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Revision 76 by tim, Thu Oct 14 23:28:09 2004 UTC vs.
Revision 1390 by gezelter, Wed Nov 25 20:02:06 2009 UTC

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1   /*
2 < * Copyright (C) 2000-2004  Object Oriented Parallel Simulation Engine (OOPSE) project
3 < *
4 < * Contact: oopse@oopse.org
5 < *
6 < * This program is free software; you can redistribute it and/or
7 < * modify it under the terms of the GNU Lesser General Public License
8 < * as published by the Free Software Foundation; either version 2.1
9 < * of the License, or (at your option) any later version.
10 < * All we ask is that proper credit is given for our work, which includes
11 < * - but is not limited to - adding the above copyright notice to the beginning
12 < * of your source code files, and to any copyright notice that you may distribute
13 < * with programs based on this work.
14 < *
15 < * This program is distributed in the hope that it will be useful,
16 < * but WITHOUT ANY WARRANTY; without even the implied warranty of
17 < * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
18 < * GNU Lesser General Public License for more details.
19 < *
20 < * You should have received a copy of the GNU Lesser General Public License
21 < * along with this program; if not, write to the Free Software
22 < * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.
2 > * Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3   *
4 + * The University of Notre Dame grants you ("Licensee") a
5 + * non-exclusive, royalty free, license to use, modify and
6 + * redistribute this software in source and binary code form, provided
7 + * that the following conditions are met:
8 + *
9 + * 1. Redistributions of source code must retain the above copyright
10 + *    notice, this list of conditions and the following disclaimer.
11 + *
12 + * 2. Redistributions in binary form must reproduce the above copyright
13 + *    notice, this list of conditions and the following disclaimer in the
14 + *    documentation and/or other materials provided with the
15 + *    distribution.
16 + *
17 + * This software is provided "AS IS," without a warranty of any
18 + * kind. All express or implied conditions, representations and
19 + * warranties, including any implied warranty of merchantability,
20 + * fitness for a particular purpose or non-infringement, are hereby
21 + * excluded.  The University of Notre Dame and its licensors shall not
22 + * be liable for any damages suffered by licensee as a result of
23 + * using, modifying or distributing the software or its
24 + * derivatives. In no event will the University of Notre Dame or its
25 + * licensors be liable for any lost revenue, profit or data, or for
26 + * direct, indirect, special, consequential, incidental or punitive
27 + * damages, however caused and regardless of the theory of liability,
28 + * arising out of the use of or inability to use software, even if the
29 + * University of Notre Dame has been advised of the possibility of
30 + * such damages.
31 + *
32 + * SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
33 + * research, please cite the appropriate papers when you publish your
34 + * work.  Good starting points are:
35 + *                                                                      
36 + * [1]  Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).            
37 + * [2]  Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).          
38 + * [3]  Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008).          
39 + * [4]  Vardeman & Gezelter, in progress (2009).                        
