ellint_rd

kyosu::ellint_rd = eve::functor<ellint_rd_t>

inlineconstexpr

Computes the Carlson's elliptic integral \( \mathbf{R}_\mathbf{D}(x, y) = \frac32 \int_{0}^{\infty} \scriptstyle[(t+x)(t+y)]^{-1/2}(t+z)^{-3/2}\;\mathrm{d}t\).

Header file

#include <eve/module/elliptic.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto ellint_rd( auto x,  auto y, auto z)                        noexcept; // 1
 
   // semantic modifier
   constexpr auto ellint_rd[threshold = tol](auto x, auto y, auto z)         noexcept; // 1
 
   // Lanes masking
   constexpr auto ellint_rd[conditional_expr auto c](auto x, auto y, auto z) noexcept; // 2
   constexpr auto ellint_rd[logical_value auto m](auto x, auto y, auto z)    noexcept; // 2
}

Parameters

• x, y, z: Can be a mix of complex and real floating values. z must be non zero
• p: floating real arguments.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. the value of the \(\mathbf{R}_\mathbf{D}\) Carlson elliptic integral: \( \frac32 \int_{0}^{\infty}\scriptstyle[(t+x)(t+y)]^{-1/2}(t+z)^{-3/2}\;\mathrm{d}t\) is returned with relative error less in magnitude than tol (tol default to eps). The integral is well defined if x, y, z lie in the complex plane cut along the nonpositive real axis, with the exception that at z must be non 0
2. The operation is performed conditionnaly

External references

• DLMF: Elliptic Integrals
• Wolfram MathWorld: Elliptic Integral

Example

// revision 1
#include <eve/module/elliptic.hpp>
#include <iostream>
 
using wide_t = eve::wide <double, eve::fixed<4>>;
using r_t = double;
wide_t re1 = { 3.0, 2.0, 1.0, 0.5};
wide_t im1 = { 2.0, -1.0,  -5.0, 0.0};
wide_t re2 = { 0.0, 1.0, 2.0, 3.0};
wide_t im2 = { 1.0 , -4.0,  -2.0, 0.0};
wide_t re3 = { 0.1, -1.0, 2.0, 4.0};
wide_t im3 = { 2.0 , -4.0,  -3.0, 0.0};
auto p = kyosu::complex_t<wide_t>(re1, im1);
auto q = kyosu::complex_t<wide_t>(re2, im2);
auto r = kyosu::complex_t<wide_t>(re3, im3);
 
int main()
{
  std::cout << "<- p = " << p << "\n";
  std::cout << "<- q = " << q << "\n";
  std::cout << "<- r = " << r << "\n";
 
   std::cout << "-> ellint_rd(p, q, r)                = " << kyosu::ellint_rd(p, q, r) << "\n";
   std::cout << "-> ellint_rd[ignore_last(2)](p, q, r)= " << kyosu::ellint_rd[eve::ignore_last(2)](p, q, r) << "\n";
   std::cout << "-> ellint_rd[q != 4.0](p, q, r)    = " << kyosu::ellint_rd[q != 4.0](p, q, r) << "\n";
}