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