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<a href="bessel_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">/* Copyright (c) 2008-2022 the MRtrix3 contributors.</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span><span class="comment"> *</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="comment"> * This Source Code Form is subject to the terms of the Mozilla Public</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="comment"> * License, v. 2.0. If a copy of the MPL was not distributed with this</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment"> * file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment"> *</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment"> * Covered Software is provided under this License on an "as is"</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment"> * basis, without warranty of any kind, either expressed, implied, or</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment"> * statutory, including, without limitation, warranties that the</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment"> * Covered Software is free of defects, merchantable, fit for a</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="comment"> * particular purpose or non-infringing.</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment"> * See the Mozilla Public License v. 2.0 for more details.</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span><span class="comment"> *</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span><span class="comment"> * For more details, see http://www.mrtrix.org/.</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="comment"> */</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span> </div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="preprocessor">#ifndef __math_bessel_h__</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="preprocessor">#define __math_bessel_h__</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span> </div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="preprocessor">#include <limits></span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span> </div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="preprocessor">#include "<a class="code" href="chebyshev_8h.html">math/chebyshev.h</a>"</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span> </div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespace_m_r.html">MR</a></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span>{</div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span> <span class="keyword">namespace </span>Math</div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span> {</div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"><a class="line" href="namespace_m_r_1_1_math_1_1_bessel.html"> 28</a></span> <span class="keyword">namespace </span>Bessel</div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span> {</div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span> </div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#ac9e4add26f811e9c03d0e6be53546c62">coef_aI0</a>[];</div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a5a2fa6d46d94526938bbff1637898875">coef_bI0</a>[];</div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a6e8393ec64725ca088412a9da1d35b9c">coef_cI0</a>[];</div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span> </div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#abc7bf52971e184a784a5acb6f463d7e5">coef_aI1</a>[];</div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a0bed364aa0cfcbce87fb5fbcd977e970">coef_bI1</a>[];</div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span> <span class="keyword">extern</span> <span class="keyword">const</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a9fcc21de1da4bf5c96cd9b433a346172">coef_cI1</a>[];</div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span> </div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span> </div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span> <span class="comment">//* Compute the scaled regular modified cylindrical Bessel function of zeroth order exp(-|x|) I_0(x). */</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"><a class="line" href="namespace_m_r_1_1_math_1_1_bessel.html#a1f8f9befd18cad7f26526f5f3e0d87a2"> 42</a></span><span class="comment"></span> <span class="keyword">template</span> <<span class="keyword">typename</span> T> <span class="keyword">inline</span> T <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_bessel.html#a1f8f9befd18cad7f26526f5f3e0d87a2">I0_scaled</a> (<span class="keyword">const</span> T x)</div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span> {</div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> assert (x >= 0.0);</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> <span class="keywordflow">if</span> (x*x < 4.0*std::numeric_limits<T>::epsilon()) <span class="keywordflow">return</span> (1.0-x);</div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> <span class="keywordflow">if</span> (x <= 3.0) <span class="keywordflow">return</span> (exp (-x) * (2.75 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#ac9e4add26f811e9c03d0e6be53546c62">coef_aI0</a>, 11, -1.0, 1.0, x*x/4.5-1.0)));</div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span> <span class="keywordflow">if</span> (x <= 8.0) <span class="keywordflow">return</span> ( (0.375 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a5a2fa6d46d94526938bbff1637898875">coef_bI0</a>, (<span class="keyword">sizeof</span> (T) >4?20:13), -1.0, 1.0, (48.0/x-11.0) /5.0)) /sqrt (x));</div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> <span class="keywordflow">return</span> ( (0.375 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a6e8393ec64725ca088412a9da1d35b9c">coef_cI0</a>, (<span class="keyword">sizeof</span> (T) >4?21:11), -1.0, 1.0, 16.0/x-1.0)) /sqrt (x));</div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> }</div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> </div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <span