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

double      hermite( unsigned int n, double x );
double      hermite( unsigned int n, float x );
double      hermite( unsigned int n, long double x );
float       hermitef( unsigned int n, float x );
long double hermitel( unsigned int n, long double x );

(1)

(since C++17)

double      hermite( unsigned int n, Integral x );

(2)

(since C++17)

1) Computes the (physicist's) Hermite polynomials of the degree n and argument x

4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

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

Return value

If no errors occur, value of the order-nHermite polynomial of x, that is (-1)n
e^x2

dxn

e^-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:

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

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

External links

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

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Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
http://en.cppreference.com/w/cpp/numeric/special_math/hermite

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