# Hermite approximation

## Functions

template<class Real >
Real Normal_Hermite_Ftns (Real r, unsigned int n, Real PI)
template<class Real >
Real Potential_QAGI (Real t, void *params)
template<class Real >
int Hermite_Potential (Hermite_data< Real > &data, TNT::Matrix< Real > &P)
template<class Real >
void Hermite_Kinetic (TNT::Matrix< Real > &K)
template<class Real >
void Hermite_Moment (int ell, TNT::Matrix< Real > &P)

## Function Documentation

template<class Real >
 Real Normal_Hermite_Ftns ( Real r, unsigned int n, Real PI )

The non-normalized Hermite functions have a numerically stable implemenation via the three-term recursion

The normalization factors are computed, likewise recursively, from the relationships

See [1] for more details.

The value (of the template parameter type) is passed as a parameter since it is required to compute the normalization factors.

template<class Real >
 Real Potential_QAGI ( Real t, void * params )

An integral on the real line is transformed to an integral over the interval via the following variable transformation:

Potential_QAGI defines the integrand

where the function as well as the indices are passed in the structure Hermite_data as parameters.

template<class Real >
 int Hermite_Potential ( Hermite_data< Real > & data, TNT::Matrix< Real > & P )

The coefficients of the "potential" energy matrix,

are computed via numerical quadrature.The integrand together with quadrature parameters, are passed in the structure Hermite_data.

template<class Real >
 void Hermite_Kinetic ( TNT::Matrix< Real > & K )

The coefficients of the "kinetic" energy matrix,

can be computed explicitly:

template<class Real >
 void Hermite_Moment ( int ell, TNT::Matrix< Real > & P )

Assembles the projection matrix of the polynomial onto the basis of Hermite polynomials. The coefficients

coincide with the coefficients in the expansion

This is a linear system of equations in whose coefficients are those of the and which can be solved via back-substitution.