Pii: S0893-9659(97)00024-4

-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. Key'words---Improper integral, Double exponential method, Trapezoidal rule. 1. I N T R O D U C T I O N A N D P R E L I M I N A R Y The asymptotic bit operation cost of a family of quadrature formulas, very suitable for evaluation of improper integl"als, is studied. Without loss of generality, the problem taken into consideration is tha t of approximating // I = f ( x ) dx, (1) where f is an even analytic function on ( co , +co); all the other cases of improper integrals with integrand function analytic in the interior part of the integration interval can be reduced to the previous one. More precisely, the family of quadrature methods under investigation (see [1-3]) is obtained by applying k > 0 times the variable transformation x = sinh(y) and then the trapezoidal rule to the integral I = fk(Y) dy, where f i (Y) = f i l ( s inh(y))cosh(y) and fo(Y) = f (Y) . The computational cost measure used in this analysis is the number of bit operations as a function of the number of exact decimal digits in the result. As should be expected, the cost of these quadrature methods is characterized by (i) the order of decay of the integrand function f , when its argument goes to infinity, (ii) the number k of variable transformations applied, and