An error functional expansion for $N$-dimensional quadrature with an integrand function singular at a point

Author:
J. N. Lyness

Journal:
Math. Comp. **30** (1976), 1-23

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1976-0408211-0

MathSciNet review:
0408211

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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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Keywords:
Multidimensional quadrature,
singularity,
error functional asymptotic expansion,
Romberg integration,
Euler-Maclaurin expansion

Article copyright:
© Copyright 1976
American Mathematical Society