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 ); |
(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) |
std::hermite for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)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)nex2
| dn |
| dxn |
, 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) |
External links
| Weisstein, Eric W. "Hermite Polynomial." From MathWorld — A Wolfram Web Resource. |