bessel.h

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/* Copyright (c) 2008-2022 the MRtrix3 contributors.
 *
 * This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/.
 *
 * Covered Software is provided under this License on an "as is"
 * basis, without warranty of any kind, either expressed, implied, or
 * statutory, including, without limitation, warranties that the
 * Covered Software is free of defects, merchantable, fit for a
 * particular purpose or non-infringing.
 * See the Mozilla Public License v. 2.0 for more details.
 *
 * For more details, see http://www.mrtrix.org/.
 */
 
#ifndef __math_bessel_h__
#define __math_bessel_h__
 
#include <limits>
 
#include "math/chebyshev.h"
 
namespace MR
{
  namespace Math
  {
    namespace Bessel
    {
 
      extern const double coef_aI0[];
      extern const double coef_bI0[];
      extern const double coef_cI0[];
 
      extern const double coef_aI1[];
      extern const double coef_bI1[];
      extern const double coef_cI1[];
 
 
      //* Compute the scaled regular modified cylindrical Bessel function of zeroth order exp(-|x|) I_0(x). */
      template <typename T> inline T I0_scaled (const T x)
      {
        assert (x >= 0.0);
        if (x*x < 4.0*std::numeric_limits<T>::epsilon()) return (1.0-x);
        if (x <= 3.0) return (exp (-x) * (2.75 + Chebyshev::eval (coef_aI0, 11, -1.0, 1.0, x*x/4.5-1.0)));
        if (x <= 8.0) return ( (0.375 + Chebyshev::eval (coef_bI0, (sizeof (T) >4?20:13), -1.0, 1.0, (48.0/x-11.0) /5.0)) /sqrt (x));
        return ( (0.375 + Chebyshev::eval (coef_cI0, (sizeof (T) >4?21:11), -1.0, 1.0, 16.0/x-1.0)) /sqrt (x));
      }
 
      //* Compute the scaled regular modified cylindrical Bessel function of first order exp(-|x|) I_1(x). */
      template <typename T> inline T I1_scaled (const T x)
      {
        assert (x >= 0.0);
        if (x == 0.0) return (0.0);
        if (x*x < 8.0*std::numeric_limits<T>::epsilon()) return (0.5*x);
        if (x <= 3.0) return (x * exp (-x) * (0.875 + Chebyshev::eval (coef_aI1, 10, -1.0, 1.0, x*x/4.5-1.0)));
        if (x <= 8.0) {
          T b = (0.375 + Chebyshev::eval (coef_bI1, (sizeof (T) >4?20:11), -1.0, 1.0, (48.0/x-11.0) /5.0)) / sqrt (x);
          return (x > 0.0 ? b : -b);
        }
        T b = (0.375 + Chebyshev::eval (coef_cI1, (sizeof (T) >4?21:9), -1.0, 1.0, 16.0/x-1.0)) / sqrt (x);
        return (x > 0.0 ? b : -b);
      }
 
    }
  }
}
 
#endif