laguerre

eve::laguerre = {}

inlineconstexpr

Computes the value of the Laguerre and associated Laguerre polynomials of order n at x:

• 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 \(\displaystyle \mbox{L}_{n}^{m} = (-1)^m\frac{d^m}{dx^m}\mbox{L}_{n+m}(x)\).

Defined in header

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
  template< eve::integral_value N, eve::floating_real_value T >
   eve::as_wide_as<T, N> laguerre(N n, T x) noexcept;                               //1
 
  template< eve::integral_value N, eve::integral_value M, eve::floating_real_value T >
   eve::as_wide_as<T, N> laguerre(N n, M m, T x) noexcept;                          //2
}

1. Evaluate the laguerre polynomial of order n
2. Evaluate the associated laguerre polynomial of orders n, m.

Parameters

• n, m : integral positive arguments.
• x : real floating argument.

Return value

The value of the polynomial at x is returned.

Example

#include <eve/module/polynomial.hpp>
#include <eve/wide.hpp>
#include <iostream>
 
using wide_ft = eve::wide<double, eve::fixed<8>>;
using wide_it = eve::wide<int   , eve::fixed<8>>;
 
int main()
{
 
  wide_ft xd = {0.5, -1.5, 0.1, -1.0, 19.0, 25.0, 21.5, 10000.0};
  wide_it n = {0, 1, 2, 3, 4, 5, 6, 7};
  wide_ft x(0.5);
 
  std::cout << "---- simd" << '\n'
            << "<- xd               = " << xd << '\n'
            << "<- n                = " << n  << '\n'
            << "<- x                = " << x  << '\n'
            << "-> laguerre(n, xd)   = " << eve::laguerre(n, xd) << '\n'
            << "-> laguerre(3, xd)   = " << eve::laguerre(3, xd) << '\n'
            << "-> laguerre(n, 0.5)  = " << eve::laguerre(n, 0.5) << '\n'
            << "-> laguerre(n, x)    = " << eve::laguerre(n, x)   << '\n';
 
  double xs = 3.0;
 
  std::cout << "---- scalar" << '\n'
            << "<- xs               = " << xs << '\n'
            << "-> laguerre(4, xs)   = " << eve::laguerre(4, xs) << '\n';
 
  return 0;
}

Semantic Modifiers

• eve::successor

  The expression successor(laguerre)(n, x, ln, lnm1) implements the three term recurrence relation for the Laguerre polynomials, \(\displaystyle \mbox{L}_{n+1} = \left((2n+1-x)\mbox{L}_{n}-n\mbox{L}_{n-1}\right)/(n+1)\)

  The expression successor(laguerre)(n, m, x, ln, lnmm1) implements the three term recurrence relation for the associated Laguerre polynomials, \(\displaystyle \mbox{L}_{n+1}^m = \left((m+2n+1-x)\mbox{L}_{n}^{m}-(m+n)\mbox{L}_{n-1}^{m}\right)/(n+1)\)

  These functions can be used to create a sequence of values evaluated at the same x (and m) and for rising n.

  Example

  #include <eve/module/polynomial.hpp>
  #include <eve/wide.hpp>
  #include <iostream>
   
  using wide_ft = eve::wide<double, eve::fixed<8>>;
  using wide_it = eve::wide<int   , eve::fixed<8>>;
   
  int main()
  {
   
    wide_ft xd = {0.5, -1.5, 0.1, -1.0, 19.0, 25.0, 21.5, 10000.0};
    wide_it n = {0, 1, 2, 3, 4, 5, 6, 7};
    std::cout << "---- simd" << '\n'
              << "<- xd             = " << xd << '\n'
              << "<- n              = " << n  << '\n';
    std::array<wide_ft, 8> h;
    h[0] = eve::laguerre(0, xd);
    std::cout << "-> h[0]          =  " <<  h[0] << '\n';
    std::cout << "-> laguerre(" << 0 << ", xd) = " << eve::laguerre(0, xd)   << '\n';
    h[1] = eve::laguerre(1, xd);
    std::cout << "-> laguerre(" << 1 << ", xd) = " << eve::laguerre(1, xd)   << '\n';
    std::cout << "-> h[1]          = " <<  h[1] << '\n';
     for(int i = 2; i <= 7; ++i)
    {
      h[i] = eve::successor(eve::laguerre)(i-1, xd, h[i-1], h[i-2]);
      std::cout << "-> h[" << i << "]           = " <<  h[i] << '\n';
      std::cout << "-> laguerre(" << i << ", xd) = " << eve::laguerre(i, xd)   << '\n';
   }
    return 0;
  }