gegenbauer

auto eve::gegenbauer = functor<gegenbauer_t>

inlineconstexpr

Defined in header

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
  // Regular overload
  constexpr auto gegenbauer(integral_value auto n, floating_value auto lambda,
                            floating_value auto x)                                                    noexcept; //1
 
   // Lanes masking
   constexpr auto gegenbauer[conditional_expr auto c](integral_value auto n, floating_value auto lambda,
                                                      floating_value auto x)                          noexcept; // 2
   constexpr auto gegenbauer[logical_value auto m](integral_value auto n, floating_value auto lambda,
                                                   floating_value auto x)                             noexcept; // 2
}

Parameters

• n: integral argument.
• lambda: real floating argument. Must be greater than \(-\frac12\).
• x: real floating argument.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. The value of \( \mathbf{C}_n^\lambda(x)\) is returned.

   The Gegenbauer polynomials are a sequence of orthogonal polynomials relative to \((1-x^2)^{\lambda-1/2}\) on the \([-1, +1]\) interval satisfying the following recurrence relation:

   • \( \mathbf{C}_0^\lambda(x) = 1\).
   • \( \mathbf{C}_1^\lambda(x) = 2\lambda x\).
   • \( \mathbf{C}_n^\lambda(x) = \left[(2x+\lambda-1)\mathbf{C}_{n-1}^\lambda(x) - (n+2\lambda-2)\mathbf{C}_{n-2}^\lambda(x)\right]/n\).
2. The operation is performed conditionally.

External references

• Wikipedia: Gegenbauer polynomials
• Wolfram MathWorld: Gegenbauer polynomial

Example

// revision 1
#include <eve/module/polynomial.hpp>
#include <iostream>
 
int main()
{
  eve::wide xd{0.5, -1.5, 0.1, -1.0, 19.0, 25.0, 21.5, 10000.0};
  eve::wide n{0, 1, 2, 3, 4, 5, 6, 7};
  double    l(-3.0/8.0);
 
  std::cout << "<- xd                                  = " << xd << "\n";
  std::cout << "<- n                                   = " << n  << "\n";
  std::cout << "<- l                                   = " << l  << "\n";
 
  std::cout << "-> gegenbauer(n, l, xd)                = " << eve::gegenbauer(n, l, xd) << '\n';
  std::cout << "-> gegenbauer[ignore_last(2)](n, l, xd)= " << eve::gegenbauer[eve::ignore_last(2)](n, l, xd) << '\n';
  std::cout << "-> gegenbauer[n > 3](n, l, xd)         = " << eve::gegenbauer[n > 3](n, l, xd) << '\n';
  std::cout << "-> gegenbauer(3, l, xd)                = " << eve::gegenbauer(3, l, xd) << '\n';
  std::cout << "-> gegenbauer(n, l, 0.3)               = " << eve::gegenbauer(n, l, 0.3) << '\n';
}