Std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel (3) Linux Manual Page

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

Synopsis

double assoc_laguerre(unsigned int n, unsigned int m, double x);

float assoc_laguerre(unsigned int n, unsigned int m, float x);

long double assoc_laguerre(unsigned int n, unsigned int m, long double x);
(1)(since C++ 17)

    float assoc_laguerref(unsigned int n, unsigned int m, float x);

long double assoc_laguerrel(unsigned int n, unsigned int m, long double x);

double assoc_laguerre(unsigned int n, unsigned int m, IntegralType x);
(2)(since C++ 17)

1) Computes the associated_Laguerre_polynomials of the degree n, order m, and argument x
2) A set of overloads or a function template accepting an argument of any integral_type. Equivalent to (1) after casting the argument to double.

Parameters

n – the degree of the polymonial, a value of unsigned integer type
m – the order of the polynomial, a value of unsigned integer type
x – the argument, a value of a floating-point or integral type

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
dxm
L
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 : *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)]

Example

// Run this code

#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.

std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel (3) Linux Manual Page