std::legendre,std::legendref,std::legendrel (3) - Linux Manuals

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

NAME

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

  (3x2
  -1)
* legendre(3, x) =

  1
  2

  (5x3
  -3x)
* legendre(4, x) =

  1
  8

  (35x4
  -30x2
  +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.