# 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);
In case one, the regular Alpert rule is used in both end intervals. In case two, the power singular Alpert rule is used in the leftmost interval. In case three, the log singular Alpert rule is used in the leftmost interval.

### 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:

alpert_rule_test, a C program which tests alpert_rule.

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:

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.

Last revised on 23 October 2018.