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

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Mathematical special functions (special functions TR)
 
 
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
ex
n!
dn
dxn
(xn
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
    [x2
    - 4x + 2]
    .
  • laguerre(3, x) =
    1
    6
    [-x3
    - 9x2
    - 18x + 6]
    .

Example

(works as shown with gcc 6.0)

#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.