40   */
41 <
41 >
42   /**
43   * @file SquareMatrix3.hpp
44   * @author Teng Lin
45   * @date 10/11/2004
46   * @version 1.0
47   */
48 < #ifndef MATH_SQUAREMATRIX#_HPP
49 < #define  MATH_SQUAREMATRIX#_HPP
50 <
48 > #ifndef MATH_SQUAREMATRIX3_HPP
49 > #define  MATH_SQUAREMATRIX3_HPP
50 > #include <vector>
51 > #include "Quaternion.hpp"
52   #include "SquareMatrix.hpp"
53 < namespace oopse {
53 > #include "Vector3.hpp"
54 > #include "utils/NumericConstant.hpp"
55 > namespace OpenMD {
56  
57 <    template<typename Real>
58 <    class SquareMatrix3 : public SquareMatrix<Real, 3> {
59 <        public:
57 >  template<typename Real>
58 >  class SquareMatrix3 : public SquareMatrix<Real, 3> {
59 >  public:
60 >
61 >    typedef Real ElemType;
62 >    typedef Real* ElemPoinerType;
63              
64 <            /** default constructor */
65 <            SquareMatrix3() : SquareMatrix<Real, 3>() {
66 <            }
64 >    /** default constructor */
65 >    SquareMatrix3() : SquareMatrix<Real, 3>() {
66 >    }
67  
68 <            /** copy  constructor */
69 <            SquareMatrix3(const SquareMatrix<Real, 3>& m)  : SquareMatrix<Real, 3>(m) {
70 <            }
68 >    /** Constructs and initializes every element of this matrix to a scalar */
69 >    SquareMatrix3(Real s) : SquareMatrix<Real,3>(s){
70 >    }
71  
72 <            /** copy assignment operator */
73 <            SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) {
74 <                if (this == &m)
53 <                    return *this;
54 <                 SquareMatrix<Real, 3>::operator=(m);
55 <            }
72 >    /** Constructs and initializes from an array */
73 >    SquareMatrix3(Real* array) : SquareMatrix<Real,3>(array){
74 >    }
75  
57 –            /**
58 –             * Sets this matrix to a rotation matrix by three euler angles
59 –             * @ param euler
60 –             */
61 –            void setupRotMat(const Vector3d& euler);
76  
77 <            /**
78 <             * Sets this matrix to a rotation matrix by three euler angles
79 <             * @param phi
80 <             * @param theta
81 <             * @psi theta
82 <             */
83 <            void setupRotMat(double phi, double theta, double psi);
77 >    /** copy  constructor */
78 >    SquareMatrix3(const SquareMatrix<Real, 3>& m)  : SquareMatrix<Real, 3>(m) {
79 >    }
80 >            
81 >    SquareMatrix3( const Vector3<Real>& eulerAngles) {
82 >      setupRotMat(eulerAngles);
83 >    }
84 >            
85 >    SquareMatrix3(Real phi, Real theta, Real psi) {
86 >      setupRotMat(phi, theta, psi);
87 >    }
88  
89 +    SquareMatrix3(const Quaternion<Real>& q) {
90 +      setupRotMat(q);
91  
92 <            /**
73 <             * Sets this matrix to a rotation matrix by quaternion
74 <             * @param quat
75 <            */
76 <            void setupRotMat(const Vector4d& quat);
92 >    }
93  
94 <            /**
95 <             * Sets this matrix to a rotation matrix by quaternion
96 <             * @param q0
97 <             * @param q1
98 <             * @param q2
99 <             * @parma q3
100 <            */
101 <            void setupRotMat(double q0, double q1, double q2, double q4);
94 >    SquareMatrix3(Real w, Real x, Real y, Real z) {
95 >      setupRotMat(w, x, y, z);
96 >    }
97 >            
98 >    /** copy assignment operator */
99 >    SquareMatrix3<Real>& operator =(const SquareMatrix<Real, 3>& m) {
100 >      if (this == &m)
101 >        return *this;
102 >      SquareMatrix<Real, 3>::operator=(m);
103 >      return *this;
104 >    }
105  
87 –            /**
88 –             * Returns the quaternion from this rotation matrix
89 –             * @return the quaternion from this rotation matrix
90 –             * @exception invalid rotation matrix
91 –            */            
92 –            Quaternion rotMatToQuat();
106  
107 <            /**
108 <             * Returns the euler angles from this rotation matrix
109 <             * @return the quaternion from this rotation matrix
110 <             * @exception invalid rotation matrix
98 <            */            
99 <            Vector3d rotMatToEuler();
100 <            
101 <            /**
102 <             * Sets the value of this matrix to  the inversion of itself.
