QUADRATURE_RULES_TANH_SINH
Quadrature Rules of Tanh-Sinh Type

QUADRATURE_RULES_TANH_SINH is a dataset directory which contains examples of quadrature rules of tanh-sinh type.

Tanh-sinh quadrature rules are designed to approximate integrals on the interval [-1,1].

Standard tanh-sinh quadrature assumes that the integrand we are considering has a form like:

        Integral ( -1 <= x <= +1 ) f(x) dx
      

A tanh-sinh quadrature rule is a set of n positive weights w and abscissas x so that

        Integral ( -1 <= x <= +1 ) f(x) dx
      

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region.

A tanh-sinh quadrature rule has two parameters, the order N and the stepsize H. Various choices are available to relate these quantities when making a family of rules. One choice, which results in a nested family, is to take the K-th rule to have order N = (2^K)-1 and parameter H = 4.0/2^K.

Another issue with tanh-sinh quadrature is that the weights don't add up to 2. Particularly for low order rules, the discrepancy is large. Since these rules are used as families, and we're looking for asymptotic accuracy, the errors in the early rules might not matter; in that case, there is a simple relationship between the weights used in successive elements of the family. We will take a different view here, and force the weights to add up to 2 by normalizing them.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which can read a set of files defining a Gauss-Legendre quadrature rule, and test it for exactness against monomial integrands.

LEGENDRE_RULE, a FORTRAN90 program which generates a Gauss-Legendre rule of the desired order.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE, a FORTRAN90 library which defines various quadrature rules.

TEST_INT, a FORTRAN90 library which defines several test integrands which may be integrated.

Sample Files:

Tanh-Sinh Rule, Order 1, H = 2:

• ts_o1_x.txt, the abscissas.
• ts_o1_w.txt, the weights.
• ts_o1_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 3, H = 1:

• ts_o3_x.txt, the abscissas.
• ts_o3_w.txt, the weights.
• ts_o3_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 7, H = 1/2:

• ts_o7_x.txt, the abscissas.
• ts_o7_w.txt, the weights.
• ts_o7_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 15, H = 1/4:

• ts_o15_x.txt, the abscissas.
• ts_o15_w.txt, the weights.
• ts_o15_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 31, H = 1/8:

• ts_o31_x.txt, the abscissas.
• ts_o31_w.txt, the weights.
• ts_o31_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 63, H = 1/16:

• ts_o63_x.txt, the abscissas.
• ts_o63_w.txt, the weights.
• ts_o63_r.txt, the range of the integration region.

Tanh-Sinh Rule, Order 127, H = 1/32:

• ts_o127_x.txt, the abscissas.
• ts_o127_w.txt, the weights.
• ts_o127_r.txt, the range of the integration region.

You can go up one level to the DATASETS page.