<ahref="matrix__utils_8h.html">Go to the documentation of this file.</a><divclass="fragment"><divclass="line"><aid="l00001"name="l00001"></a><spanclass="lineno"> 1</span><spanclass="comment">// Copyright 2010-2021 Google LLC</span></div>
<divclass="line"><aid="l00002"name="l00002"></a><spanclass="lineno"> 2</span><spanclass="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div>
<divclass="line"><aid="l00003"name="l00003"></a><spanclass="lineno"> 3</span><spanclass="comment">// you may not use this file except in compliance with the License.</span></div>
<divclass="line"><aid="l00004"name="l00004"></a><spanclass="lineno"> 4</span><spanclass="comment">// You may obtain a copy of the License at</span></div>
<divclass="line"><aid="l00008"name="l00008"></a><spanclass="lineno"> 8</span><spanclass="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
<divclass="line"><aid="l00009"name="l00009"></a><spanclass="lineno"> 9</span><spanclass="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div>
<divclass="line"><aid="l00010"name="l00010"></a><spanclass="lineno"> 10</span><spanclass="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
<divclass="line"><aid="l00011"name="l00011"></a><spanclass="lineno"> 11</span><spanclass="comment">// See the License for the specific language governing permissions and</span></div>
<divclass="line"><aid="l00012"name="l00012"></a><spanclass="lineno"> 12</span><spanclass="comment">// limitations under the License.</span></div>
<divclass="line"><aid="l00023"name="l00023"></a><spanclass="lineno"> 23</span><spanclass="comment">// Finds the proportional columns of the given matrix: all the pairs of columns</span></div>
<divclass="line"><aid="l00024"name="l00024"></a><spanclass="lineno"> 24</span><spanclass="comment">// such that one is a non-zero scalar multiple of the other. The returned</span></div>
<divclass="line"><aid="l00025"name="l00025"></a><spanclass="lineno"> 25</span><spanclass="comment">// mapping for a given column will either be:</span></div>
<divclass="line"><aid="l00026"name="l00026"></a><spanclass="lineno"> 26</span><spanclass="comment">// - The index of the first column which is proportional with this one.</span></div>
<divclass="line"><aid="l00027"name="l00027"></a><spanclass="lineno"> 27</span><spanclass="comment">// - Or kInvalidCol if this column is not proportional to any other.</span></div>
<divclass="line"><aid="l00029"name="l00029"></a><spanclass="lineno"> 29</span><spanclass="comment">// Because of precision errors, two columns are declared proportional if the</span></div>
<divclass="line"><aid="l00030"name="l00030"></a><spanclass="lineno"> 30</span><spanclass="comment">// multiples (>= 1.0) defined by all the couple of coefficients of the same row</span></div>
<divclass="line"><aid="l00031"name="l00031"></a><spanclass="lineno"> 31</span><spanclass="comment">// are equal up to the given absolute tolerance.</span></div>
<divclass="line"><aid="l00033"name="l00033"></a><spanclass="lineno"> 33</span><spanclass="comment">// The complexity is in most cases O(num entries of the matrix). However,</span></div>
<divclass="line"><aid="l00034"name="l00034"></a><spanclass="lineno"> 34</span><spanclass="comment">// compared to the less efficient algorithm below, it is highly unlikely but</span></div>
<divclass="line"><aid="l00035"name="l00035"></a><spanclass="lineno"> 35</span><spanclass="comment">// possible that some pairs of proportional columns are not detected.</span></div>
<divclass="line"><aid="l00039"name="l00039"></a><spanclass="lineno"> 39</span><spanclass="comment">// A simple version of FindProportionalColumns() that compares all the columns</span></div>
<divclass="line"><aid="l00040"name="l00040"></a><spanclass="lineno"> 40</span><spanclass="comment">// pairs one by one. This is slow, but here for reference. The complexity is</span></div>
<divclass="line"><aid="l00045"name="l00045"></a><spanclass="lineno"> 45</span><spanclass="comment">// Returns true iff the two given matrices have exactly the same first num_rows</span></div>
<divclass="line"><aid="l00046"name="l00046"></a><spanclass="lineno"> 46</span><spanclass="comment">// entries on the first num_cols columns. The two given matrices must be ordered</span></div>
<divclass="line"><aid="l00047"name="l00047"></a><spanclass="lineno"> 47</span><spanclass="comment">// by rows (this is DCHECKed, but only for the first one at this point).</span></div>
<divclass="line"><aid="l00052"name="l00052"></a><spanclass="lineno"> 52</span><spanclass="comment">// Returns true iff the rightmost square matrix is an identity matrix.</span></div>
<divclass="ttc"id="anamespaceoperations__research_html"><divclass="ttname"><ahref="namespaceoperations__research.html">operations_research</a></div><divclass="ttdoc">Collection of objects used to extend the Constraint Solver library.</div><divclass="ttdef"><b>Definition:</b><ahref="dense__doubly__linked__list_8h_source.html#l00021">dense_doubly_linked_list.h:21</a></div></div>
