KRONROD
Gauss-Kronrod Quadrature Rules

KRONROD is a MATLAB 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.

Related Data and Programs:

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

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

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

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

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

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:

• abwe1.m, calculates a Kronrod abscissa and weight.
• abwe2.m, calculates a Gaussian abscissa and two weights.
• kronrod.m, adds N+1 points to an N-point Gaussian rule.
• timestamp.m, prints out the current YMDHMS date as a timestamp.

Examples and Tests:

• kronrod_test.m, a sample calling program.
• kronrod_test01.m, calls kronrod for a Gauss rule of order 3;
• kronrod_test02.m, calls kronrod for a Gauss rule of order 4;
• kronrod_test03.m, shows how to use kronrod to estimate an integral, and to increase the order of the approximation until an error tolerance is satisfied.
• kronrod_prb_output.txt, the output file.

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