std:: laguerre, std:: laguerref, std:: laguerrel

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

double laguerre ( unsigned int n, float x ) ;
double laguerre ( unsigned int n, long double x ) ;
float laguerref ( unsigned int n, float x ) ;

long double laguerrel ( unsigned int n, long double x ) ;
(1)
double laguerre ( unsigned int n, IntegralType x ) ;
(2)
1) Computes the non-associated Laguerre polynomials of the degree n 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, 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
x - the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the nonassociated Laguerre polynomial of x , that is
e x
n!
d n
dx n
(x n e -x ) , is returned.

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 is greater or equal than 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 Laguerre polynomials are the polynomial solutions of the equation xy ,, + (1 - x)y , + ny = 0 .

The first few are:

  • laguerre(0, x) = 1.
  • laguerre(1, x) = -x + 1 .
  • laguerre(2, x) =
    1
    2
    [x 2 - 4x + 2]     .
  • laguerre(3, x) =
    1
    6
    [-x 3 - 9x 2 - 18x + 6]     .

Example

(works as shown with gcc 6.0)

Run this code 
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#include <cmath>
#include <iostream>
 
double L1(double x)
{
    return -x + 1;
}
 
double L2(double x)
{
    return 0.5 * (x * x - 4 * x + 2);
}
 
int main()
{
    // spot-checks
    std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
              << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n';
}

Output: 

0.5=0.5
0.125=0.125

See also

associated Laguerre polynomials
(function)

External links

Weisstein, Eric W. "Laguerre Polynomial." From MathWorld--A Wolfram Web Resource.

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