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

From cppreference.com
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) (since C++17)
double      laguerre( unsigned int n, Integral x );
(2) (since C++17)
1) Computes the non-associated Laguerre polynomials of the degree n and argument x
4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Contents

[edit] Parameters

n - the degree of the polymonial, a value of unsigned integer type
x - the argument, a value of a floating-point or integral type

[edit] 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.

[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 is greater or equal than 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 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]

[edit] Example

#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

[edit] See also

associated Laguerre polynomials
(function) [edit]

[edit] External links

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