An efficient composite formula for multidimensional quadrature

ABSTRACT

A (2s+1)-point, second-degree quadrature formula for integration over an s-dimensional hyper-rectangle is presented. All but one of the points lie on the surface with weights of opposite sign attached to points on opposite faces. When a large volume is subdivided into congruent rectangular subdivisions, only one point is required in each interior subdivision to achieve second-degree accuracy.

REFERENCES

• 1: A. Ralston, A Family of Quadrature Formulas Which Achieve High Accuracy in Composite Rules, Journal of the ACM (JACM), v.6 n.3, p.384-394, July 1959 [doi>10.1145/320986.320993]
• 2: VON MISES, R. Numerische berechnung mehrdimensionaler integrale. ZAMM 34 (1954), 201-210.
• 3: STROUD, A. H. Quadrature methods for functions of more than one independent variable. Ann. N. Y. Acad. Sci. 86 (1960), 776-791.
• 4: R. Don Freeman, Jr., Algorithm 32: MULTINT, Communications of the ACM, v.4 n.2, p.106, Feb. 1961 [doi>10.1145/366105.366181]
• 5: THACHER, H. C., JR. Optimum quadrature formulas in s dimensions. MTAC 11 (1957), 189-194.

INDEX TERMS

Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Design, Performance, Theory