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

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Defined in header <cmath>
(1)
float laguerre ( unsigned int n, float x ) ;

double laguerre ( unsigned int n, double x ) ;

long double laguerre ( unsigned int n, long double x ) ;
(since C++17)
(until C++23)
/* floating-point-type */ laguerre ( unsigned int n,
/* floating-point-type */ x ) ;
(since C++23)
float laguerref ( unsigned int n, float x ) ;
(2) (since C++17)
long double laguerrel ( unsigned int n, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Integer >
double laguerre ( unsigned int n, Integer x ) ;
(A) (since C++17)
1-3) Computes the non-associated Laguerre polynomials of the degree n and argument x . The library provides overloads of std::laguerre for all cv-unqualified floating-point types as the type of the parameter x . (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double .

Parameters

n - the degree of the polynomial, an unsigned integer value
x - the argument, a floating-point or integer value

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 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 .

The first few are:

Function Polynomial
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)

The additional overloads are not required to be provided exactly as (A) . They only need to be sufficient to ensure that for their argument num of integer type, std :: laguerre ( int_num, num ) has the same effect as std :: laguerre ( int_num, static_cast < double > ( num ) ) .

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

See also

(C++17) (C++17) (C++17)
associated Laguerre polynomials
(function)

External links

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