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

Defined in header `<cmath>`
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 );	(1)	(since C++17) (until C++23)
/* floating-point-type / assoc_laguerre( unsigned int n, unsigned int m, / floating-point-type */ x );	(1)		(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)

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
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)

^mdm/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

External links

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

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