(Student Research Project) From a practical point of view, investigate some new algorithms for numerical integration that are associated with the names Clenshaw and Curtis [1960], Kronrod [1964], and Patterson [1968]. The later two are adaptive Gaussian quadrature methods that provide error estimates based on the evaluation and reuse of the results at Kronrod points. See QUADPACK byPessens, de Doncker, Uberhuber, and Kahaner [1983] and also Laurie [1997], Ammar, Calvetti, and Reichel [1999], and Calvetti, Golub, Gragg, and Reichel [2000] for examples.
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