alpert_rule

alpert_rule, a MATLAB code 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 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

matlab_rule, a MATLAB 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:

• a_log.m returns the value of A for a given log singular rule.
• a_power.m returns the value of A for a given power singular rule.
• a_regular.m returns the value of A for a given regular rule.
• integral_log.m returns the exact integral of the log singular test integrand.
• integral_power.m returns the exact integral of the power singular test integrand.
• integral_regular.m returns the exact integral of the regular test integrand.
• integrand_log.m evaluates the log singular test integrand.
• integrand_power.m evaluates the power singular test integrand.
• integrand_regular.m evaluates the regular test integrand.
• j_log.m returns the number of abscissas for a log singular Alpert rule.
• j_power.m returns the number of abscissas for a power singular Alpert rule.
• j_regular.m returns the number of abscissas for a regular Alpert rule.
• num_log.m returns the number of log singular Alpert rules.
• num_power.m returns the number of power singular Alpert rules.
• num_regular.m returns the number of regular Alpert rules.
• order_log.m returns the convergence rate of log singular Alpert rules.
• order_power.m returns the convergence rate of power singular Alpert rules.
• order_regular.m returns the convergence rate of regular Alpert rules.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• rule_log.m returns abscissas and weights for a log singular Alpert rule.
• rule_power.m returns abscissas and weights for a power singular Alpert rule.
• rule_regular.m returns abscissas and weights for a regular Alpert rule.