Hermite Polynomials

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#### Hermite Polynomials

##### Synopsis

    #include <boost/math/special_functions/hermite.hpp>

    namespace boost{ namespace math{

    template <class T>
    calculated-result-type hermite(unsigned n, T x);

    template <class T, class Policy>
    calculated-result-type hermite(unsigned n, T x, const Policy&);

    template <class T1, class T2, class T3>
    calculated-result-type hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);

    }} // namespaces

##### Description

The return type of these functions is computed using the result type calculation rules: note than when there is a single template argument the result is the same type as that argument or `double` if the template argument is an integer type.

    template <class T>
    calculated-result-type hermite(unsigned n, T x);

    template <class T, class Policy>
    calculated-result-type hermite(unsigned n, T x, const Policy&);

Returns the value of the Hermite Polynomial of order n at point x:

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

The following graph illustrates the behaviour of the first few Hermite Polynomials:

    template <class T1, class T2, class T3>
    calculated-result-type hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);

Implements the three term recurrence relation for the Hermite polynomials, this function can be used to create a sequence of values evaluated at the same x, and for rising n.

For example we could produce a vector of the first 10 polynomial values using:

    double x = 0.5;  // Abscissa value
    vector<double> v;
    v.push_back(hermite(0, x)).push_back(hermite(1, x));
    for(unsigned l = 1; l < 10; ++l)
       v.push_back(hermite_next(l, x, v[l], v[l-1]));

Formally the arguments are:

n

  The degree n of the last polynomial calculated.

x

  The abscissa value

Hn

  The value of the polynomial evaluated at degree n.

Hnm1

  The value of the polynomial evaluated at degree n-1.

##### Accuracy

The following table shows peak errors (in units of epsilon) for various domains of input arguments. Note that only results for the widest floating point type on the system are given as narrower types have effectively zero error.

Table 34. Peak Errors In the Hermite Polynomial

Significand Size | Platform and Compiler | Errors in range 0 < l < 20
--- | --- | ---
53 | Win32, Visual C++ 8 | Peak=4.5 Mean=1.5
64 | Red Hat Linux IA32, g++ 4.1 | Peak=6 Mean=2
64 | Red Hat Linux IA64, g++ 3.4.4 | Peak=6 Mean=2
113 | HPUX IA64, aCC A.06.06 | Peak=6 Mean=4

Note that the worst errors occur when the degree increases, values greater than ~120 are very unlikely to produce sensible results, especially in the associated polynomial case when the order is also large. Further the relative errors are likely to grow arbitrarily large when the function is very close to a root.

##### Testing

A mixture of spot tests of values calculated using functions.wolfram.com, and randomly generated test data are used: the test data was computed using NTL::RR at 1000-bit precision.

##### Implementation

These functions are implemented using the stable three term recurrence relations. These relations guarentee low absolute error but cannot guarentee low relative error near one of the roots of the polynomials.