alpert_rule

alpert_rule, a C 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 computer code and data files described and made available on this web page are 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

line_fekete_rule, a C code 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 code 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 code 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 code 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 code 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.h, the include file.
• a_log.c, returns the value of A for an Alpert rule for log singular functions.
• a_power.c, returns A for an Alpert rule for power singular functions.
• a_regular.c, returns the value of A for an Alpert rule for regular functions.
• integral_log.c, evaluates the test integral with logarithmic singularity.
• integral_power.c, evaluates the test integral with power singularity.
• integral_regular.c, evaluates the regular test integral.
• integrand_log.c, evaluates the test integrand with logarithmic singularity.
• integrand_power.c, evaluates the test integrand with power singularity.
• integrand_regular.c, evaluates the regular test integrand.
• j_log.c, returns the value of J for an Alpert rule for log singular functions.
• j_power.c, returns J for an Alpert rule for power singular functions.
• j_regular.c, returns the value of J for an Alpert rule for regular functions.
• num_log.c, returns the number of Alpert rules for log singular functions.
• num_power.c, returns the number of Alpert rules for power singular functions.
• num_regular.c, returns the number of Alpert rules for regular functions.
• order_log.c, returns the order of an Alpert rule for log singular functions.
• order_power.c, returns the order of an Alpert rule for power singular functions.
• order_regular.c, returns the order of an Alpert rule for regular functions.
• r8vec_copy.c, copies an R8VEC.
• r8vec_dot_product.c, computes the dot product of two R8VECs.
• r8vec_linspace_new.c, creates a vector of linearly spaced values.
• r8vec_sum.c, sums an R8VEC.
• r8vec_uniform_01_new.c, returns a unit pseudorandom R8VEC.
• rule_log.c, returns an Alpert rule for log singular functions.
• rule_power.c, returns an Alpert rule for power singular functions.
• rule_regular.c, returns an Alpert rule for regular functions.
• timestamp.c, prints the current YMDHMS date as a time stamp.