scholarly journals Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

2020 ◽  
Vol 53 ◽  
pp. 352-382
Author(s):  
D. Lj. Djukić ◽  
R. M. Mutavdžić Djukić ◽  
A. V. Pejčev ◽  
M. M. Spalević
Keyword(s):  
Error Estimates ◽  
Analytic Functions ◽  
Quadrature Formulae ◽  
Gaussian Type

scholarly journals Error estimates of Gaussian quadrature formulae with the third class of Bernstein-Szegő weights

2017 ◽  
Vol 11 (2) ◽  
pp. 451-469
Author(s):  
Aleksandar Pejcev
Keyword(s):  
Error Estimates ◽  
Error Bounds ◽  
Analytic Functions ◽  
Gaussian Quadrature ◽  
Gauss Quadrature ◽  
Weight Functions ◽  
Quadrature Formulae ◽  
The Third ◽  
Exact Degree ◽  
Gauss Quadrature Rules

For analytic functions we study the remainder terms of Gauss quadrature rules with respect to Bernstein-Szeg? weight functions w(t) = w?,?,?(t) = ?1+t/ 1-t/?(?-2?)t2+2?(?-?)t+?2+?2, t?(-1,1), where 0 < ? < ?, ??2?, ??? < ?-?, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1,1]. The subcase ?=1, ?= 2/(1+?), -1 < ? < 0 and ?=0 has been considered recently by M. M. Spalevic, Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szeg? weights, Math. Comp., 82 (2013), 1037-1056.

scholarly journals Error estimates for Gaussian quadratures of analytic functions

2009 ◽  
Vol 233 (3) ◽  
pp. 802-807 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Miodrag M. Spalević ◽  
Miroslav S. Pranić
Keyword(s):  
Error Estimates ◽  
Analytic Functions ◽  
Gaussian Quadratures

scholarly journals Error estimates for asymptotic representations of integrals

1972 ◽  
Vol 13 (4) ◽  
pp. 395-410 ◽  
Author(s):  
J. J. Mahony
Keyword(s):  
Error Estimates ◽  
Analytic Functions ◽  
A Priori ◽  
Singular Points ◽  
Optimum Point ◽  
Image Position ◽  
Asymptotic Representations ◽  
Marked Points ◽  
Path Of Integration

Abstract. For large real positive values of the parameter k asymptotic representations of integrals of the form , where f and g are analytic functions, can be obtained by using methods such as steepest desents. Here methods are considered for obtaining estimates, for fixed values of k, of the minimum errors achievable when such asymptotic representations are appropriately curtailed. A priori criteria are derived for the optimum point at which to curtail such asymptotic representations are appropriately curtailed. A priori criteria are derived for the optimum point at which to curtail such asymptotic representations. Both the curtailment points and the minimum errors are related to the distance between certain marked points on the path of integration and the singular points of f(u) and the zeros of g(u). The analysis permits the determination of errors whose presence is not indicated by the numerical behaviour of the asymptotic representations. It is also capable of extension to complex parameters k and to the derivation of asymptotic representations for the most significant errors. It can therefore be used to extend the domain of k for which asymptotic representations are available.

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