toms672


toms672, a FORTRAN90 code which computes a quadrature rule which is a maximally accurate extension of a given quadrature rule.

That is, we suppose we have a quadrature rule that uses N points, and that we wish to compute a new quadrature rule, which uses N+M points. The new rule is required to include the original N points of the old rule.

This kind of procedure is analogous to the process by which the nested Gauss-Patterson rules were developed. Thus, one use of the software is to try to develop nested families of rules for other weight functions, or starting from other initial rules.

In certain cases, this algorithm may be able to extend the given rule in a way which produces a rule with the maximum possible precision. (However, it is also possible that the extension cannot be made.)

By calling this algorithm repeatedly, it is possible to compute a family of nested quadrature rules of any order. Nested rules can be efficient when function evaluations are expensive, and a series of integral estimates must be made to estimate accuracy, or to detect convergence.

The text of many ACM TOMS algorithms is available online through ACM: http://calgo.acm.org/ or NETLIB: http://www.netlib.org/toms/index.html.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

toms672 is available in a FORTRAN90 version.

Related Data and Programs:

INT_EXACTNESS, a FORTRAN90 code which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

KRONROD, a FORTRAN90 code which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

PATTERSON_RULE, a FORTRAN90 code which returns a Gauss-Patterson quadrature rule of specified order.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRULE, a FORTRAN90 code which defines quadrature rules on a variety of intervals with different weight functions.

toms672_test

Reference:

  1. Gene Golub, Thomas Robertson,
    A generalized Bairstow Algorithm,
    Communications of the ACM,
    Volume 10, Number 6, June 1967, pages 371-373.
  2. Thomas Patterson,
    The Optimal Addition of Points to Quadrature Formulae,
    Mathematics of Computation,
    Volume 22, Number 104, October 1968, pages 847-856.
  3. Thomas Patterson,
    An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions,
    Transactions on Mathematical Software,
    Volume 15, Number 2, June 1989, pages 123-136.
  4. Thomas Patterson,
    Algorithm 672: EXTEND: generation of interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions,
    Transactions on Mathematical Software,
    Volume 15, Number 2, June 1989, pages 137-143.

Source Code:


Last revised on 14 March 2021.