NAME¶

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

Synopsis¶

double laguerre( unsigned int n, double x );

double laguerre( unsigned int n, float x );
double laguerre( unsigned int n, long double x ); (1)
float laguerref( unsigned int n, float x );

long double laguerrel( unsigned int n, long double x );
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!

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)

// 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¶

assoc_laguerre associated Laguerre polynomials
assoc_laguerref (function)
assoc_laguerrel

External links¶

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