E.V.E
v2023.02.15

◆ jacobi

eve::jacobi = {}
inlineconstexpr

Computes the value of the Jacobi polynomials \(P^{\alpha, \beta}_n(x)\).

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: \(\displaystyle P^{\alpha, \beta}_n(x) = \frac{(-1)^n}{2^n n!}(1-x)^{-\alpha} (1+x)^{-\beta} \frac{d}{dx^n}\left\{ (1-x)^{\alpha}(1+x)^{\beta}(1-x^2)^n \right\}\).

Defined in header

#include <eve/module/polynomial.hpp>

Callable Signatures

namespace eve
{
template< eve::integral_value N
eve::as_wide_as<common_value_t<T, A, B>, N>
jacobi(N n, T x, A alpha, B beta) noexcept;
}
Definition: value.hpp:114
Definition: value.hpp:42
constexpr callable_jacobi_ jacobi
Computes the value of the Jacobi polynomials .
Definition: jacobi.hpp:66
constexpr callable_beta_ beta
Computes the beta function: is returned.
Definition: beta.hpp:66
Definition: abi.hpp:18

Parameters

Return value

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

Example

#include <eve/module/polynomial.hpp>
#include <eve/wide.hpp>
#include <iostream>
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;
}
Wrapper for SIMD registers.
Definition: wide.hpp:65