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