# HERMITE_TEST_INT Quadrature Tests for Infinite Intervals

HERMITE_TEST_INT is a FORTRAN90 library which defines integration problems over infinite intervals of the form (-oo,+oo).

The test integrands would normally be used to testing one dimensional quadrature software. It is possible to invoke a particular function by index, or to try out all available functions, as demonstrated in the sample calling program.

For a given integrand function f(x), the problem is to estimate

```        I(f) = integral ( -oo < x < +oo ) w(x) * f(x) dx
```

We consider three variations of the problem, depending on the form of the weight factor w(x):

• option = 0, the unweighted integral:
```            Integral ( -oo < x < +oo ) f(x) dx
```
• option = 1, the physicist weighted integral:
```            Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx
```
• option = 2, the probabilist weighted integral:
```            Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
```

For option 0, the test integrands have the form:

1. f1(x) = exp(-x*x) * cos(2*omega*x);
2. f2(x) = exp(-x*x);
3. f3(x) = exp(-px)/(1+exp(-qx));
4. f4(x) = sin ( x^2 );
5. f5(x) = 1 / (1+x^2) sqrt (4+3x^2) );
6. f6(x) = exp(-x*x) * x^m;
7. f7(x) = x^2 cos(x) exp(-x*x);
8. f8(x) = sqrt ( 1 + x * x / 2 ) * exp(-x*x/2);

For option 1, the test integrands have the form:

1. f1(x) = cos(2*omega*x);
2. f2(x) = 1
3. f3(x) = exp(x*x) * exp(-px)/(1+exp(-qx));
4. f4(x) = exp(x*x) * sin ( x^2 );
5. f5(x) = exp(x*x) * 1 / (1+x^2) sqrt (4+3x^2) );
6. f6(x) = x^m;
7. f7(x) = x^2 cos(x);
8. f8(x) = sqrt ( 1 + x * x / 2 ) * exp(+x*x/2);

For option 2, the test integrands have the form:

1. f1(x) = exp(-x*x/2) * cos(2*omega*x);
2. f2(x) = exp(-x*x/2);
3. f3(x) = exp(+x*x/2) * exp(-px)/(1+exp(-qx));
4. f4(x) = exp(+x*x/2) * sin ( x^2 );
5. f5(x) = exp(+x*x/2) * 1 / (1+x^2) sqrt (4+3x^2) );
6. f6(x) = exp(-x*x/2) * x^m;
7. f7(x) = x^2 cos(x) exp(-x*x/2);
8. f8(x) = sqrt ( 1 + x * x / 2 );

The library includes not just the integrand, but also the exact value of the integral (or, typically, an estimate of this value), and a title for the problem. Thus, for each integrand function, several routines are supplied. For instance, for function #1, we have the routines:

• P01_FUN evaluates the integrand for problem 1.
• P01_EXACT returns the estimated integral for problem 1.
• P01_TITLE returns a title for problem 1.
So once you have the calling sequences for these routines, you can easily evaluate the function, or integrate it on the appropriate interval, or compare your estimate of the integral to the exact value.

Moreover, since the same interface is used for each function, if you wish to work with problem 5 instead, you simply change the "01" to "05" in your routine calls.

If you wish to call all of the functions, then you simply use the generic interface, which requires you to specify the problem number as an extra input argument:

• P00_FUN evaluates the integrand for any problem.
• P00_EXACT returns the exact integral for any problem.
• P00_TITLE returns a title for any problem.

### Languages:

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

### Related Data and Programs:

HERMITE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

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

LAGUERRE_TEST_INT, a FORTRAN90 library which defines test integrands for quadrature rules for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

QUADRATURE_RULES_HERMITE_PHYSICIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).

QUADRATURE_RULES_HERMITE_PROBABILIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).

QUADRATURE_RULES_HERMITE_UNWEIGHTED, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function 1.

TEST_INT, a FORTRAN90 library which defines some test integration problems over finite intervals.

### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
2. Prem Kythe, Michael Schaeferkotter,
Handbook of Computational Methods for Integration,
Chapman and Hall, 2004,
ISBN: 1-58488-428-2,
LC: QA299.3.K98.
3. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
QUADPACK: A Subroutine Package for Automatic Integration,
Springer, 1983,
ISBN: 3540125531,
LC: QA299.3.Q36.
4. William Squire,
Comparison of Gauss-Hermite and Midpoint Quadrature with Application to the Voigt Function,
in Numerical Integration: Recent Developments, Software and Applications,
edited by Patrick Keast, Graeme Fairweather,
Reidel, 1987, pages 337-340,
ISBN: 9027725144,
LC: QA299.3.N38.
5. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.
6. Alan Turing,
A Method for the Calculation of the Zeta Function,
Proceedings of the London Mathematical Society,
Volume 48, 1943, pages 180-197.

### List of Routines:

• HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_INTEGRAL returns the value of a Hermite polynomial integral.
• HERMITE_RECUR finds the value and derivative of a Hermite polynomial.
• HERMITE_ROOT improves an approximate root of a Hermite polynomial.
• I4_FACTORIAL2 computes the double factorial function.
• P00_EXACT returns the exact integral for any problem.
• P00_FUN evaluates the integrand for any problem.
• P00_GAUSS_HERMITE applies a Gauss-Hermite quadrature rule.
• P00_MONTE_CARLO applies a Monte Carlo procedure to Hermite integrals.
• P00_PROBLEM_NUM returns the number of test integration problems.
• P00_TITLE returns the title for any problem.
• P00_TURING applies the Turing quadrature rule.
• P01_EXACT returns the exact integral for problem 1.
• P01_FUN evaluates the integrand for problem 1.
• P01_TITLE returns the title for problem 1.
• P02_EXACT returns the exact integral for problem 2.
• P02_FUN evaluates the integrand for problem 2.
• P02_TITLE returns the title for problem 2.
• P03_EXACT returns the exact integral for problem 3.
• P03_FUN evaluates the integrand for problem 3.
• P03_TITLE returns the title for problem 3.
• P04_EXACT returns the exact integral for problem 4.
• P04_FUN evaluates the integrand for problem 4.
• P04_TITLE returns the title for problem 4.
• P05_EXACT returns the exact integral for problem 5.
• P05_FUN evaluates the integrand for problem 5.
• P05_TITLE returns the title for problem 5.
• P06_EXACT returns the exact integral for problem 6.
• P06_FUN evaluates the integrand for problem 6.
• P06_PARAM gets or sets parameters for problem 6.
• P06_TITLE returns the title for problem 6.
• P07_EXACT returns the exact integral for problem 7.
• P07_FUN evaluates the integrand for problem 7.
• P07_TITLE returns the title for problem 7.
• P08_EXACT returns the exact integral for problem 8.
• P08_FUN evaluates the integrand for problem 8.
• P08_TITLE returns the title for problem 8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 30 July 2010.