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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_legendre_h__
#define __math_legendre_h__
#include "math/math.h"
namespace MR
{
namespace Math
{
namespace Legendre
{
template <typename ValueType>
inline ValueType factorial (const ValueType n)
{
return (n < 2.0 ? 1.0 : n*factorial (n-1.0));
}
template <typename ValueType>
inline ValueType double_factorial (const ValueType n)
{
return (n < 2.0 ? 1.0 : n*double_factorial (n-2.0));
}
template <typename ValueType>
inline ValueType Plm (const int l, const int m, const ValueType x)
{
if (m && abs (x) >= 1.0) return (0.0);
ValueType v0 = 1.0;
if (m > 0) v0 = double_factorial (ValueType (2*m-1)) * pow (1.0-pow2 (x), m/2.0);
if (m & 1) v0 = -v0; // (-1)^m
if (l == m) return (v0);
ValueType v1 = x * (2*m+1) * v0;
if (l == m+1) return (v1);
for (int n = m+2; n <= l; n++) {
ValueType v2 = ( (2.0*n-1.0) *x*v1 - (n+m-1) *v0) / (n-m);
v0 = v1;
v1 = v2;
}
return (v1);
}
namespace
{
template <typename ValueType>
inline ValueType Plm_sph_helper (const ValueType x, const ValueType m)
{
return (m < 1.0 ? 1.0 : x * (m-1.0) / m * Plm_sph_helper (x, m-2.0));
}
}
template <typename ValueType>
inline ValueType Plm_sph (const int l, const int m, const ValueType x)
{
ValueType x2 = pow2 (x);
if (m && x2 >= 1.0) return (0.0);
ValueType v0 = 0.282094791773878;
if (m) v0 *= std::sqrt ((2*m+1) * Plm_sph_helper (1.0-x2, 2.0*m));
if (m & 1) v0 = -v0;
if (l == m) return (v0);
ValueType f = std::sqrt (ValueType (2*m+3));
ValueType v1 = x * f * v0;
for (int n = m+2; n <= l; n++) {
ValueType v2 = x*v1 - v0/f;
f = std::sqrt (ValueType (4*pow2 (n)-1) / ValueType (pow2 (n)-pow2 (m)));
v0 = v1;
v1 = f*v2;
}
return (v1);
}
//* compute array of normalised associated Legendre functions
/** \note upon completion, the (l,m) value will be stored in \c array[l]. Entries in \a array for l<m will be left undefined. */
template <typename VectorType>
inline void Plm_sph (VectorType& array, const int lmax, const int m, const typename VectorType::Scalar x)
{
using value_type = typename VectorType::Scalar;
value_type x2 = pow2 (x);
if (m && x2 >= 1.0) {
for (int n = m; n <= lmax; ++n)
array[n] = 0.0;
return;
}
array[m] = 0.282094791773878;
if (m) array[m] *= std::sqrt (value_type (2*m+1) * Plm_sph_helper (1.0-x2, 2.0*m));
if (m & 1) array[m] = -array[m];
if (lmax == m) return;
value_type f = std::sqrt (value_type (2*m+3));
array[m+1] = x * f * array[m];
for (int n = m+2; n <= lmax; n++) {
array[n] = x*array[n-1] - array[n-2]/f;
f = std::sqrt (value_type (4*pow2 (n)-1) / value_type (pow2 (n)-pow2 (m)));
array[n] *= f;
}
}
//* compute derivatives of normalised associated Legendre functions
/** \note this function expects the previously computed array of associated Legendre functions to be stored in \a array,
* (as computed by Plm_sph (VectorType& array, const int lmax, const int m, const ValueType x))
* and will overwrite the values in \a array with the derivatives */
template <typename VectorType>
inline void Plm_sph_deriv (VectorType& array, const int lmax, const int m, const typename VectorType::Scalar x)
{
using value_type = typename VectorType::Scalar;
value_type x2 = pow2 (x);
if (x2 >= 1.0) {
for (int n = m; n <= lmax; n++)
array[n] = NaN;
return;
}
x2 = 1.0 / (x2-1.0);
for (int n = lmax; n > m; n--)
array[n] = x2 * (n*x*array[n] - (n+m) * std::sqrt ( (2.0*n+1.0) * (n-m) / ( (2.0*n-1.0) * (n+m))) *array[n-1]);
array[m] *= x2*m*x;
}
}
}
}
#endif
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