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Methods for approximate calculus of sums and series

Abstract

The main objective of the project is to develop methods for precise numerical approximation of either sums with large number or of convergent series by finite sums with small number of term. Numerical Integration, called also the Theory of Quadrature Formulae is one of the most fundamental areas of Numerical Analysis, due to the vats number of its applications in other branches of Applied Mathematics, Physics and Engineering. However, there are any results and algorithms for approximate calculation of sums with large number of terms and of series, despite that this topic is very close to numerical integration from mathematical point of view. On the other hand, the problem for numerical calculation of sums appear in practice as often as the one for calculation integrals. One of the fundamental applications of calculation of sums is in the solution of the problem for construction the least squares approximations, which itself is the most commonly used method for approximate recovery of data. Our proposal is to construct analogs of the Gaussian quadrature formulae for precise approximation of sums. The construction of such Gaussian type formulae will be developed on the basis of some results about the behavior of zeros of orthogonal polynomials of discrete variable, the use of the so-called Weiertrass-Dochev-Durand-Kerner method for simultaneous calculus of zeros of polynomials, numerical methods for eigenvalues of matrices and other numerical techniques. We plan to investigate thoroughly all aspects of the proposed algorithms: precision of the calculations, the error analysis when the general term of the approximated sum is a function which belongs to a certain normed space, as well as the convergence and stability of the algorithms. Some of the fundamental tools in the analysis are results from the Theory of Special Functions and Orthogonal Polynomials and Optimal Recovery. The team which has initiated the work on this topic consists of Professors Eduardo Godoy and Ivan Area from University of Vigo, Spain, Dr. Vanessa Gonçalves Paschoa Ferraz, from UNIFESP, São José dos Campos, and the author of the project. (AU)

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Scientific publications (4)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
AREA, IVAN; DIMITROV, DIMITAR K.; GODOY, EDUARDO; PASCHOA, VANESSA G. APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE. SIAM JOURNAL ON NUMERICAL ANALYSIS, v. 54, n. 4, p. 2210-2227, 2016. Web of Science Citations: 2.
AREA, IVAN; DIMITROV, DIMITAR K.; GODOY, EDUARDO; PASCHOA, VANESSA. Bounds for the zeros of symmetric Kravchuk polynomials. NUMERICAL ALGORITHMS, v. 69, n. 3, p. 611-624, JUL 2015. Web of Science Citations: 1.
AREA, IVAN; DIMITROV, DIMITAR K.; GODOY, EDUARDO. Zero sets of bivariate Hermite polynomials. Journal of Mathematical Analysis and Applications, v. 421, n. 1, p. 830-841, JAN 1 2015. Web of Science Citations: 2.
AREA, IVAN; DIMITROV, DIMITAR K.; GODOY, EDUARDO; PASCHOA, VANESSA. APPROXIMATE CALCULATION OF SUMS I: BOUNDS FOR THE ZEROS OF GRAM POLYNOMIALS. SIAM JOURNAL ON NUMERICAL ANALYSIS, v. 52, n. 4, p. 1867-1886, 2014. Web of Science Citations: 6.

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