul TÜRK ˙ IYE (TURKEY) ercan.gurvit@be.itu.edu.tr N.A. BAYKARA Marmara University Mathematics Department Göztepe Campus ˙ Istanbul TÜRK ˙ IYE (TURKEY) nbaykara@marmara.edu.tr MET ˙ IN DEM ˙ IRALP ˙ Istanbul Technical University Informatics Institute Maslak, 34469, ˙ Istanbul TÜRK ˙ IYE (TURKEY) metin.demiralp@be.itu.edu.tr Abstract: According to the Fluctuationlessness Theorem, the matrix representation of a function is approxi- mately equal to the image of the matrix representation of its independent variable under the same function, in the Hilbert space of square integrable functions. In this work Extended Fluctuationlessness theorem applied to the finite integral of the Taylor expansion is taken into consideration to form a new quadrature. Key–Words: Fluctuationlessness theorem, quadrature, numerical approximation, numerical integration, Fluctu- ation expansion, Taylor expansion, matrix representation, extended fluctuationlessness theorem 1 Introduction The Fluctuationlessness theorem tells us that the ma- trix representation of a univariate function can be ap- proximated by the image of its independent variable’s matrix representation under the same function given that this function is analytic over the interval in ques- tion [2–20]. By choosing the unit constant function as the first basis function of the Hilbert space of square integrable functions a quadrature whose nodes appear as the eigenvalues of the matrix can be formed. And the corresponding weights are the squares of the first elements of the eigenvectors [5, 6, 11–20, 23]. Even though the gaussian quadrature and the Fluctuationlessness theorem [14–18, 22, 23] seem to be similar, the latter has the advantage of being able to use arbitrary basis sets which may lead to more ac- curate results 2 Fluctuationlessness Theorem and Numerical Integration The matrix representation of the operator multiplying its operand by the function f (x) which is analytic on the working interval can be expressed as  f = ∞  k=0 f k  x k (1) * All authors are member of the Group for Science and Meth- ods of Computing We are working in the Hilbert space H. An inner prod- uct is defined as (f,g)=  b a dx w(x)f (x)g(x) (2) where w(x) is a typical weight function. Any or- thonormal set of functions can be chosen as a basis set. The integration interval can be transformed into the [0,1] interval via an affine transformation. Finally the weight function satisfies the below mentioned prop- erty.  b a dx w(x)=1 (3) The norm of a function in the Hilbert space in question is defined as ||f || =  (f,f ) (4) The n × n matrix representation of an operator or a function is denoted by M and the choice of n reflects the dimension of the truncated Hilbert space as well as the number of nodes of the quadrature. We can now write down the finite matrix representation of ˆ f . e T i M   f  e j =  b a dx u i (x)f (x)u j (x), 1 ≤ i, j ≤ n (5) where e k is the unit vector in the k-th direction. u k (x) is the kth basis function of the truncated Hilbert space which means that the image on the truncated subspace is formed by considering the projection on the first n basis functions. Proceedings of the International Conference on Applied Computer Science 362