TANH_SINH_RULE
Tanh-Sinh Quadrature Rules

TANH_SINH_RULE is a C++ 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
      

        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.

Usage:

tanh_sinh_rule order prefix where

order

the number of points in the quadrature rule. A typical value might be 1, 3, 7, 15, 31, 63, 127 or (2^K)-1.

prefix

the common prefix for the three output files:

• prefix_x.txt, will contain the abscissas;
• prefix_w.txt, will contain the weights;
• prefix_r.txt, contains the region definition.

Licensing:

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

Languages:

TANH_SINH_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CCN_RULE, a C++ program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a C++ program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, a C++ program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a C++ program which computes a Gauss-Legendre quadrature rule.

PATTERSON_RULE, a C++ program which computes a Gauss-Patterson quadrature rule.

QUADRATURE_RULES_TANH_SINH, a dataset directory which contains triples of files defining tanh-sinh quadrature rules.

TRUNCATED_NORMAL_RULE, a C++ program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• tanh_sinh_rule.cpp, the source code.
• tanh_sinh_rule.sh, commands to compile the source code.

Examples and Tests:

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:

• ts_o31_x.txt, the abscissa file;
• ts_o31_w.txt, the weight file;
• ts_o31_r.txt, the region file;

List of Routines:

• MAIN is the main program for TANH_SIH_RULE.
• DTABLE_WRITE0 writes information to a DTABLE file.
• TANH_SINH_COMPUTE computes a tanh-sinh quadrature rule.
• TANH_SINH_HANDLE computes the requested tanh-sinh rule and outputs it.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.