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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 González
^{1} and G. López Lagomasino^{2}
J.R. Illán González, G. López Lagomasino, "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
. This
paper shows how the technique of changing the
variable to increase the efficiency of a Gaussian rational
quadrature formula (GRQF) can be applied. We are especially interested in
evaluating
when
f is analytic but has poles
close to [a,b]. We say that a pole of f is difficult
when it is so close to [a,b] that instability shows up. We
mainly consider the case in which the integrand f has real
difficult poles and the weight function possibly has
integrable singularities at the end points of the integration
interval.
Gautschi [1] has developed routines to calculate nodes and
coefficients for a GRQF when some poles of
Let be the array defined as
,
,
where , is the The rational function given below
is adopted as the smoothing transformation of [a,b] which is
fitted into the modified moments . One expects that the
process of changing the variable produces the effect of moving
real poles away from I. The singularities of W are also
annihilated, so that the integrals can be computed
accurately by applying a composite Gauss-Legendre rule.
If and are the coefficients of the recurrence relation which the polynomials fulfil, then we can easily derive a relationship as follows:
Thus, we only have to calculate the moments ,
, to obtain, step by step, all the recurrence
coefficients , , For every 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
. It consists of fitting a substitution
,
, into . This mapping is defined as
, where 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
- 1
- W. Gautschi, "Algorithm 793: GQRAT-Gauss quadrature for rational functions", ACM Trans. Math. Software, 25, 213-239, 1999. doi:10.1145/317275.317282
- 2
- U. Fidalgo Prieto, J.R. Illán González and G. López Lagomasino, "Convergence and computation of simultaneous rational quadrature formulas", Submitted to Numer. Math. doi:10.1007/s00211-006-0056-8
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