legendre

auto eve::legendre = functor<legendre_t>

inlineconstexpr

• The Legendre 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 legendre 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
{
   // Regular overload
   constexpr auto legendre(integral_value auto n, floating_value auto x)             noexcept; // 1
 
   // Semantic options
   constexpr auto legendre[p_kind](integral_value auto n, floating_value auto x)     noexcept; // 1
   constexpr auto legendre[q_kind](integral_value auto n, floating_value auto x)     noexcept; // 2
   constexpr auto legendre[associated](integral_value auto n,
                                       integral_value auto m,
                                       floating_value auto x)                        noexcept; // 3
   constexpr auto legendre[condon_shortley](integral_value auto n,
                                            floating_value auto x)                   noexcept; // 4
   constexpr auto legendre[sph](integral_value auto l, integral_value auto m,
                                floating_value auto theta)                           noexcept; // 5
   constexpr auto legendre[successor](integral_value auto l,
                                      floating_value auto x,
                                      integral_value auto ln, integral_value lnm1)   noexcept; // 6
   constexpr auto legendre[successor](integral_value auto l,integral_value auto m,
                                      floating_value auto x,
                                      integral_value auto ln, integral_value lnm1)   noexcept; // 6
}

Parameters

• n, m, ln, lnm1 : integral positive arguments.
• x : floating argument. (prefer double elements for accuracy)

Return value

1. The value of the Legendre polynomial of order n at x is returned.
2. The value of the Legendre polynomial of order n at x of the second kind is returned.
3. The value of the associated legendre polynomial of orders n, m at x is returned.
4. multiplies the associated legendre polynomial value by the Condon-Shortley phase \((-1)^m\) to match the definition given by Abramowitz and Stegun (8.6.6). This is currently the version implemented in boost::math.
5. returns the spherical associated Legendre function of degree l, order m, and polar angle theta in radian (that is the classical spherical harmonic with \(\phi = 0\)), i.e. \(\displaystyle (-1)^m\frac{(2l+1)(l-m)!}{4\pi(l+m)!}\mbox{P}^m_{l}(\cos\theta)\)
6. The successor option implements the three term recurrence relation for the (associated) Legendre polynomials, \(\displaystyle \mbox{P}^m_{l+1} = \left((2l+1)\mbox{P}^m_{l}(x)-l\mbox{P}^m_{l-1}(x)\right)/(l+m+1)\) ( \(m = 0\) and no \(m\) in call are equivalent here).

External references

• DLMF: Classical Orthogonal Polynomials
• C++ standard reference: legendre
• Wolfram MathWorld: Legendre 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 xd = {-0.1, -0.2, -0.3, -0.5, 0.0, 0.2, 0.3, 2.0};
  double    x(0.5);
  double    z(1);
 
  std::cout << "<- xd                                = " << xd << '\n';
  std::cout << "<- n                                 = " << n  << '\n';
  std::cout << "<- x                                 = " << x  << '\n';
  std::cout << "<- z                                 = " << z  << '\n';
 
  std::cout << "-> legendre(n, xd)                   = " << eve::legendre(n, xd) << '\n';
  std::cout << "-> legendre[ignore_last(2)](n, xd)   = " << eve::legendre[eve::ignore_last(2)](n, xd) << "\n";
  std::cout << "-> legendre[n > 3](n, xd)            = " << eve::legendre[n > 3](n, xd) << "\n";
  std::cout << "-> legendre(3, xd)                   = " << eve::legendre(3, xd) << '\n';
  std::cout << "-> legendre(n, 0.5)                  = " << eve::legendre(n, 0.5) << '\n';
  std::cout << "-> legendre(n, x)                    = " << eve::legendre(n, x)   << '\n';
  std::cout << "-> legendre[p_kind](n, xd)           = " << eve::legendre[eve::p_kind](n, xd) << '\n';
  std::cout << "-> legendre[q_kind](n, xd)           = " << eve::legendre[eve::q_kind](n, xd) << '\n';
  std::cout << "-> legendre[spherical](n, z, xd)     = " << eve::legendre[eve::spherical](n, z, xd) << '\n';
  std::cout << "-> legendre[associated(n, z, xd)     = " << eve::legendre[eve::associated](n, z, xd) << '\n';
  std::cout << "-> legendre[condon_shortley(n, z, xd = " << eve::legendre[eve::condon_shortley](n, z, xd) << '\n';
}