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Some extrapolation processes for integrands with endpoint singularities. (English) Zbl 0597.65011
Summary: A study of some numerical quadratures for endpoint singularity integrands, linear and nonlinear extrapolation processes which include the Romberg method, Aitken’s \(\Delta^ 2\) process and the \(\epsilon\)- algorithm is reported. These acceleration processes can be shown to belong to the Neville type extrapolation. The \(\epsilon\)-algorithm is not as good as the modified Romberg method but better than the unmodified Romberg and Aitken’s \(\Delta^ 2\) process. This report may help the users to select the appropriate method for evaluating the integrand with singularity efficiently.
65D32 Numerical quadrature and cubature formulas
65B05 Extrapolation to the limit, deferred corrections
41A55 Approximate quadratures
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane