tchebytchev

eve::tchebytchev = functor<tchebytchev_t>

inlineconstexpr

• The Tchebytchev polynomial of order n is given by $\displaystyle \mbox{T}_{n}(x) = \cos(n\arccos(x))$ on $[-1, +1]$

Header file

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
  constexpr auto tchebytchev(integral_value auto n, floating_value auto x)                           noexcept; // 1
 
   // Lanes masking
   constexpr auto tchebytchev[conditional_expr auto c](integral_value auto n, floating_value auto x) noexcept; // 2
   constexpr auto tchebytchev[logical_value auto m](integral_value auto n, floating_value auto x)    noexcept; // 2
 
   // Semantic options
   constexpr auto tchebytchev[kind_1](integral_value auto n, floating_value auto x)                  noexcept; // 1
   constexpr auto tchebytchev[kind_2](integral_value auto n, floating_value auto x)                  noexcept; // 3
   constexpr auto tchebytchev[successor](floating_value auto x, integral_value auto tn,
                                                                integral_value auto tnm1)            noexcept; // 4
}

Parameters

• n : integral positive arguments. (note that negative values return a NaN)
• x : real floating argument.

Return value

1.The value of the polynomial at x is returned.

1. The operation is performed conditionnaly.
2. evaluates the nth polynomial of tchebytchev of second kind $\displaystyle U_n(x) = \frac{\sin(n\arccos x)}{\sin(\arccos x)}$. on $[-1, +1]$.
3. computes the value of $T_{n+1}(x)$ knowing the values tn = $T_n(x)$ and tnm1 = $T_{n-1}(x)$, This call can be used to create a sequence of values evaluated at the same x and for rising n.

External references

• Wikipedia: Chebyshev_polynomials
• Wolfram MathWorld: first kind
• Wolfram MathWorld: second kind

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'
            << "-> tchebytchev(n, xd)      = " << eve::tchebytchev(n, xd) << '\n'
            << "-> tchebytchev(3, xd)      = " << eve::tchebytchev(3, xd) << '\n'
            << "-> tchebytchev(n, 2.0)     = " << eve::tchebytchev(n, 2.0) << '\n'
            << "-> tchebytchev(n, x)       = " << eve::tchebytchev(n, x)   << '\n'
 
    ;
 
  double xs = 3.0;
 
  std::cout << "---- scalar" << '\n'
            << "<- xs               = " << xs << '\n'
            << "-> tchebytchev(4, xs)   = " << eve::tchebytchev(4.0, xs) << '\n';
 
  return 0;
}