103 <             * @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the
104 <             * implementation of inverse in SquareMatrix class
105 <             */
106 <            void  inverse();
107 >    SquareMatrix3<Real>& operator =(const Quaternion<Real>& q) {
108 >      this->setupRotMat(q);
109 >      return *this;
110 >    }
111  
112 <            void diagonalize();
112 >    /**
113 >     * Sets this matrix to a rotation matrix by three euler angles
114 >     * @ param euler
115 >     */
116 >    void setupRotMat(const Vector3<Real>& eulerAngles) {
117 >      setupRotMat(eulerAngles[0], eulerAngles[1], eulerAngles[2]);
118 >    }
119  
120 +    /**
121 +     * Sets this matrix to a rotation matrix by three euler angles
122 +     * @param phi
123 +     * @param theta
124 +     * @psi theta
125 +     */
126 +    void setupRotMat(Real phi, Real theta, Real psi) {
127 +      Real sphi, stheta, spsi;
128 +      Real cphi, ctheta, cpsi;
129 +
130 +      sphi = sin(phi);
131 +      stheta = sin(theta);
132 +      spsi = sin(psi);
133 +      cphi = cos(phi);
134 +      ctheta = cos(theta);
135 +      cpsi = cos(psi);
136 +
137 +      this->data_[0][0] = cpsi * cphi - ctheta * sphi * spsi;
138 +      this->data_[0][1] = cpsi * sphi + ctheta * cphi * spsi;
139 +      this->data_[0][2] = spsi * stheta;
140 +                
141 +      this->data_[1][0] = -spsi * ctheta - ctheta * sphi * cpsi;
142 +      this->data_[1][1] = -spsi * stheta + ctheta * cphi * cpsi;
143 +      this->data_[1][2] = cpsi * stheta;
144 +
145 +      this->data_[2][0] = stheta * sphi;
146 +      this->data_[2][1] = -stheta * cphi;
147 +      this->data_[2][2] = ctheta;
148      }
149  
112 –    };
150  
151 < }
152 < #endif // MATH_SQUAREMATRIX#_HPP
151 >    /**
152 >     * Sets this matrix to a rotation matrix by quaternion
153 >     * @param quat
154 >     */
155 >    void setupRotMat(const Quaternion<Real>& quat) {
156 >      setupRotMat(quat.w(), quat.x(), quat.y(), quat.z());
157 >    }
158 >
159 >    /**
160 >     * Sets this matrix to a rotation matrix by quaternion
161 >     * @param w the first element
162 >     * @param x the second element
163 >     * @param y the third element
164 >     * @param z the fourth element
165 >     */
166 >    void setupRotMat(Real w, Real x, Real y, Real z) {
167 >      Quaternion<Real> q(w, x, y, z);
168 >      *this = q.toRotationMatrix3();
169 >    }
170 >
171 >    void setupSkewMat(Vector3<Real> v) {
172 >        setupSkewMat(v[0], v[1], v[2]);
173 >    }
174 >
175 >    void setupSkewMat(Real v1, Real v2, Real v3) {
176 >        this->data_[0][0] = 0;
177 >        this->data_[0][1] = -v3;
178 >        this->data_[0][2] = v2;
179 >        this->data_[1][0] = v3;
180 >        this->data_[1][1] = 0;
181 >        this->data_[1][2] = -v1;
182 >        this->data_[2][0] = -v2;
183 >        this->data_[2][1] = v1;
184 >        this->data_[2][2] = 0;
185 >        
186 >        
187 >    }
188 >
189 >
190 >
191 >    /**
192 >     * Returns the quaternion from this rotation matrix
193 >     * @return the quaternion from this rotation matrix
194 >     * @exception invalid rotation matrix
195 >     */            
196 >    Quaternion<Real> toQuaternion() {
197 >      Quaternion<Real> q;
198 >      Real t, s;
199 >      Real ad1, ad2, ad3;    
200 >      t = this->data_[0][0] + this->data_[1][1] + this->data_[2][2] + 1.0;
