Towards a General Error Theory of the Trapezoidal Rule

The trapezoidal rule is the method of choice for numerical quadrature of analytic functions over the real line ℝ. Other intervals and slowly decaying integrands may elegantly be handled by means of simple analytic transformations of the integration variable. In the case of an integrand analytic in an open strip containing ℝ the discretization error is exponentially small in the reciprocal step size. If the integrand has singularities arbitrarily close to ℝ, the discretization error is larger and its theory is more complicated. We present examples illustrating possible error laws of the trapezoidal rule.

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