APPROXIMATE CALCULATION OF SUMS I: BOUNDS FOR THE ZEROS OF GRAM POLYNOMIALS

Author(s)

Area, Ivan ^[1] ; Dimitrov, Dimitar K. ^[2] ; Godoy, Eduardo ^[1] ; Paschoa, Vanessa ^[3]

Affiliation

^[1] Univ Vigo, EE Telecomunicac, Dept Matematica Aplicada 2, Vigo 36310 - Spain

^[2] Univ Estadual Paulista, IBILCE, Dept Matematica Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[3] Univ Fed Sao Paulo, Dept Ciencia & Tecnol, BR-12231280 Sao Jose Do Rio Preto, SP - Brazil

• Document type: Journal article
• Source: SIAM JOURNAL ON NUMERICAL ANALYSIS; v. 52, n. 4, p. 1867-1886, 2014.
• Web of Science Citations: 6

Abstract

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. (AU)

• FAPESP's process: 09/13832-9 - Orthogonal polynomials, special functions and applications
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 13/23606-1 - Methods for approximate calculus of sums and series
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International