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std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel

Defined in header <cmath>
(1) 
float       assoc_laguerre ( unsigned int n, unsigned int m, float x );
double      assoc_laguerre ( unsigned int n, unsigned int m, double x );
long double assoc_laguerre ( unsigned int n, unsigned int m, long double x );
 (since C++17)
(until C++23) 
/* floating-point-type */ assoc_laguerre( unsigned int n, unsigned int m,
                                          /* floating-point-type */ x );
 (since C++23) 
float       assoc_laguerref( unsigned int n, unsigned int m, float x );
 (2) (since C++17) 
long double assoc_laguerrel( unsigned int n, unsigned int m, long double x );
 (3) (since C++17)
Additional overloads
Defined in header <cmath> 
template< class Integer >
double      assoc_laguerre ( unsigned int n, unsigned int m, Integer x );
 (A) (since C++17)
1-3) Computes the associated Laguerre polynomials of the degree n, order m, and argument x. The library provides overloads of std::assoc_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
m - the order 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 associated Laguerre polynomial of x, that is \((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{L}_{n+m}(x)\)(-1)m
dm/dxmL
n+m(x), is returned (where \(\mathsf{L}_{n+m}(x)\)L
n+m(x) is the unassociated Laguerre polynomial, std::laguerre(n + m, x)).

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 or m is greater or equal to 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 associated Laguerre polynomials are the polynomial solutions of the equation \(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)xy,,
+(m+1-x)y,
+ny = 0.

The first few are:

Function Polynomial
assoc_laguerre(0, m, x) 1
assoc_laguerre(1, m, x) -x + m + 1
assoc_laguerre(2, m, x) 1/2[x2
 - 2(m + 2)x + (m + 1)(m + 2)]
assoc_laguerre(3, m, x) 1/6[-x3
 - 3(m + 3)x2
 - 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)]

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::assoc_laguerre(int_num1, int_num2, num) has the same effect as std::assoc_laguerre(int_num1, int_num2, static_cast<double>(num)).

Example

#include <cmath>
#include <iostream>
 
double L1(unsigned m, double x)
{
    return -x + m + 1;
}
 
double L2(unsigned m, double x)
{
    return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2));
}
 
int main()
{
    // spot-checks
    std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n'
              << std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n';
}

Output:

10.5=10.5
60.125=60.125

See also

(C++17)(C++17)(C++17)
 Laguerre polynomials
(function)
Weisstein, Eric W. "Associated Laguerre Polynomial." From MathWorld — A Wolfram Web Resource.

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