std::legendre, std::legendref, std::legendrel

Since C++23

Header: <cmath>

1-3) Computes the unassociated Legendre polynomials of the degree n and argument x.The library provides overloads of std::legendre for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)

# Declarations

float legendre ( unsigned int n, float x );
double legendre ( unsigned int n, double x );
long double legendre ( unsigned int n, long double x );

(since C++17) (until C++23)

/* floating-point-type */ legendre( unsigned int n,
/* floating-point-type */ x );

(since C++23)

float legendref( unsigned int n, float x );

(since C++17)

long double legendrel( unsigned int n, long double x );

(since C++17)

Additional overloads

template< class Integer >
double legendre ( unsigned int n, Integer x );

(since C++17)

# Parameters

# 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 first few Legendre polynomials are:

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::legendre(int_num, num) has the same effect as std::legendre(int_num, static_cast(num)).

# Example

#include <cmath>
#include <iostream>
 
double P3(double x)
{
    return 0.5 * (5 * std::pow(x, 3) - 3 * x);
}
 
double P4(double x)
{
    return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
}
 
int main()
{
    // spot-checks
    std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
              << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

# See also