std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel
| double assoc_laguerre ( unsigned int n, unsigned int m, double x ); double assoc_laguerre ( unsigned int n, unsigned int m, float x ); |
(1) | |
| double assoc_laguerre ( unsigned int n, unsigned int m, IntegralType x ); |
(2) | |
As all special functions, assoc_laguerre 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.
Parameters
| n | - | the degree of the polynomial, a value of unsigned integer type |
| m | - | the order of the polynomial, a value of unsigned integer type |
| x | - | the argument, a value of a floating-point or integral type |
Return value
If no errors occur, value of the associated Laguerre polynomial of x, that is (-1)m| dm |
| dxm |
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.
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 associated Laguerre polynomials are the polynomial solutions of the equation xy,,
+ (m + 1 - x)y,
+ ny = 0.
The first few are:
-
assoc_laguerre(0, m, x)= 1. -
assoc_laguerre(1, m, x)= -x + m + 1. -
assoc_laguerre(2, m, x)=
[x21 2
- 2(m + 2)x + (m + 1)(m + 2)]. -
assoc_laguerre(3, m, x)=
[-x31 6
- 3(m + 3)x2
- 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)].
Example
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1 #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
See also
| Laguerre polynomials (function) |
External links
| Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld — A Wolfram Web Resource. |