201 >
202 >      if( t > NumericConstant::epsilon ){
203 >
204 >        s = 0.5 / sqrt( t );
205 >        q[0] = 0.25 / s;
206 >        q[1] = (this->data_[1][2] - this->data_[2][1]) * s;
207 >        q[2] = (this->data_[2][0] - this->data_[0][2]) * s;
208 >        q[3] = (this->data_[0][1] - this->data_[1][0]) * s;
209 >      } else {
210 >
211 >        ad1 = this->data_[0][0];
212 >        ad2 = this->data_[1][1];
213 >        ad3 = this->data_[2][2];
214 >
215 >        if( ad1 >= ad2 && ad1 >= ad3 ){
216 >
217 >          s = 0.5 / sqrt( 1.0 + this->data_[0][0] - this->data_[1][1] - this->data_[2][2] );
218 >          q[0] = (this->data_[1][2] - this->data_[2][1]) * s;
219 >          q[1] = 0.25 / s;
220 >          q[2] = (this->data_[0][1] + this->data_[1][0]) * s;
221 >          q[3] = (this->data_[0][2] + this->data_[2][0]) * s;
222 >        } else if ( ad2 >= ad1 && ad2 >= ad3 ) {
223 >          s = 0.5 / sqrt( 1.0 + this->data_[1][1] - this->data_[0][0] - this->data_[2][2] );
224 >          q[0] = (this->data_[2][0] - this->data_[0][2] ) * s;
225 >          q[1] = (this->data_[0][1] + this->data_[1][0]) * s;
226 >          q[2] = 0.25 / s;
227 >          q[3] = (this->data_[1][2] + this->data_[2][1]) * s;
228 >        } else {
229 >
230 >          s = 0.5 / sqrt( 1.0 + this->data_[2][2] - this->data_[0][0] - this->data_[1][1] );
231 >          q[0] = (this->data_[0][1] - this->data_[1][0]) * s;
232 >          q[1] = (this->data_[0][2] + this->data_[2][0]) * s;
233 >          q[2] = (this->data_[1][2] + this->data_[2][1]) * s;
234 >          q[3] = 0.25 / s;
235 >        }
236 >      }            
237 >
238 >      return q;
239 >                
240 >    }
241 >
242 >    /**
243 >     * Returns the euler angles from this rotation matrix
244 >     * @return the euler angles in a vector
245 >     * @exception invalid rotation matrix
246 >     * We use so-called "x-convention", which is the most common definition.
247 >     * In this convention, the rotation given by Euler angles (phi, theta,
248 >     * psi), where the first rotation is by an angle phi about the z-axis,
249 >     * the second is by an angle theta (0 <= theta <= 180) about the x-axis,
250 >     * and the third is by an angle psi about the z-axis (again).
251 >     */            
252 >    Vector3<Real> toEulerAngles() {
253 >      Vector3<Real> myEuler;
254 >      Real phi;
255 >      Real theta;
256 >      Real psi;
257 >      Real ctheta;
258 >      Real stheta;
259 >                
260 >      // set the tolerance for Euler angles and rotation elements
261 >
262 >      theta = acos(std::min((RealType)1.0, std::max((RealType)-1.0,this->data_[2][2])));
263 >      ctheta = this->data_[2][2];
264 >      stheta = sqrt(1.0 - ctheta * ctheta);
265 >
266 >      // when sin(theta) is close to 0, we need to consider
267 >      // singularity In this case, we can assign an arbitary value to
268 >      // phi (or psi), and then determine the psi (or phi) or
269 >      // vice-versa. We'll assume that phi always gets the rotation,
270 >      // and psi is 0 in cases of singularity.
271 >      // we use atan2 instead of atan, since atan2 will give us -Pi to Pi.
272 >      // Since 0 <= theta <= 180, sin(theta) will be always
273 >      // non-negative. Therefore, it will never change the sign of both of
274 >      // the parameters passed to atan2.
