Std::legendre,std::legendref,std::legendrel (3) Linux Manual Page

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

Synopsis

double legendre(unsigned int n, double x);

float legendre(unsigned int n, float x);

long double legendre(unsigned int n, long double x);
(1)(since C++ 17)

    float legendref(unsigned int n, float x);

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

double legendre(unsigned int n, IntegralType x);
(2)(since C++ 17)

1) Computes the unassociated Legendre_polynomials of the degree n and argument x
2) 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-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)
1
2n
n!
dn
dxn
(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:

* legendre(0, x) = 1

* legendre(1, x) = x

* legendre(2, x) =

  1
  2
  (3×2
  -1)
* legendre(3, x) =
  1
  2
  (5×3
  -3x)
* legendre(4, x) =
  1
  8
  (35×4
  -30×2
  +3)

Example

// Run this code

#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

External links

Weisstein,_Eric_W._"Legendre_Polynomial." From MathWorld–A Wolfram Web Resource.

std::legendre,std::legendref,std::legendrel (3) Linux Manual Page