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32 lines
1.5 KiB
32 lines
1.5 KiB
The sequence of poles is [2:1:10].
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The quadrature formula should be exact for integrals of the form:
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int_{-1}^{1} ( (x-a).^(-1) ) dx, for a = 2,...,10.
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Exact maximal relative error on the approximations: 1.8657659241435624e-15
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We now reverse the order of the poles:
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The new sequence of poles is [10:-1:2].
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The quadrature formula should be exact for the same integrals as before.
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Exact maximal relative error on the approximations: 1.7413815292006581e-15
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Theoretically the nodes and weights do not depend on the
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order of the poles. Hence, the weights should be identical
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for both sequences of poles.
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In practice, however, the weights can contain large errors.
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When comparing the computed weights for both sequences of poles,
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we obtain a minimal relative distance: 7.5336888536851936e-05
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Despite the large errors, both quadrature formulae perform
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equally well for the approximation of integrals of the form:
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int_{-1}^{1} f(x) dx, where the function f is arbitrary.
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Consider the case in which f(x) = (x-1.8).^(-1).
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Note that none of poles coincides with the pole of f at x = 1.8.
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The exact value for the integral: -1.2527629684953678e+00
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The approximation obtained from the first sequence of poles: -1.2527629716740507e+00
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with exact relative error: 2.5373378303887123e-09
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The approximation obtained from the second sequence of poles: -1.2527629716389368e+00
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with exact relative error: 2.5093086559468681e-09
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The relative distance between the two approximations: 2.8029174370724948e-11
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