Numerical quadrature by the 𝜖-algorithm

Kahaner, David K.

doi:10.1090/S0025-5718-1972-0329210-X

Numerical quadrature by the $\varepsilon$-algorithm
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Math. Comp. 26 (1972), 689-693 Request permission

Abstract:

Modifications of Romberg’s integration method are applicable to functions with endpoint singularities. Several authors have reached different conclusions on the usefulness of these schemes due to the potential difficulties in calculating certain exponents in the asymptotic error expansion. In this paper, we consider a method based on the $\epsilon$-algorithm which does not require the user to supply these parameters. Tests show this luxury produces a method substantially better than the unmodified Romberg’s method, but not as good as the modified procedure which assumes all the exponents are known. It is possible to design a scheme which incorporates the best of both methods and allows the user to decide how much information he wishes to provide. The algorithm is stable numerically in quadrature applications and converges for functions with endpoint singularities of an algebraic or logarithmic nature.

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Numerical Quadrature by the $\epsilon$-Algorithm

Upon a Conjecture Concerning a Method for Solving Linear Equations, and Certain Other Matters