ALPERT_RULE
Alpert Quadrature for regular, log singular, and power singular functions

ALPERT_RULE is a Python library which has tabulated values that define Alpert quadrature rules of a number of orders of accuracy for functions that are regular, log singular, or power singular.

The rules defined here assume that the integral is to be taken over the interval [0,1]. The interval is divided into N+1 intervals. The leftmost and rightmost intervals are handled in a special way, depending on whether a particular kind of singularity is expected.

A singularity may exist at the left endpoint, x = 0. The cases are:

• regular, no singularity;
• power, the integrand has the form g(x)=x^(-1/2)*phi(x)+psi(x);
• log, the integrand has the form g(x)=phi(x)*log(x)+psi(x);

Licensing:

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

Languages:

ALPERT_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

DISK_RULE, a Python library which computes quadrature rules over the interior of the general disk in 2D, with radius RC and center (XC,YC).

DISK01_RULE, a Python library which computes quadrature rules over the interior of the unit disk in 2D, with center (0,0) and radius 1.

QUADRULE, a Python library which contains 1-dimensional quadrature rules.

Reference:

1. Bradley Alpert,
   Hybrid Gauss-Trapezoidal Quadrature Rules,
   SIAM Journal on Scientific Computing,
   Volume 20, Number 5, pages 1551-1584, 1999.

Source Code:

• alpert_log.py defines an Alpert rule for a log singular integrand.
• alpert_power.py defines an Alpert rule for a log singular integrand.
• alpert_regular.py defines an Alpert rule for a regular integrand.
• monte_carlo.py applies the Monte Carlo method to the various test integrands.
• r8vec_linspace2.py, returns an R8VEC of values evenly spaced between given limits, omitting the endpoints.
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_01.py, returns a unit pseudorandom R8VEC.
• test_functions.py defines the log-singular, power-singular, and regular test integrands and their integrals from 0 to 1.
• timestamp.py prints the YMDHMS date as a timestamp.
• trapezoid.py applies the trapezoid rule to the various integrands.

Examples and Tests:

• alpert_rule_test.py calls all the tests.
• alpert_rule_test.sh runs all the tests.
• alpert_rule_test.txt the output file.

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