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
```
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.

### 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.

### 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,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### 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:

### 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.

Last revised on 28 June 2009.