APPROXIMATE CALCULATION OF SUMS I: BOUNDS FOR THE ZEROS OF GRAM POLYNOMIALS
Nenhuma Miniatura disponível
Data
2014-01-01
Autores
Area, Ivan
Dimitrov, Dimitar K. [UNESP]
Godoy, Eduardo
Paschoa, Vanessa
Título da Revista
ISSN da Revista
Título de Volume
Editor
Siam Publications
Resumo
Let N be a positive integer and x(j) be N equidistant points. We propose an algorithmic approach for approximate calculation of sums of the form Sigma(N)(j=1) F(x(j)). The method is based on the Gaussian type quadrature formula for sums, Sigma F-N(j =1)(x(j)) approximate to Sigma B-n(k=1)n,k F(g(n,k)(N)), n << N,where g(n,k)(N) are the zeros of the so-called Gram polynomials. This allows the calculation of sums with very large number of terms N to be reduced to sums with a much smaller number of summands n. The first task in constructing such a formula is to calculate its nodes g(n,k)(N). In this paper we obtain precise lower and upper bounds for g(n,k)(N). Numerical experiments show that the estimates for the zeros g(n,k)(N) are very sharp and that the proposed method for calculation of sums is efficient.
Descrição
Palavras-chave
approximate calculation of sums, Gaussian type quadrature formula for sums, orthogonal Gram polynomials, zeros of Gram polynomials
Como citar
Siam Journal On Numerical Analysis. Philadelphia: Siam Publications, v. 52, n. 4, p. 1867-1886, 2014.