A Family of Quadrature Formulas Which Achieve High Accuracy in Composite Rules

ABSTRACT

A class of quadrature formulas is derived which achieve higher accuracy in composite rules (i.e., where the interval of integration is broken up into a number of subintervals) than analogous Newton-Cotes or Gaussian formulas. The cost of this higher accuracy is the computation of one or two more ordinates over the whole interval of integration. The high accuracy is obtained by using Gaussian techniques in the interior of each subinterval and by using the endpoints of each subinterval as abscissas with weights of equal magnitude and opposite sign. In this way when the subintervals are put together only the endpoints of the whole interval of integration remain. It is proved that the abscissas are all real and interior to the subinterval and that the weights corresponding to the interior abscissas are positive. Since the abscissas are not equally spaced, the method is not suited to tabular functions but rather to analytically given functions. The roundoff properties of the formulas are discussed and are shown to be quite good.

REFERENCES

• 1: Begnaud Francis Hildebrand, Introduction to numerical analysis: 2nd edition, Dover Publications, Inc., New York, NY, 1987
• 2: KoPaL, Z., Numerical Analysi% John Wiley (1955).

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Gaussian quadrature

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Error analysis

General Terms:
Measurement, Performance, Reliability, Theory