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Showing content from https://en.cppreference.com/w/cpp/algorithm/../../cpp/numeric/special_functions/assoc_laguerre.html below:

std::assoc_laguerre, std::assoc_laguerref, std::assoc_laguerrel - cppreference.com

(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)

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.

[edit] 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 [edit] 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 Ln+m(x)

, is returned (where

\(\mathsf{L}_{n+m}(x)\)Ln+m(x)

is the unassociated Laguerre polynomial,

std::laguerre(n + m, x)

[edit] 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

[edit] 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)).

[edit] 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:

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