TANH_SINH_RULE
Tanh-Sinh Quadrature Rules

TANH_SINH_RULE is a FORTRAN90 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

• order is 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 is 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 FORTRAN90 program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.

INTLIB, a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a FORTRAN90 program which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LOGNORMAL_RULE, a FORTRAN90 program which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.

PATTERSON_RULE, a FORTRAN90 program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 program which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

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

QUADRULE, a FORTRAN90 library which defines 1-dimensional quadrature rules.

TANH_QUAD, a FORTRAN90 library which sets up the tanh quadrature rule;

TRUNCATED_NORMAL_RULE, a FORTRAN90 program which computes a quadrature rule for a normal 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.f90, 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_SINH_RULE.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_WRITE0 writes a DTABLE file with no headers.
• GET_UNIT returns a free FORTRAN unit number.
• S_TO_I4 reads an I4 from a string.
• TANH_SINH_HANDLE computes the requested tanh-sinh rule and outputs it.
• TANH_SINH_COMPUTE computes a tanh-sinh quadrature rule.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.