Algorithm 655: IQPACK: FORTRAN subroutines for the weights of interpolatory quadratures

ABSTRACT

We present FORTRAN subroutines that implement the method described in [3] for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots. Given a set of knots and their multiplicities, the package generates the weights by using the zeroth moment &mgr;0 of w, the weight function in the integrand, and the (symmetric tridiagonal) Jacobi matrix J associated with the polynomials orthogonal on (a, b) with respect to w. There are utility routines that generate &mgr;0 and J for classical weight functions, but quadratures can be generated for any &mgr;0 and J supplied by the user. Utility routines are also provided that (1) evaluate a computed quadrature, applied to a user-supplied integrand, (2) check the polynomial order of precision of a quadrature formula, and (3) compute the knots and weights of simple Gaussian quadrature formula.

REFERENCES

• 1: GOLUB, G. H., AND KAUTSKY, J. Calculation of Gauss quadratures with multiple free and fixed knots. Numer. Math. 41 (1983), 147-163.
• 2: GOLUB, G. H., AND WELSCH, Z.H. Calculation of Gauss quadrature rules. Math. Comput. 23 (1969), 221-230.
• 3: KAUTSKY, J., AND ELHAY, S. Calculation of the weights of interpolatory quadratures. Numer. Math. 40 (1982), 407-422.
• 4: SMITH, B. T., BOYLE, J., GARBOW, B., IKEBE, Y., KLEMA, V., AND MOLER, C. Matrix Eigensystem Routines--EISPACK Guide, 2nd ed., Lecture Notes in Computer Science, vol. 6. Springer Verlag, New York, 1976.

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Gaussian quadrature

General Terms:
Algorithms

Keywords:
approximate integration, interpolatory quadrature, multiple knot quadratures, quadrature coefficients, simple Gaussian quadrature