Quadratic Convergence of the Tanh-sinh Quadrature Rule

In [5] and [2] the authors describe the remarkable effectiveness of the doubly exponential ‘tanh-sinh’ transformation for numerical integration —even for quite unruly integrands. Our intention in this note is to provide a theoretical underpinning when the integrand is analytic for the observed superlinear convergence of the method. Our analysis rests on the corresponding but somewhat easier analysis by Haber [1] of the less numerically effective ‘tanh’ substitution. 1. Preliminaries The standard trapezoidal rule for numerical integration when the integrand is defined on the interval (−∞,∞) is: (1.1) ∫ ∞ −∞ f(t) dt ≈ h ∞ ∑ n=−∞ f(nh) On changing variables, we can approximate the definite integral via: (1.2) ∫ 1 −1 f(t) dt = ∫ ∞ −∞ f(ψ(x))ψ′(x) dx ≈ h N ∑ n=−N ψ′(nh)f(ψ(nh)). Here, ψ is any absolutely continuous monotonic increasing function mapping (−∞,∞) onto (−1, 1), and without loss of generality, we assume the integrand is defined over (−1, 1). The tanh-sinh rule use the doubly-exponential transformation (1.3) ψ(x) = tanh (π 2 sinh(x) ) with ψ′(x) = π cosh(x) 2 cosh ( π 2 sinh(x) ) . Correspondingly, ψ(x) := tanh(x) gives rise to the scheme analyzed by Haber in [1]. Received by the editor January 24, 2006. 2000 Mathematics Subject Classification. Primary 65D30; Secondary 42B35, 65Y20.