275 >
276 >      if (fabs(stheta) < 1e-6){
277 >        psi = 0.0;
278 >        phi = atan2(-this->data_[1][0], this->data_[0][0]);  
279 >      }
280 >      // we only have one unique solution
281 >      else{    
282 >        phi = atan2(this->data_[2][0], -this->data_[2][1]);
283 >        psi = atan2(this->data_[0][2], this->data_[1][2]);
284 >      }
285 >
286 >      //wrap phi and psi, make sure they are in the range from 0 to 2*Pi
287 >      if (phi < 0)
288 >        phi += 2.0 * M_PI;
289 >
290 >      if (psi < 0)
291 >        psi += 2.0 * M_PI;
292 >
293 >      myEuler[0] = phi;
294 >      myEuler[1] = theta;
295 >      myEuler[2] = psi;
296 >
297 >      return myEuler;
298 >    }
299 >            
300 >    /** Returns the determinant of this matrix. */
301 >    Real determinant() const {
302 >      Real x,y,z;
303 >
304 >      x = this->data_[0][0] * (this->data_[1][1] * this->data_[2][2] - this->data_[1][2] * this->data_[2][1]);
305 >      y = this->data_[0][1] * (this->data_[1][2] * this->data_[2][0] - this->data_[1][0] * this->data_[2][2]);
306 >      z = this->data_[0][2] * (this->data_[1][0] * this->data_[2][1] - this->data_[1][1] * this->data_[2][0]);
307 >
308 >      return(x + y + z);
309 >    }            
310 >
311 >    /** Returns the trace of this matrix. */
312 >    Real trace() const {
313 >      return this->data_[0][0] + this->data_[1][1] + this->data_[2][2];
314 >    }
315 >            
316 >    /**
317 >     * Sets the value of this matrix to  the inversion of itself.
318 >     * @note since simple algorithm can be applied to inverse the 3 by 3 matrix, we hide the
319 >     * implementation of inverse in SquareMatrix class
320 >     */
321 >    SquareMatrix3<Real>  inverse() const {
322 >      SquareMatrix3<Real> m;
323 >      RealType det = determinant();
324 >      if (fabs(det) <= OpenMD::epsilon) {
325 >        //"The method was called on a matrix with |determinant| <= 1e-6.",
326 >        //"This is a runtime or a programming error in your application.");
327 >        std::vector<int> zeroDiagElementIndex;
328 >        for (int i =0; i < 3; ++i) {
329 >            if (fabs(this->data_[i][i]) <= OpenMD::epsilon) {
330 >                zeroDiagElementIndex.push_back(i);
331 >            }
332 >        }
333 >
334 >        if (zeroDiagElementIndex.size() == 2) {
335 >            int index = zeroDiagElementIndex[0];
336 >            m(index, index) = 1.0 / this->data_[index][index];
337 >        }else if (zeroDiagElementIndex.size() == 1) {
338 >
339 >            int a = (zeroDiagElementIndex[0] + 1) % 3;
340 >            int b = (zeroDiagElementIndex[0] + 2) %3;
341 >            RealType denom = this->data_[a][a] * this->data_[b][b] - this->data_[b][a]*this->data_[a][b];
342 >            m(a, a) = this->data_[b][b] /denom;
343 >            m(b, a) = -this->data_[b][a]/denom;
344 >
345 >            m(a,b) = -this->data_[a][b]/denom;
346 >            m(b, b) = this->data_[a][a]/denom;
347 >                
348 >        }
349 >      
350 > /*
351 >        for(std::vector<int>::iterator iter = zeroDiagElementIndex.begin(); iter != zeroDiagElementIndex.end() ++iter) {
352 >            if (this->data_[*iter][0] > OpenMD::epsilon || this->data_[*iter][1] ||this->data_[*iter][2] ||
353 >                this->data_[0][*iter] > OpenMD::epsilon || this->data_[1][*iter] ||this->data_[2][*iter] ) {
354 >                std::cout << "can not inverse matrix" << std::endl;
355 >            }
356 >        }
357 > */
358 >      } else {
359 >
360 >          m(0, 0) = this->data_[1][1]*this->data_[2][2] - this->data_[1][2]*this->data_[2][1];
361 >          m(1, 0) = this->data_[1][2]*this->data_[2][0] - this->data_[1][0]*this->data_[2][2];
362 >          m(2, 0) = this->data_[1][0]*this->data_[2][1] - this->data_[1][1]*this->data_[2][0];
363 >          m(0, 1) = this->data_[2][1]*this->data_[0][2] - this->data_[2][2]*this->data_[0][1];
364 >          m(1, 1) = this->data_[2][2]*this->data_[0][0] - this->data_[2][0]*this->data_[0][2];
365 >          m(2, 1) = this->data_[2][0]*this->data_[0][1] - this->data_[2][1]*this->data_[0][0];
366 >          m(0, 2) = this->data_[0][1]*this->data_[1][2] - this->data_[0][2]*this->data_[1][1];
367 >          m(1, 2) = this->data_[0][2]*this->data_[1][0] - this->data_[0][0]*this->data_[1][2];
368 >          m(2, 2) = this->data_[0][0]*this->data_[1][1] - this->data_[0][1]*this->data_[1][0];
369 >
370 >          m /= det;
371 >        }
372 >      return m;
373 >    }
374 >
375 >    SquareMatrix3<Real> transpose() const{
376 >      SquareMatrix3<Real> result;
377 >                
378 >      for (unsigned int i = 0; i < 3; i++)
379 >        for (unsigned int j = 0; j < 3; j++)              
380 >          result(j, i) = this->data_[i][j];
381 >
382 >      return result;
383 >    }
384 >    /**
385 >     * Extract the eigenvalues and eigenvectors from a 3x3 matrix.
