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

Defined in header <cmath>
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 );
(2) (since C++17)
long double legendrel( unsigned int n, long double x );
(3) (since C++17)
Defined in header <cmath>
template< class Integer >
double      legendre ( unsigned int n, Integer x );
(A) (since C++17)
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)
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 unassociated Legendre polynomial of x, that is
, 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
  • The function is not required to be defined for |x|>1
  • 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 first few Legendre polynomials are:

Function Polynomial
    legendre(0, x)     1
legendre(1, x) x
legendre(2, x)
- 1)
legendre(3, x)
- 3x)
legendre(4, x)     
- 30x2
+ 3)

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<double>(num)).

[edit] 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';



[edit] See also

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

[edit] External links

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