| 1 |
/* |
| 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] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010). |
| 40 |
* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011). |
| 41 |
*/ |
| 42 |
|
| 43 |
/** |
| 44 |
* @file SquareMatrix.hpp |
| 45 |
* @author Teng Lin |
| 46 |
* @date 10/11/2004 |
| 47 |
* @version 1.0 |
| 48 |
*/ |
| 49 |
#ifndef MATH_SQUAREMATRIX_HPP |
| 50 |
#define MATH_SQUAREMATRIX_HPP |
| 51 |
|
| 52 |
#include "math/RectMatrix.hpp" |
| 53 |
#include "utils/NumericConstant.hpp" |
| 54 |
|
| 55 |
namespace OpenMD { |
| 56 |
|
| 57 |
/** |
| 58 |
* @class SquareMatrix SquareMatrix.hpp "math/SquareMatrix.hpp" |
| 59 |
* @brief A square matrix class |
| 60 |
* @template Real the element type |
| 61 |
* @template Dim the dimension of the square matrix |
| 62 |
*/ |
| 63 |
template<typename Real, int Dim> |
| 64 |
class SquareMatrix : public RectMatrix<Real, Dim, Dim> { |
| 65 |
public: |
| 66 |
typedef Real ElemType; |
| 67 |
typedef Real* ElemPoinerType; |
| 68 |
|
| 69 |
/** default constructor */ |
| 70 |
SquareMatrix() { |
| 71 |
for (unsigned int i = 0; i < Dim; i++) |
| 72 |
for (unsigned int j = 0; j < Dim; j++) |
| 73 |
this->data_[i][j] = 0.0; |
| 74 |
} |
| 75 |
|
| 76 |
/** Constructs and initializes every element of this matrix to a scalar */ |
| 77 |
SquareMatrix(Real s) : RectMatrix<Real, Dim, Dim>(s){ |
| 78 |
} |
| 79 |
|
| 80 |
/** Constructs and initializes from an array */ |
| 81 |
SquareMatrix(Real* array) : RectMatrix<Real, Dim, Dim>(array){ |
| 82 |
} |
| 83 |
|
| 84 |
|
| 85 |
/** copy constructor */ |
| 86 |
SquareMatrix(const RectMatrix<Real, Dim, Dim>& m) : RectMatrix<Real, Dim, Dim>(m) { |
| 87 |
} |
| 88 |
|
| 89 |
/** copy assignment operator */ |
| 90 |
SquareMatrix<Real, Dim>& operator =(const RectMatrix<Real, Dim, Dim>& m) { |
| 91 |
RectMatrix<Real, Dim, Dim>::operator=(m); |
| 92 |
return *this; |
| 93 |
} |
| 94 |
|
| 95 |
/** Retunrs an identity matrix*/ |
| 96 |
|
| 97 |
static SquareMatrix<Real, Dim> identity() { |
| 98 |
SquareMatrix<Real, Dim> m; |
| 99 |
|
| 100 |
for (unsigned int i = 0; i < Dim; i++) |
| 101 |
for (unsigned int j = 0; j < Dim; j++) |
| 102 |
if (i == j) |
| 103 |
m(i, j) = 1.0; |
| 104 |
else |
| 105 |
m(i, j) = 0.0; |
| 106 |
|
| 107 |
return m; |
| 108 |
} |
| 109 |
|
| 110 |
/** |
| 111 |
* Retunrs the inversion of this matrix. |
| 112 |
* @todo need implementation |
| 113 |
*/ |
| 114 |
SquareMatrix<Real, Dim> inverse() { |
| 115 |
SquareMatrix<Real, Dim> result; |
| 116 |
|
| 117 |
return result; |
| 118 |
} |
| 119 |
|
| 120 |
/** |
| 121 |
* Returns the determinant of this matrix. |
| 122 |
* @todo need implementation |
| 123 |
*/ |
| 124 |
Real determinant() const { |
| 125 |
Real det; |
| 126 |
return det; |
| 127 |
} |
| 128 |
|
| 129 |
/** Returns the trace of this matrix. */ |
| 130 |
Real trace() const { |
| 131 |
Real tmp = 0; |
| 132 |
|
| 133 |
for (unsigned int i = 0; i < Dim ; i++) |
| 134 |
tmp += this->data_[i][i]; |
| 135 |
|
| 136 |
return tmp; |
| 137 |
} |
| 138 |
|
| 139 |
/** |
| 140 |
* Returns the tensor contraction (double dot product) of two rank 2 |
| 141 |
* tensors (or Matrices) |
| 142 |
* @param t1 first tensor |
| 143 |
* @param t2 second tensor |
| 144 |
* @return the tensor contraction (double dot product) of t1 and t2 |
| 145 |
*/ |
| 146 |
Real doubleDot( const SquareMatrix<Real, Dim>& t1, const SquareMatrix<Real, Dim>& t2 ) { |
| 147 |
Real tmp; |
| 148 |
tmp = 0; |
| 149 |
|
| 150 |
for (unsigned int i = 0; i < Dim; i++) |
| 151 |
for (unsigned int j =0; j < Dim; j++) |
| 152 |
tmp += t1[i][j] * t2[i][j]; |
| 153 |
|
| 154 |
return tmp; |
| 155 |
} |
| 156 |
|
| 157 |
/** Tests if this matrix is symmetrix. */ |
| 158 |
bool isSymmetric() const { |
| 159 |
for (unsigned int i = 0; i < Dim - 1; i++) |
| 160 |
for (unsigned int j = i; j < Dim; j++) |
| 161 |
if (fabs(this->data_[i][j] - this->data_[j][i]) > epsilon) |
| 162 |
return false; |
| 163 |
|
| 164 |
return true; |
| 165 |
} |
| 166 |
|
| 167 |
/** Tests if this matrix is orthogonal. */ |
| 168 |
bool isOrthogonal() { |
| 169 |
SquareMatrix<Real, Dim> tmp; |
| 170 |
|
| 171 |
