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

From cppreference.com
double      legendre( unsigned int n, double x );

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

long double legendrel( unsigned int n, long double x );
(1) (since C++17)
double      legendre( unsigned int n, Integral x );
(2) (since C++17)
1) Computes the unassociated Legendre 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.

Contents

[edit] Parameters

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

[edit] Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is
1
2n
n!
dn
dxn
(x2
-1)n
, 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:

  • 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)

[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';
}

Output:

-0.335938=-0.335938
0.157715=0.157715

[edit] See also

(C++17)(C++17)(C++17)
Laguerre polynomials
(function) [edit]
(C++17)(C++17)(C++17)
Hermite polynomials
(function) [edit]

[edit] External links

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