hermite

auto eve::hermite = functor<hermite_t>

inlineconstexpr

The physicists Hermite polynomials are a sequence of orthogonal polynomials relative to \(e^{-x^2}\) on the \([-\infty, +\infty]\) interval satisfying the following recurrence relation:

• \( \mathbf{H}_0(x) = 1\).
• \( \mathbf{H}_1(x) = 2x\).
• \( \mathbf{H}_n(x) = 2x\mathbf{H}_{n-1}(x) -2(n-1)\mathbf{H}_{n-2}x\).

Callable Signatures

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto hermite(integral_value auto n, floating_value auto x)                          noexcept; //1
 
   // Lanes masking
   constexpr auto hermite[conditional_expr auto c](integral_value auto n, floating_value auto x) noexcept; // 2
   constexpr auto hermite[logical_value auto m](integral_value auto n, floating_value auto x)    noexcept; // 2
 
   // Semantic option
   constexpr auto hermite(integral_value auto n, floating_value auto x,
                         floating_value auto hn, floating_value auto hnm1)                      noexcept; // 3
}

Parameters

• n: integral positive argument.
• x: real floating argument.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. The value of the 'physicists' hermite polynomial \( \displaystyle \mathbf{H}_n(x) = (-1)^n e^{x^2}\frac{d}{dx^n}e^{-x^2}\) is returned.
2. The operation is performed conditionnaly.
3. implements the three terms recurrence relation for the physicists Hermite polynomials, \(\displaystyle \mbox{H}_{n+1} = (2*x)\mbox{H}_{n}-2*n\mbox{H}_{n-1}\). This call can be used to create a sequence of values evaluated at the same x, and for rising n.

External references

• DLMF: Classical Orthogonal Polynomials
• C++ standard reference: hermite
• Wolfram MathWorld: Hermite Polynomial

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(2.0);
 
  std::cout << "---- simd" << '\n'
            << "<- xd                  = " << xd << '\n'
            << "<- n                   = " << n  << '\n'
            << "<- x                   = " << x  << '\n'
            << "-> hermite(n, xd)      = " << eve::hermite(n, xd) << '\n'
            << "-> hermite(3, xd)      = " << eve::hermite(3, xd) << '\n'
            << "-> hermite(n, 2.0)     = " << eve::hermite(n, 2.0) << '\n'
            << "-> hermite(n, x)       = " << eve::hermite(n, x)   << '\n'
            ;
 
  double xs = 3.0;
 
  std::cout << "---- scalar" << '\n'
            << "<- xs               = " << xs << '\n'
            << "-> hermite(4, xs)   = " << eve::hermite(4, xs) << '\n';
 
  return 0;
}