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# hermitian<T,N>

### Description

Hermitian<T,N> is a family of Kernels. Specifically, it is a family of Mercer Kernels. The N-th Hermitian kernel is equal to the N-th order derivative of the Gaussian kernel. They are named after Hermite polynomials, which may be used to identify the derivatives of the Gaussian kernels.

If k is an object of class hermite<T,N>, u and v are objects of class T, and N is a integral constant, then k(u,v) returns

(1) where is Rodrigues’ formula for Hermite polynomials

(2) Figure 1 shows the Hermitian kernel for N=0 through N=5 for a scalar input type. Figure 1: Hermitian kernels, parametrised by , with N=0 at the top left through N=5 at the bottom right.

### Example

```vector< double > x(10);
vector< double > v(10);
hermitian< vector< double >, 2 > kernel(1.0);
cout << kernel( x, v ) << endl; ```

### Definition

Defined in the KML header <kml/hermitian.hpp>.

### Template Parameters

 Parameter Description Default T The hermitian argument type N The order of the hermitian

### Type requirements

T must be a vector type or a scalar type.

### Members

 Member Where defined Description hermitian() Default Constructible The default constructor result_type Input value The type of the result: input_value