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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 31
A Numerical Approach for Gaussian Rational Formulas to Handle Difficult Poles J.R. Illán González1 and G. López Lagomasino2
1Department of Applied Mathematics I, University of Vigo, Spain
Full Bibliographic Reference for this paper
, "A Numerical Approach for Gaussian Rational Formulas to Handle Difficult Poles", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 31, 2006. doi:10.4203/ccp.84.31
Keywords: Gauss rational quadrature formula, smoothing transformation, difficult poles, substitution mapping, meromorphic integrand.
Summary
Let
![]() ![]() ![]()
Gautschi [1] has developed routines to calculate nodes and
coefficients for a GRQF when some poles of f are difficult. The authors and Fidalgo [2] found a
method different from Gautschi's which has been successfully
applied to compute simultaneous rational quadrature formulas
(SRQF). This paper presents a version of the SRQF approach adapted
to GRQF for evaluating
Let
The rational
function
is adopted as the smoothing transformation of [a,b] which is fitted into the modified moments ![]() ![]()
If
Thus, we only have to calculate the moments Table 1 is a sample of the results produced when the following integral is evaluated by the smoothing method and by Gautschi's. In comparison, one observes that this approach is superior to that reported in [1].
We present a slight variant of the smoothing method to improve
the accuracy when some poles are very difficult, that is, when the
distance from the pole to the integration interval is less than
One of the experimental conclusions is that the smoothing
method also works when the non real difficult poles are in the
region
The results of some numerical tests are shown in the paper to illustrate the power of this approach when they are compared with those produced by other polynomial and rational methods. References
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