KRONROD
Gauss-Kronrod Quadrature Rules

KRONROD is a FORTRAN77 library which computes both a Gauss quadrature rule of order N, and the Gauss-Kronrod rule of order 2*N+1.

A pair of Gauss and Gauss-Kronrod quadrature rules are typically used to provide an error estimate for an integral. The integral is estimated using the Gauss rule, and then the Gauss-Kronrod rule provides a higher precision estimate. The difference between the two estimates is taken as an approximation to the level of error.

The advantage of using a Gauss and Gauss-Kronrod pair is that the second rule, which uses 2*N+1 points, actually includes the N points in the previous Gauss rule. This means that the function values from that computation can be reused. This efficiency comes at the cost of a mild reduction in the degree of polynomial precision of the Gauss-Kronrod rule.

Licensing:

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

Languages:

KRONROD is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a PYTHON version.

Related Data and Programs:

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

PATTERSON_RULE, a FORTRAN90 program which computes a Gauss-Patterson quadrature rule.

QUADPACK, a FORTRAN77 library which contains a variety of routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES_PATTERSON, a dataset directory which contains Gauss-Patterson quadrature rules for the interval [-1,+1].

QUADRULE, a FORTRAN77 library which defines quadrature rules for 1D domains.

TEST_INT, a FORTRAN90 library which contains a number of functions that may be used as test integrands for quadrature rules in 1D.

TOMS672, a FORTRAN77 library which generates an interpolatory quadrature rule of highest possible order, given a set of preassigned abscissas;
this library is commonly called EXTEND;
this is ACM TOMS algorithm 672.

TOMS699, a FORTRAN77 library which implements a new representation of Patterson's quadrature formula;
this is ACM TOMS algorithm 699.

Reference:

1. Robert Piessens, Maria Branders,
   A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto,
   Mathematics of Computation,
   Volume 28, Number 125, January 1974, pages 135-139.

Source Code:

• kronrod.f, the source code.
• kronrod.sh, commands to compile the source code.

Examples and Tests:

• kronrod_prb.f, a sample calling program.
• kronrod_prb.sh, commands to compile and run the sample program.
• kronrod_prb_output.txt, the output file.

List of Routines:

• KRONROD adds N+1 points to an N-point Gaussian rule.
• ABWE1 calculates a Kronrod abscissa and weight.
• ABWE2 calculates a Gaussian abscissa and two weights.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.