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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An error functional expansion for $N$-dimensional quadrature with an integrand function singular at a point
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by J. N. Lyness PDF
Math. Comp. 30 (1976), 1-23 Request permission

Abstract:

Let If be the integral of $f(\vec x)$ over an N-dimensional hypercube and ${Q^{(m)}}f$ be the approximation to If obtained by subdividing the hypercube into ${m^N}$ equal subhypercubes and applying the same quadrature rule Q to each. In order to extrapolate efficiently for If on the basis of several different approximations ${Q^{({m_i})}}f$, it is necessary to know the form of the error functional ${Q^{(m)}}f - If$ as an expansion in m. When $f(\vec x)$ has a singularity, the conventional form (with inverse even powers of m) is not usually valid. In this paper we derive the expansion in the case in which $f(\vec x)$ has the form \[ f(\vec x) = {r^\alpha }\varphi (\vec \theta )h(r)g(\vec x),\quad \alpha > - N,\] the only singularity being at the origin, a vertex of the unit hypercube of integration. Here $(r,\vec \theta )$ represents the hyperspherical coordinates of $(\vec x)$. It is shown that for this integrand the error function expansion includes only terms ${A_{\alpha + N + t}}/{m^{\alpha + N + t}},{B_t}/{m^t},{C_{\alpha + N + t}}\ln m/{m^{\alpha + N + t}},t = 1,2, \ldots$ . The coefficients depend only on the integrand function $f(\vec x)$ and the quadrature rule Q. For several easily recognizable classes of integrand function and for most familiar quadrature rules some of these coefficients are zero. An analogous expansion for the error functional with integrand function $F(\vec x) = \ln rf(\vec x)$ is also derived.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Math. Comp. 30 (1976), 1-23
  • MSC: Primary 65D30
  • DOI: https://doi.org/10.1090/S0025-5718-1976-0408211-0
  • MathSciNet review: 0408211