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20 lines
1.0 KiB
20 lines
1.0 KiB
Integrand with unknown singularities.
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int_{-1}^{1} exp(-x^2) dx
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The integrand tends to infinity when the imaginary part of x tends
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to infinity. Hence, a rational quadrature rule with all poles at
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infinity (i.e., a classical polynomial quadrature rule) will work
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perfectly well for this example. But let us assume for a moment that
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the exact location of the singularities are unknown. One usually
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relies then on the singularities of a [n/m] Pade approximant, where
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n and m denote the degree of the numerator and denominator polynomial
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respectively.
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For the poles we use the zeros of the Maclaurin polynomial of
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degree 20 of exp(x^2):
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1+x^2+(1/2)*x^4+(1/6)*x^6+(1/24)*x^8+(1/120)*x^10+(1/720)*x^12+
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+(1/5040)*x^14+(1/40320)*x^16+(1/362880)*x^18+(1/3628800)*x^20
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(which corresponds to using the singularities of the [1/20] Pade
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approximant of exp(-x^2)).
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The maximal number of iterations is 11.000
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Computed value: 1.4936482656248531e+00
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Estimated relative error : 3.8651400473026821e-15
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Exact relative error : 5.9463693035425850e-16
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