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std::legendre, std::legendref, std::legendrel - cppreference.com

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Defined in header <cmath>

(1)

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.

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 unassociated Legendre polynomial of x, that is

(x2
-1)n

, 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
  • The function is not required to be defined for |x|>1
  • 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 first few Legendre polynomials are:

Function Polynomial
    legendre(0, x)     1
legendre(1, x) x
legendre(2, x)
1
2
(3x2
 - 1)
legendre(3, x)
1
2
(5x3
 - 3x)
legendre(4, x)     
1
8
(35x4
 - 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)).

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';
}

Output: 

-0.335938=-0.335938
0.157715=0.157715

See also

External links