TANH_SINH_RULE is a MATLAB program which generates a specific tanh-sinh quadrature rule, based on user input.
The rule is output as three files for easy use as input to other programs.
The tanh-sinh quadrature rule is designed for the interval [-1,+1].
The tanh-sinh quadrature assumes that the integrand has the form:
Integral ( -1 <= x <= +1 ) f(x) dx
The tanh-sinh quadrature rule is used as follows:
Integral ( -1 <= x <= +1 ) f(x) dx
is to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))
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.
tanh_sinh_rule ( order, 'prefix' )where
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
TANH_SINH_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
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QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_LEGENDRE, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.
QUADRULE, a MATLAB library which defines 1-dimensional quadrature rules.
TANH_QUAD, a MATLAB library which sets up the tanh quadrature rule;
The directory QUADRATURE_RULES_TANH_SINH contains a number of tanh-sinh quadrature rules created by this program. Here is a pointer to the rule of order 31:
You can go up one level to the MATLAB source codes.