Comparison of Birkhoff type quadrature formulae

Bojanov, Borislav; Nikolov, Geno

doi:10.1090/S0025-5718-1990-1010595-7

Comparison of Birkhoff type quadrature formulae
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by Borislav Bojanov and Geno Nikolov PDF

Math. Comp. 54 (1990), 627-648 Request permission

Abstract:

The classical approach to the theory of quadrature formulae is based on the concept of algebraic degree of precision (ADP). A quadrature formula ${Q_1}$ is considered to be "better" than ${Q_2}$ if ${\text {ADP}}({Q_1}) > {\text {ADP}}({Q_2})$. However, there are many quadratures that use the same number of evaluations of the integrand and have the same ADP. Then, how should one compare such formulae? We show in this paper that the error of the quadrature depends monotonically on the type of data used. Roughly speaking, the lower the order of the derivatives used, the smaller is the error. As a consequence of the main result we demonstrate the existence of Birkhoff quadrature formulae of double precision.

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