std::legendre, std::legendref, std::legendrel
<cmath>
double legendre ( unsigned int n, double x );
(until C++23)
/* floating-point-type */ x );
<cmath>
double legendre ( unsigned int n, Integer x );
std::legendre for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)[edit] Parameters
[edit] Return value
If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)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 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:
| Function | Polynomial |
|---|---|
| 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) |
The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::legendre(int_num, num) has the same effect as std::legendre(int_num, static_cast<double>(num)).
[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