A generalized discrepancy and quadrature error bound

1998/01/01 by Fred Hickernell, Fred J. Hickernell · 24 citations
Mathematics · Materials Science · #Mathematical Approximation and Integration #Mathematical functions and polynomials #Radiation Shielding Materials Analysis

paper · pdf · doi:10.1090/s0025-5718-98-00894-1

Abstract

An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -star discrepancy and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript alpha"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">P_α</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that arises in the study of lattice rules.

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