386 >     * The eigenvectors (the columns of V) will be normalized.
387 >     * The eigenvectors are aligned optimally with the x, y, and z
388 >     * axes respectively.
389 >     * @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is
390 >     *     overwritten            
391 >     * @param w will contain the eigenvalues of the matrix On return of this function
392 >     * @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are
393 >     *    normalized and mutually orthogonal.              
394 >     * @warning a will be overwritten
395 >     */
396 >    static void diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w, SquareMatrix3<Real>& v);
397 >  };
398 >  /*=========================================================================
399 >
400 >  Program:   Visualization Toolkit
401 >  Module:    $RCSfile: SquareMatrix3.hpp,v $
402 >
403 >  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
404 >  All rights reserved.
405 >  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
406 >
407 >  This software is distributed WITHOUT ANY WARRANTY; without even
408 >  the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
409 >  PURPOSE.  See the above copyright notice for more information.
410 >
411 >  =========================================================================*/
412 >  template<typename Real>
413 >  void SquareMatrix3<Real>::diagonalize(SquareMatrix3<Real>& a, Vector3<Real>& w,
414 >                                        SquareMatrix3<Real>& v) {
415 >    int i,j,k,maxI;
416 >    Real tmp, maxVal;
417 >    Vector3<Real> v_maxI, v_k, v_j;
418 >
419 >    // diagonalize using Jacobi
420 >    jacobi(a, w, v);
421 >    // if all the eigenvalues are the same, return identity matrix
422 >    if (w[0] == w[1] && w[0] == w[2] ) {
423 >      v = SquareMatrix3<Real>::identity();
424 >      return;
425 >    }
426 >
427 >    // transpose temporarily, it makes it easier to sort the eigenvectors
428 >    v = v.transpose();
429 >        
430 >    // if two eigenvalues are the same, re-orthogonalize to optimally line
431 >    // up the eigenvectors with the x, y, and z axes
432 >    for (i = 0; i < 3; i++) {
433 >      if (w((i+1)%3) == w((i+2)%3)) {// two eigenvalues are the same
434 >        // find maximum element of the independant eigenvector
435 >        maxVal = fabs(v(i, 0));
436 >        maxI = 0;
437 >        for (j = 1; j < 3; j++) {
438 >          if (maxVal < (tmp = fabs(v(i, j)))){
439 >            maxVal = tmp;
440 >            maxI = j;
441 >          }
442 >        }
443 >            
444 >        // swap the eigenvector into its proper position
445 >        if (maxI != i) {
446 >          tmp = w(maxI);
447 >          w(maxI) = w(i);
448 >          w(i) = tmp;
449 >
450 >          v.swapRow(i, maxI);
451 >        }
452 >        // maximum element of eigenvector should be positive
453 >        if (v(maxI, maxI) < 0) {
454 >          v(maxI, 0) = -v(maxI, 0);
455 >          v(maxI, 1) = -v(maxI, 1);
456 >          v(maxI, 2) = -v(maxI, 2);
457 >        }
458 >
459 >        // re-orthogonalize the other two eigenvectors
460 >        j = (maxI+1)%3;
461 >        k = (maxI+2)%3;
462 >
463 >        v(j, 0) = 0.0;
464 >        v(j, 1) = 0.0;
