NAME

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

Synopsis

Defined in header <cmath>
double laguerre( unsigned int n, double x );

float laguerre( unsigned int n, float x );
long double laguerre( unsigned int n, long double x ); (1) (since C++17)
float laguerref( unsigned int n, float x );

long double laguerrel( unsigned int n, long double x );
double laguerre( unsigned int n, IntegralType x ); (2) (since C++17)

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.

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

Return value

If no errors occur, value of the nonassociated Laguerre polynomial of x, that is

e^x
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 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]

Example

// Run this code

#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'
<< std::laguerre(3, 0.0) << '=' << 1.0 << '\n';
}

Output:

0.5=0.5
0.125=0.125
1=1

External links

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