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

Journal Article · Thu Jan 01 04:00:00 EST 1976 · Math. Comput.; (United States)

DOI:https://doi.org/10.1090/S0025-5718-1976-0408211-0· OSTI ID:7136626

Lyness, J N

Let If be the integral of f(x) over an N-dimensional hypercube and Q/sup (m)/f be the approximation to If obtained by subdividing the hypercube into m/sup 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/sup (m/sub i/)/f, it is necessary to know the form of the error functional Q/sup (m)/f-If as an expansion in m. When f(x) has a singularity, the conventional form (with inverse even powers of m) is not usually valid. The expansion in the case in which f(x) has the form f(x) = r/sup ..cap alpha../phi(theta)h(r)g(x), ..cap alpha..>-N, the only singularity being at the origin, a vertex of the unit hypercube of integration is derived. Here (r, theta) represents the hyperspherical coordinates of (x). It is shown that for this integrand the error function expansion includes only terms A/sub ..cap alpha..+N+t//m/sup ..cap alpha..+N+t/,B/sub r//m/sup t/, C/sub ..cap alpha..+N+t/ln m/m/sup ..cap alpha..+N+t/, t = 1,2,.... The coefficients depend only on the integrand function f(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 rf(x) is also derived.

• OSTI ID: 7136626
• Journal Information: Math. Comput.; (United States), Journal Name: Math. Comput.; (United States) Vol. 30:133; ISSN MCMPA
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COORDINATES
ERRORS
FUNCTIONALS
FUNCTIONS
INTEGRALS
MANY-DIMENSIONAL CALCULATIONS
SERIES EXPANSION
SINGULARITY