class="comment">//* Compute the scaled regular modified cylindrical Bessel function of first order exp(-|x|) I_1(x). */</span></div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="namespace_m_r_1_1_math_1_1_bessel.html#a3deeb383576f3d7a352a4ba53578c503"> 53</a></span><span class="comment"></span> <span class="keyword">template</span> <<span class="keyword">typename</span> T> <span class="keyword">inline</span> T <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_bessel.html#a3deeb383576f3d7a352a4ba53578c503">I1_scaled</a> (<span class="keyword">const</span> T x)</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> {</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> assert (x >= 0.0);</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> <span class="keywordflow">if</span> (x == 0.0) <span class="keywordflow">return</span> (0.0);</div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> <span class="keywordflow">if</span> (x*x < 8.0*std::numeric_limits<T>::epsilon()) <span class="keywordflow">return</span> (0.5*x);</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordflow">if</span> (x <= 3.0) <span class="keywordflow">return</span> (x * exp (-x) * (0.875 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#abc7bf52971e184a784a5acb6f463d7e5">coef_aI1</a>, 10, -1.0, 1.0, x*x/4.5-1.0)));</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> <span class="keywordflow">if</span> (x <= 8.0) {</div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> T b = (0.375 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a0bed364aa0cfcbce87fb5fbcd977e970">coef_bI1</a>, (<span class="keyword">sizeof</span> (T) >4?20:11), -1.0, 1.0, (48.0/x-11.0) /5.0)) / sqrt (x);</div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> <span class="keywordflow">return</span> (x > 0.0 ? b : -b);</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> }</div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> T b = (0.375 + <a class="code hl_function" href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">Chebyshev::eval</a> (<a class="code hl_variable" href="namespace_m_r_1_1_math_1_1_bessel.html#a9fcc21de1da4bf5c96cd9b433a346172">coef_cI1</a>, (<span class="keyword">sizeof</span> (T) >4?21:9), -1.0, 1.0, 16.0/x-1.0)) / sqrt (x);</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <span class="keywordflow">return</span> (x > 0.0 ? b : -b);</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> }</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> </div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> }</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> }</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span>}</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> </div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> </div>
<div class="ttc" id="achebyshev_8h_html"><div class="ttname"><a href="chebyshev_8h.html">chebyshev.h</a></div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a0bed364aa0cfcbce87fb5fbcd977e970"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a0bed364aa0cfcbce87fb5fbcd977e970">MR::Math::Bessel::coef_bI1</a></div><div class="ttdeci">const double coef_bI1[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a1f8f9befd18cad7f26526f5f3e0d87a2"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a1f8f9befd18cad7f26526f5f3e0d87a2">MR::Math::Bessel::I0_scaled</a></div><div class="ttdeci">T I0_scaled(const T x)</div><div class="ttdef"><b>Definition:</b> <a href="bessel_8h_source.html#l00042">bessel.h:42</a></div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a3deeb383576f3d7a352a4ba53578c503"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a3deeb383576f3d7a352a4ba53578c503">MR::Math::Bessel::I1_scaled</a></div><div class="ttdeci">T I1_scaled(const T x)</div><div class="ttdef"><b>Definition:</b> <a href="bessel_8h_source.html#l00053">bessel.h:53</a></div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a5a2fa6d46d94526938bbff1637898875"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a5a2fa6d46d94526938bbff1637898875">MR::Math::Bessel::coef_bI0</a></div><div class="ttdeci">const double coef_bI0[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a6e8393ec64725ca088412a9da1d35b9c"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a6e8393ec64725ca088412a9da1d35b9c">MR::Math::Bessel::coef_cI0</a></div><div class="ttdeci">const double coef_cI0[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_a9fcc21de1da4bf5c96cd9b433a346172"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#a9fcc21de1da4bf5c96cd9b433a346172">MR::Math::Bessel::coef_cI1</a></div><div class="ttdeci">const double coef_cI1[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_abc7bf52971e184a784a5acb6f463d7e5"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#abc7bf52971e184a784a5acb6f463d7e5">MR::Math::Bessel::coef_aI1</a></div><div class="ttdeci">const double coef_aI1[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_bessel_html_ac9e4add26f811e9c03d0e6be53546c62"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_bessel.html#ac9e4add26f811e9c03d0e6be53546c62">MR::Math::Bessel::coef_aI0</a></div><div class="ttdeci">const double coef_aI0[]</div></div>
<div class="ttc" id="anamespace_m_r_1_1_math_1_1_chebyshev_html_a1a35b12f5587b6dbe60b22a7e86e1498"><div class="ttname"><a href="namespace_m_r_1_1_math_1_1_chebyshev.html#a1a35b12f5587b6dbe60b22a7e86e1498">MR::Math::Chebyshev::eval</a></div><div class="ttdeci">T eval(const double *coef, const int order, const T lower, const T upper, const T x)</div><div class="ttdef"><b>Definition:</b> <a href="chebyshev_8h_source.html#l00029">chebyshev.h:29</a></div></div>
<div class="ttc" id="anamespace_m_r_html"><div class="ttname"><a href="namespace_m_r.html">MR</a></div><div class="ttdef"><b>Definition:</b> <a href="core_2adapter_2base_8h_source.html#l00023">base.h:24</a></div></div>
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