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

ALPERT_RULE is a C++ 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:

LINE_FEKETE_RULE, a C++ library which returns the points and weights of a Fekete quadrature rule over the interior of a line segment in 1D.

LINE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCC_RULE, a C++ library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

LINE_NCO_RULE, a C++ library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

QUADRATURE_WEIGHTS_VANDERMONDE, a C++ library which computes the weights of a quadrature rule using the Vandermonde matrix, assuming that the points have been specified.

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.cpp, the source code.
• alpert_rule.hpp, the include file.

Examples and Tests:

• alpert_rule_test.cpp, a sample calling program.
• alpert_rule_test_output.txt, the output file.

List of Routines:

• A_LOG returns the value of A for an Alpert rule for log singular functions.
• A_POWER returns A for an Alpert rule for power singular functions.
• A_REGULAR returns the value of A for an Alpert rule for regular functions.
• INTEGRAL_LOG evaluates the test integral with logarithmic singularity.
• INTEGRAL_POWER evaluates the test integral with power singularity.
• INTEGRAL_REGULAR evaluates the regular test integral.
• INTEGRAND_LOG evaluates the test integrand with logarithmic singularity.
• INTEGRAND_POWER evaluates the test integrand with power singularity.
• INTEGRAND_REGULAR evaluates the regular test integrand.
• J_LOG returns the value of J for an Alpert rule for log singular functions.
• J_POWER returns J for an Alpert rule for power singular functions.
• J_REGULAR returns the value of J for an Alpert rule for regular functions.
• NUM_LOG returns the number of Alpert rules for log singular functions.
• NUM_POWER returns the number of Alpert rules for power singular functions.
• NUM_REGULAR returns the number of Alpert rules for regular functions.
• ORDER_LOG returns the order of an Alpert rule for log singular functions.
• ORDER_POWER returns the order of an Alpert rule for power singular functions.
• ORDER_REGULAR returns the order of an Alpert rule for regular functions.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• RULE_LOG returns an Alpert rule for log singular functions.
• RULE_POWER returns an Alpert rule for power singular functions.
• RULE_REGULAR returns an Alpert rule for regular functions.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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