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

Defined in header <cmath>
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)
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.


[edit] Parameters

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

[edit] Return value

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

[edit] 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.

[edit] 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,,
= -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)).

[edit] 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';



[edit] See also

Laguerre polynomials
(function) [edit]
Legendre polynomials
(function) [edit]

[edit] External links

Weisstein, Eric W. "Hermite Polynomial." From MathWorld — A Wolfram Web Resource.