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.
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:
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:

Bradley Alpert,
Hybrid GaussTrapezoidal Quadrature Rules,
SIAM Journal on Scientific Computing,
Volume 20, Number 5, pages 15511584, 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.