std::legendre, std::legendref, std::legendrel
double legendre( unsigned int n, float x );
double legendre( unsigned int n, long double x );
float legendref( unsigned int n, float x );
As all special functions, legendre is only guaranteed to be available in <cmath> 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.
[edit] Parameters
[edit] Return value
If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is n!
- 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 TR 29124 but support TR 19768, 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) = 12(3x2
- 1). - legendre(3, x) = 12(5x3
- 3x). - legendre(4, x) = 18(35x4
- 30x2
+ 3).
[edit] Example
(works as shown with gcc 6.0)
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1 #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