Asymptotic bit cost of quadrature formulas obtained by variable transformation

In this paper, the asymptotic bit operation cost of a family of quadrature formulas, especially suitable for evaluation of improper integrals, is studied. More precisely, we consider the family of quadrature formulas obtained by applying k times the variable transformation x = sinh(y) and then the trapezoidal rule to the transformed integral. We prove that, if the integrand function is analytic in the interior part of the integration interval and approaches zero at a rate which is at least the reciprocal of a polynomial, then the computational bit cost is bounded above by a polynomial function of the number of exact digits in the result. Moreover, disregarding logarithmic terms, the double exponential transformation (k = 2) leads to the optimal cost among the methods of this family.