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

Abstract

Let If be the integral of f ( x → ) f(\vec x) over an N-dimensional hypercube and Q ( m ) f {Q^{(m)}}f be the approximation to If obtained by subdividing the hypercube into m N {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 {Q^{({m_i})}}f , it is necessary to know the form of the error functional Q ( m ) f − I f {Q^{(m)}}f - If as an expansion in m. When f ( x → ) 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 ( x → ) f(\vec x) has the form \[ f ( x → ) = r α φ ( θ → ) h ( r ) g ( x → ) , α > − N , f(\vec x) = {r^\alpha }\varphi (\vec \theta )h(r)g(\vec x),\quad \alpha > - N, \] the only singularity being at the origin, a vertex of the unit hypercube of integration. Here ( r , θ → ) (r,\vec \theta ) represents the hyperspherical coordinates of ( x → ) (\vec x) . It is shown that for this integrand the error function expansion includes only terms A α + N + t / m α + N + t , B t / m t , C α + N + t ln ⁡ m / m α + N + t , t = 1 , 2 , … {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 ( x → ) 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 ( x → ) = ln ⁡ r f ( x → ) F(\vec x) = \ln rf(\vec x) is also derived.