# HERMITE_INTEGRANDS Quadrature Tests for Infinite Intervals

HERMITE_INTEGRANDS, a MATLAB 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.

Some demonstration routines are built in for simple quadrature methods:

• P00_GAUSS_HERMITE uses a Gauss-Hermite quadrature formula;
• P00_MONTE_CARLO uses a Monte Carlo scheme, with sample points selected according to the standard normal probability distribution;
• P00_TURING applies a simple equally spaced method of Turing.

### Languages:

HERMITE_INTEGRANDS 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 MATLAB program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

hermite_integrands, a MATLAB library which defines test integrands for Hermite integrals with interval (-oo,+oo) and density exp(-x^2).

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

LAGUERRE_TEST_INT, a C 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 MATLAB 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.

### Examples and Tests:

Last revised on 29 January 2019.