Journal of Computational and Applied Mathematics 112 (1999) 121–145 www.elsevier.nl/locate/cam d2lri: A nonadaptive algorithm for two-dimensional cubature Michael Hill a , Ian Robinson a ; * a Department of Computer Science and Computer Engineering, La Trobe University, Bundoora, Vic 3083, Australia Received 1 October 1997; received in revised form 30 March 1999 Abstract We describe a nonadaptive automatic cubature routine for integration over a wide variety of two-dimensional domains, including innite regions. The underlying algorithm rst maps the region onto the unit square, applies a periodizing sixth-order Sidi transformation and then generates a sequence of approximations based on embedded lattice rules. Numerical experiments suggest that the routine is reliable and ecient for a wide range of integrand types and that it is often more eective than other published routines for integrands with a singularity along the boundary and for integration over innite domains. c  1999 Elsevier Science B.V. All rights reserved. Keywords: Automatic integration; Two-dimensional cubature; Lattice rules; Sidi transformation; Boundary singularities; Innite domain 1. Introduction In their book “Lattice Methods for Multiple Integration” [15, p. 141], Sloan and Joe observe in relation to lattice rules: “Although the two-dimensional case may not be of great practical interest, it does have a special charm”. Indeed, it has been a commonly held view amongst researchers interested in the development of general-purpose software for numerical cubature that in low dimensions, techniques based on the use of lattice rules are unlikely to be competitive with sophisticated adaptive algorithms such as Berntsen, Espelid and Genz’s DCUHRE [3], or a library of routines like Cools, Laurie and Pluym’s CUBPACK++ [5]. In this paper, we demonstrate that the charm of two-dimensional lattices rules can, in fact, be exploited to produce a reliable and ecient general purpose integrator. * Corresponding author. E-mail addresses: hillmj@latcs1.cs.latrobe.edu.au (M. Hill), i.robinson@latrobe.edu.au (I. Robinson) 0377-0427/99/$ - see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0377-0427(99)00217-4