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27 lines
1.2 KiB
27 lines
1.2 KiB
Integrands that are more expensive to evaluate:
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We consider the integral int_{-1}^{1}f(x)dx, for a vector-valued
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function of the form:
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f(x)=(A-x*I)^{-1}B,
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where
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A is a random antisymmetric matrix of size 2Nx2N,
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I is the 2Nx2N identity matrix,
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B is a random column vector of length 2N, and
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N=25*2^{PARAM},
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which has complex conjugate singularities on the imaginary axis.
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Figure 1 shows the 'exact' maximal relative error against
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execution time of MATLAB's built-in automatic integrator integral.m
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(respectively quadv.m) compared with respectively
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1. the (2n+1)-point rational Fejer without error estimates,
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2. the (2n+1)-point rational Fejer with error estimates,
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3. the semi-automatic ( (3:2:2n+1)-points ) rational Fejer,
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4. the (n+1)-point rational Gauss-Legendre, and
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5. the (2:1:n+1)-points rational Gauss-Legendre,
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for n=1,...,18.
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Figure 2 shows the exact maximal relative errors against n together
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with the estimated maximal relative error for the n-point
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rational Fejer.
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Figure 3 shows the imaginary part of the singularities of f(x) in
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the upper half-plane.
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The exact value of the integral is obtained from the MATLAB's
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built-in automatic integrators themselves with highest possible
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precision.
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