std:: assoc_laguerre, std:: assoc_laguerref, std:: assoc_laguerrel

From cppreference.com
C++
Experimental
Mathematical special functions
double assoc_laguerre ( unsigned int n, unsigned int m, double x ) ;

double assoc_laguerre ( unsigned int n, unsigned int m, float x ) ;
double assoc_laguerre ( unsigned int n, unsigned int m, long double x ) ;
float assoc_laguerref ( unsigned int n, unsigned int m, float x ) ;

long double assoc_laguerrel ( unsigned int n, unsigned int m, long double x ) ;
(1)
double assoc_laguerre ( unsigned int n, unsigned int m, IntegralType x ) ;
(2)
1) Computes the associated Laguerre polynomials of the degree n , order m , and argument x .
2) A set of overloads or a function template accepting an argument of any integral type . Equivalent to (1) after casting the argument to double .

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
d m
dx m
L n + m (x) , is returned (where L n + m (x) is the unassociated Laguerre polynomial, std:: laguerre ( n + m, x ) ).

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) =
    1
    2
    [x 2 - 2(m + 2)x + (m + 1)(m + 2)]     .
  • assoc_laguerre(3, m, x) =
    1
    6
    [-x 3 - 3(m + 3)x 2 - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)]     .

Example

Run this code 
#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.
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