laguerre

auto eve::laguerre = functor<laguerre_t>

inlineconstexpr

• The Laguerre polynomial of order n is given by \(\displaystyle \mbox{L}_{n}(x) = \frac{e^x}{n!}\frac{d^n}{dx^n}(x^ne^{-x})\).
• The associated laguerre polynomial is given by \(\displaystyle \mbox{L}_{n}^{m} = (-1)^m\frac{d^m}{dx^m}\mbox{L}_{n+m}(x)\).

Header file

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto laguerre(integral_value auto n, floating_value auto x)                 noexcept; //1
 
   // Semantic options
   constexpr auto laguerre[associated](integral_value auto n, integral_value auto m,
                                       floating_value auto x)                            noexcept; // 2
   constexpr auto laguerre[successor](integral_value auto n,
                                       floating_value auto x,
                                       integral_value auto ln, integral_value auto lnm1) noexcept; // 3
   constexpr auto laguerre[successor](integral_value auto n, integral_value auto l,
                                       floating_value auto x,
                                       integral_value auto ln, integral_value auto lnm1) noexcept; // 3
   constexpr auto laguerre[associated][successor](integral_value auto n,
                                       integral_value auto m, floating_value auto x)     noexcept; // 4
}

Parameters

• n, m, l, ln, lnm1 : integral positive arguments.
• x : floating argument.

Return value

1. The value of the Laguerre polynomial of order n at x is returned.
2. The value of the associated Laguerre polynomial of orders n, m at x is returned. constexpr auto legendre(integral_value auto n, floating_value auto x) noexcept; //1
3. implements the three term recurrence relation for the Laguerre polynomials, \(\displaystyle \mbox{L}_{n+1} = \left((2n+1-x)\mbox{L}_{n}-n\mbox{L}_{n-1}\right)/(n+1)\)
4. implements the three term recurrence relation for the associated Laguerre polynomials, \(\displaystyle \mbox{L}_{n+1}^m = \left((m+2n+1-x)\mbox{L}_{n}^{m}-(m+n)\mbox{L}_{n-1}^{m}\right)/(n+1)\)

External references

• DLMF: Classical Orthogonal Polynomials
• C++ standard reference: laguerre
• Wolfram MathWorld: Laguerre Polynomial

Example

// revision 1
#include <eve/module/polynomial.hpp>
#include <iostream>
 
int main()
{
  eve::wide n{0, 1, 2, 3, 4, 5, 6, 7};
  eve::wide m{0, 1, 2, 3, 4, 5, 6, 7};
  eve::wide xd{0.5, -1.5, 0.1, -1.0, 19.0, 25.0, 21.5, 10000.0};
  double    x(0.5);
 
  std::cout << "<- n                                = " << n  << '\n';
  std::cout << "<- m                                = " << m  << '\n';
  std::cout << "<- xd                               = " << xd << '\n';
  std::cout << "<- x                                = " << x  << '\n';
 
  std::cout << "-> laguerre(n, xd)                  = " << eve::laguerre(n, xd)  << '\n';
  std::cout << "-> laguerre[ignore_last(2)](n, xd)  = " << eve::laguerre[eve::ignore_last(2)](n, xd)  << '\n';
  std::cout << "-> laguerre[n > 3](n, xd)           = " << eve::laguerre[n > 3](n, xd)  << '\n';
  std::cout << "-> laguerre(3, xd)                  = " << eve::laguerre(3, xd)  << '\n';
  std::cout << "-> laguerre(n, x)                   = " << eve::laguerre(n, x)   << '\n';
 
  std::cout << "-> laguerre[associated](n, m, xd)   = " << eve::laguerre[eve::associated](n, m, xd) << '\n';
  std::cout << "-> laguerre[associated](3, m, xd)   = " << eve::laguerre[eve::associated](3, m, xd) << '\n';
  std::cout << "-> laguerre[associated](n, 3, xd)   = " << eve::laguerre[eve::associated](n, 3, xd) << '\n';
  std::cout << "-> laguerre[associated](n, m, x)    = " << eve::laguerre[eve::associated](n, m, x) << '\n';
 
}