alpert_rule

alpert_rule, a Fortran90 code which defines 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; the regular Alpert rule is used in both end intervals.
• power, the integrand has the form g(x)=x^(-1/2)*phi(x)+psi(x); the power singular Alpert rule is used in the leftmost interval.
• log, the integrand has the form g(x)=phi(x)*log(x)+psi(x); the log singular Alpert rule is used in the leftmost interval.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

alpert_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

alpert_rule_test

f90_rule, a Fortran90 code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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_rule.f90, the source code.
• alpert_rule.sh, compiles the source code.