std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel
<cmath>
double assoc_laguerre ( unsigned int n, unsigned int m, double x );
(until C++23)
/* floating-point-type */ x );
<cmath>
double assoc_laguerre ( unsigned int n, unsigned int m, Integer x );
std::assoc_laguerre 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 associated Laguerre polynomial of x, that is \((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} ,円 \mathsf{L}_{n+m}(x)\)(-1)m[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
- If x is negative, a domain error may occur
- If n or m is greater or equal to 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 associated Laguerre polynomials are the polynomial solutions of the equation \(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)xy,,
+(m+1-x)y,
+ny = 0.
The first few are:
| Function | Polynomial |
|---|---|
| assoc_laguerre(0, m, x) | 1 |
| assoc_laguerre(1, m, x) | -x + m + 1 |
| assoc_laguerre(2, m, x) | 1 2 [x2- 2(m + 2)x + (m + 1)(m + 2)] |
| assoc_laguerre(3, m, x) | 1 6 [-x3- 3(m + 3)x2 - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 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::assoc_laguerre(int_num1, int_num2, num) has the same effect as std::assoc_laguerre(int_num1, int_num2, static_cast<double>(num)).
[edit] Example
#include <cmath> #include <iostream> double L1(unsigned m, double x) { return -x + m + 1; } double L2(unsigned m, double x) { return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2)); } int main() { // spot-checks std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n' << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n'; }
Output:
10.5=10.5 60.125=60.125