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

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)
Additional overloads
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) 
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 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 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(x2
 - 4x + 2)
laguerre(3, x) 1/6(-x3
 - 9x2
 - 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

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

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