jacobi

eve::jacobi = functor<jacobi_t>

inlineconstexpr

Defined in header

#include <eve/module/polynomial.hpp>

Callable Signatures

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

Parameters

• n : integral positive argument.
• x : real floating argument.
• alpha, beta: floating arguments.

Return value

The Jacobi polynomials are a sequence of orthogonal polynomials relative to \((1-x)^{\alpha}(1+x)^{\beta}\), for \(\alpha \) and \(\beta \) greater than -1, on the \([-1, +1]\) interval.

They can be defined via a Rodrigues formula: \(\displaystyle P^{\alpha, \beta}_n(x) = \frac{(-1)^n}{2^n n!}(1-x)^{-\alpha} (1+x)^{-\beta} \frac{d}{dx^n}\left\{ (1-x)^{\alpha}(1+x)^{\beta}(1-x^2)^n \right\}\).

1. The value of the polynomial \(P^{\alpha, \beta}_n(x)\) is returned.
2. The operation is performed conditionnaly.

External references

• DLMF: Classical Orthogonal Polynomials
• Wikipedia:Jacobi polynomial
• Wolfram MathWorld:Jacobi 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<std::int64_t   , 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);
  wide_ft aa{-0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0};
  double a = -3/8.0;
  double b = 0.25;
 
  std::cout << "---- simd" << '\n'
            << "<- xd                             = " << xd << '\n'
            << "<- n                              = " << n  << '\n'
            << "<- x                              = " << x  << '\n'
            << "-> jacobi(n, a, b, xd)            = " << eve::jacobi(n, a, b, xd) << '\n'
            << "-> jacobi(4, a, b, xd)            = " << eve::jacobi(4, a, b, xd) << '\n'
            << "-> jacobi(4, a, b, x)             = " << eve::jacobi(4, a, b, x) << '\n'
            << "-> jacobi(n, a, b 0.5)            = " << eve::jacobi(n, a, b, 0.5) << '\n'
            << "-> jacobi(n, a, b, x)             = " << eve::jacobi(n, a, b, x)   << '\n'
            << "-> jacobi(n, aa, b, x)            = " << eve::jacobi(n, aa, b, x)   << '\n'
            
            
    ;
 
  double xs = 0.5;
 
  std::cout << "---- scalar" << '\n'
            << "<- xs               = " << xs << '\n'
            << "-> jacobi(4, xs)   = " << eve::jacobi(4, a, b, xs) << '\n';
 
  return 0;
}