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std::hermite, std::hermitef, std::hermitel

Defined in header <cmath>
(1) 
double      hermite ( unsigned int n, double x );
float       hermite ( unsigned int n, float x );
long double hermite ( unsigned int n, long double x );
 (since C++17)
(until C++23)		 
/* floating-point-type */ hermite( unsigned int n,
                                   /* floating-point-type */ x );
 (since C++23) 
float       hermitef( unsigned int n, float x );
 (2) (since C++17) 
long double hermitel( unsigned int n, long double x );
 (3) (since C++17)
Additional overloads
Defined in header <cmath> 
template< class Integer >
double      hermite ( unsigned int n, Integer x );
 (A) (since C++17)
1-3) Computes the (physicist's) Hermite polynomials of the degree n and argument x. The library provides overloads of std::hermite for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

Parameters

n - the degree of the polynomial
x - the argument, a floating-point or integer value

Return value

If no errors occur, value of the order-n Hermite polynomial of x, that is (-1)nex2dn/dxne-x2, is returned.

Error handling

Errors may be reported as specified in math_errhandling.

  • If the argument is NaN, NaN is returned and domain error is not reported.
  • If n is greater or equal than 128, the behavior is implementation-defined.

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

The Hermite polynomials are the polynomial solutions of the equation u,,-2xu, = -2nu.

The first few are:

Function Polynomial
hermite(0, x) 1
hermite(1, x) 2x
hermite(2, x) 4x2 - 2
hermite(3, x) 8x3 - 12x
hermite(4, x) 16x4 - 48x2 + 12

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::hermite(int_num, num) has the same effect as std::hermite(int_num, static_cast<double>(num)).

Example

#include <cmath>
#include <iostream>
 
double H3(double x)
{
    return 8 * std::pow(x, 3) - 12 * x;
}
 
double H4(double x)
{
    return 16 * std::pow(x, 4) - 48 * x * x + 12;
}
 
int main()
{
    // spot-checks
    std::cout << std::hermite(3, 10) << '=' << H3(10) << '\n'
              << std::hermite(4, 10) << '=' << H4(10) << '\n';
}

Output:

7880=7880
155212=155212

See also

(C++17)(C++17)(C++17)
Laguerre polynomials
(function)
(C++17)(C++17)(C++17)
Legendre polynomials
(function)
Weisstein, Eric W. "Hermite Polynomial." From MathWorld — A Wolfram Web Resource.

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