std::legendre, std::legendref, std::legendrel
double legendre( unsigned int n, double x ); double legendre( unsigned int n, float x ); |
(1) | (since C++17) |
double legendre( unsigned int n, IntegralType x ); |
(2) | (since C++17) |
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 1 |
2n n! |
dn |
dxn |
-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) =
(3x21 2
-1) - legendre(3, x) =
(5x31 2
-3x) - legendre(4, x) =
(35x41 8
-30x2
+3)
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
(C++17)(C++17)(C++17) |
Laguerre polynomials (function) |
(C++17)(C++17)(C++17) |
Hermite polynomials (function) |
External links
Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource.