Interpolatory integration formulas for optimal composition

ABSTRACT

A set of symmetric, closed, interpolatory integration formulas on the interval [-1, 1] with positive weights and increasing degree of precision is introduced. These formulas, called recursive monotone stable (RMS) formulas, allow applying higher order or compound rules without wasting previously computed functional values. An exhaustive search shows the existence of 27 families of RMS formulas, stemming from the simple trapezoidal rule.

REFERENCES

• 1: DAVIS, P. J., AND RABINOWITZ, P. Methods of Numerical Integratwn. Academic Press, New York, 1984.
• 2: DEBooR, C. CADRE: An algorithm for numerical quadrature. In Mathematical Software, J. R. Rice, Ed., Academic Press, New York, 1971, pp. 417-449.
• 3: Paola Favati , Grazia Lotti , Francesco Romani, Algorithm 691; Improving QUADPACK automatic integration routines, ACM Transactions on Mathematical Software (TOMS), v.17 n.2, p.218-232, June 1991 [doi>10.1145/108556.108580]
• 4: PmSSENS, R. An algorithm for automatic integration. Angewandte Informatik 9 (1973), 399-401.
• 5: PmSSENS, R., ET AL. QUADPACK: A Subroutine Package for Automatic Integration. Springer-Verlag, Berlin, 1983.

INDEX TERMS

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

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Interpolation formulas

General Terms:
Algorithms, Theory

Keywords:
integration, interpolatory quadrature, program testing