465 >        v(j, 2) = 0.0;
466 >        v(j, j) = 1.0;
467 >
468 >        /** @todo */
469 >        v_maxI = v.getRow(maxI);
470 >        v_j = v.getRow(j);
471 >        v_k = cross(v_maxI, v_j);
472 >        v_k.normalize();
473 >        v_j = cross(v_k, v_maxI);
474 >        v.setRow(j, v_j);
475 >        v.setRow(k, v_k);
476 >
477 >
478 >        // transpose vectors back to columns
479 >        v = v.transpose();
480 >        return;
481 >      }
482 >    }
483 >
484 >    // the three eigenvalues are different, just sort the eigenvectors
485 >    // to align them with the x, y, and z axes
486 >
487 >    // find the vector with the largest x element, make that vector
488 >    // the first vector
489 >    maxVal = fabs(v(0, 0));
490 >    maxI = 0;
491 >    for (i = 1; i < 3; i++) {
492 >      if (maxVal < (tmp = fabs(v(i, 0)))) {
493 >        maxVal = tmp;
494 >        maxI = i;
495 >      }
496 >    }
497 >
498 >    // swap eigenvalue and eigenvector
499 >    if (maxI != 0) {
500 >      tmp = w(maxI);
501 >      w(maxI) = w(0);
502 >      w(0) = tmp;
503 >      v.swapRow(maxI, 0);
504 >    }
505 >    // do the same for the y element
506 >    if (fabs(v(1, 1)) < fabs(v(2, 1))) {
507 >      tmp = w(2);
508 >      w(2) = w(1);
509 >      w(1) = tmp;
510 >      v.swapRow(2, 1);
511 >    }
512 >
513 >    // ensure that the sign of the eigenvectors is correct
514 >    for (i = 0; i < 2; i++) {
515 >      if (v(i, i) < 0) {
516 >        v(i, 0) = -v(i, 0);
517 >        v(i, 1) = -v(i, 1);
518 >        v(i, 2) = -v(i, 2);
519 >      }
520 >    }
521 >
522 >    // set sign of final eigenvector to ensure that determinant is positive
523 >    if (v.determinant() < 0) {
524 >      v(2, 0) = -v(2, 0);
525 >      v(2, 1) = -v(2, 1);
526 >      v(2, 2) = -v(2, 2);
527 >    }
528 >
529 >    // transpose the eigenvectors back again
530 >    v = v.transpose();
531 >    return ;
532 >  }
533 >
534 >  /**
535 >   * Return the multiplication of two matrixes  (m1 * m2).
536 >   * @return the multiplication of two matrixes
537 >   * @param m1 the first matrix
538 >   * @param m2 the second matrix
539 >   */
540 >  template<typename Real>
541 >  inline SquareMatrix3<Real> operator *(const SquareMatrix3<Real>& m1, const SquareMatrix3<Real>& m2) {
542 >    SquareMatrix3<Real> result;
543 >
544 >    for (unsigned int i = 0; i < 3; i++)
545 >      for (unsigned int j = 0; j < 3; j++)
546 >        for (unsigned int k = 0; k < 3; k++)
547 >          result(i, j)  += m1(i, k) * m2(k, j);                
548 >
549 >    return result;
550 >  }
551 >
552 >  template<typename Real>
553 >  inline SquareMatrix3<Real> outProduct(const Vector3<Real>& v1, const Vector3<Real>& v2) {
554 >    SquareMatrix3<Real> result;
555 >
556 >    for (unsigned int i = 0; i < 3; i++) {
557 >      for (unsigned int j = 0; j < 3; j++) {
558 >        result(i, j)  = v1[i] * v2[j];                
559 >      }
560 >    }
561 >            
562 >    return result;        
563 >  }
564 >
565 >    
566 >  typedef SquareMatrix3<RealType> Mat3x3d;
567 >  typedef SquareMatrix3<RealType> RotMat3x3d;
568 >
569 > } //namespace OpenMD
570 > #endif // MATH_SQUAREMATRIX_HPP
571 >

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