tchebytchev

eve::tchebytchev = {}

inlineconstexpr

Computes the value of the Tchebytchev polynomial of order n at x:

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

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> tchebytchev(N n, T x) noexcept;
}

Parameters

• n : 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'
            << "-> 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;
}

Semantic Modifiers

• eve::kind_1, eve::kind_2

  The expression kind_1(tchebytchev)(n,x) is identical to tchebytchev(n,x). The expression kind_2(tchebytchev)(n,x) evaluates the nth polynomial of tchebytchev of second kind \( \displaystyle U_n(x) = \frac{\sin(n\arccos x)}{\sin(\arccos x)}\) at x on \([-1, +1]\).

  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'
              << "-> kind_2(tchebytchev)(n, xd)      = " << eve::kind_2(eve::tchebytchev)(n, xd) << '\n'
              << "-> kind_2(tchebytchev)(3, xd)      = " << eve::kind_2(eve::tchebytchev)(3, xd) << '\n'
              << "-> kind_2(tchebytchev)(n, 2.0)     = " << eve::kind_2(eve::tchebytchev)(n, 2.0) << '\n'
              << "-> kind_2(tchebytchev)(n, x)       = " << eve::kind_2(eve::tchebytchev)(n, x)   << '\n'
   
      ;
   
    double xs = 3.0;
   
    std::cout << "---- scalar" << '\n'
              << "<- xs                              = " << xs << '\n'
              << "-> kind_2(tchebytchev)(4, xs)       = " << eve::kind_2(eve::tchebytchev)(4, xs) << '\n';
   
    return 0;
  }

• eve::successor

  The expression successor(tchebytchev)( x, tn, tnm1) computes the value of \(T_{n+1}(x)\) knowing the values tn = \(T_n(x)\) and tnm1 = \(T_{n-1}(x)\),

  This function can be used to create a sequence of values evaluated at the same x 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> t;
    t[0] = eve::tchebytchev(0, xd);
    std::cout << "-> t[0]          =  " <<  t[0] << '\n';
    std::cout << "-> tchebytchev(" << 0 << ", xd) = " << eve::tchebytchev(0, xd)   << '\n';
    t[1] = eve::tchebytchev(1, xd);
    std::cout << "-> tchebytchev(" << 1 << ", xd) = " << eve::tchebytchev(1, xd)   << '\n';
    std::cout << "-> t[1]          = " <<  t[1] << '\n';
     for(int i = 2; i <= 7; ++i)
    {
      t[i] = eve::successor(eve::tchebytchev)(xd, t[i-1], t[i-2]);
      std::cout << "-> t[" << i << "]           = " <<  t[i] << '\n';
      std::cout << "-> tchebytchev(" << i << ", xd) = " << eve::tchebytchev(i, xd)   << '\n';
   }
    return 0;
  }