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On numerical improvement of Gauss-Lobatto quadrature rules. (English) Zbl 1070.65019
Summary: It is well known that the Gauss-Lobatto quadrature rule $\int^1_{-1} f(x)\,dx\simeq \sum^n_{i=1} w_if(x_i)+ pf(-1)+ qf(1),$ is exact for polynomials of degree at most $$2n+ 1$$. We are going to find a formula which is approximately exact for monomials $$x^j$$, $$j= 0,1,\dots, 2n+ 3$$ instead of being analytically exact for monomials $$x^j$$, $$j= 0,1,\dots, 2n+ 1$$. We also consider a class of functions for which the new formula produces better results.