tmp = *this * transpose(); |
| 172 |
|
| 173 |
return tmp.isDiagonal(); |
| 174 |
} |
| 175 |
|
| 176 |
/** Tests if this matrix is diagonal. */ |
| 177 |
bool isDiagonal() const { |
| 178 |
for (unsigned int i = 0; i < Dim ; i++) |
| 179 |
for (unsigned int j = 0; j < Dim; j++) |
| 180 |
if (i !=j && fabs(this->data_[i][j]) > epsilon) |
| 181 |
return false; |
| 182 |
|
| 183 |
return true; |
| 184 |
} |
| 185 |
|
| 186 |
/** |
| 187 |
* Returns a column vector that contains the elements from the |
| 188 |
* diagonal of m in the order R(0) = m(0,0), R(1) = m(1,1), and so |
| 189 |
* on. |
| 190 |
*/ |
| 191 |
Vector<Real, Dim> diagonals() const { |
| 192 |
Vector<Real, Dim> result; |
| 193 |
for (unsigned int i = 0; i < Dim; i++) { |
| 194 |
result(i) = this->data_[i][i]; |
| 195 |
} |
| 196 |
return result; |
| 197 |
} |
| 198 |
|
| 199 |
/** Tests if this matrix is the unit matrix. */ |
| 200 |
bool isUnitMatrix() const { |
| 201 |
if (!isDiagonal()) |
| 202 |
return false; |
| 203 |
|
| 204 |
for (unsigned int i = 0; i < Dim ; i++) |
| 205 |
if (fabs(this->data_[i][i] - 1) > epsilon) |
| 206 |
return false; |
| 207 |
|
| 208 |
return true; |
| 209 |
} |
| 210 |
|
| 211 |
/** Return the transpose of this matrix */ |
| 212 |
SquareMatrix<Real, Dim> transpose() const{ |
| 213 |
SquareMatrix<Real, Dim> result; |
| 214 |
|
| 215 |
for (unsigned int i = 0; i < Dim; i++) |
| 216 |
for (unsigned int j = 0; j < Dim; j++) |
| 217 |
result(j, i) = this->data_[i][j]; |
| 218 |
|
| 219 |
return result; |
| 220 |
} |
| 221 |
|
| 222 |
/** @todo need implementation */ |
| 223 |
void diagonalize() { |
| 224 |
//jacobi(m, eigenValues, ortMat); |
| 225 |
} |
| 226 |
|
| 227 |
/** |
| 228 |
* Jacobi iteration routines for computing eigenvalues/eigenvectors of |
| 229 |
* real symmetric matrix |
| 230 |
* |
| 231 |
* @return true if success, otherwise return false |
| 232 |
* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is |
| 233 |
* overwritten |
| 234 |
* @param w will contain the eigenvalues of the matrix On return of this function |
| 235 |
* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are |
| 236 |
* normalized and mutually orthogonal. |
| 237 |
*/ |
| 238 |
|
| 239 |
static int jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& d, |
| 240 |
SquareMatrix<Real, Dim>& v); |
| 241 |
};//end SquareMatrix |
| 242 |
|
| 243 |
|
| 244 |
/*========================================================================= |
| 245 |
|
| 246 |
Program: Visualization Toolkit |
| 247 |
Module: $RCSfile: SquareMatrix.hpp,v $ |
| 248 |
|
| 249 |
Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen |
| 250 |
All rights reserved. |
| 251 |
See Copyright.txt or http://www.kitware.com/Copyright.htm for details. |
| 252 |
|
| 253 |
This software is distributed WITHOUT ANY WARRANTY; without even |
| 254 |
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
| 255 |
PURPOSE. See the above copyright notice for more information. |
| 256 |
|
| 257 |
=========================================================================*/ |
| 258 |
|
| 259 |
#define VTK_ROTATE(a,i,j,k,l) g=a(i, j);h=a(k, l);a(i, j)=g-s*(h+g*tau); \ |
| 260 |
a(k, l)=h+s*(g-h*tau) |
| 261 |
|
| 262 |
#define VTK_MAX_ROTATIONS 20 |
| 263 |
|
| 264 |
// Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn |
| 265 |
// real symmetric matrix. Square nxn matrix a; size of matrix in n; |
| 266 |
// output eigenvalues in w; and output eigenvectors in v. Resulting |
| 267 |
// eigenvalues/vectors are sorted in decreasing order; eigenvectors are |
| 268 |
// normalized. |
| 269 |
template<typename Real, int Dim> |
| 270 |
int SquareMatrix<Real, Dim>::jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w, |
| 271 |
SquareMatrix<Real, Dim>& v) { |
| 272 |
const int n = Dim; |
| 273 |
int i, j, k, iq, ip, numPos; |
| 274 |
Real tresh, theta, tau, t, sm, s, h, g, c, tmp; |
| 275 |
Real bspace[4], zspace[4]; |
| 276 |
Real *b = bspace; |
| 277 |
Real *z = zspace; |
| 278 |
|
| 279 |
// only allocate memory if the matrix is large |
| 280 |
if (n > 4) { |
| 281 |
b = new Real[n]; |
| 282 |
z = new Real[n]; |
| 283 |
} |
| 284 |
|
| 285 |
// initialize |
| 286 |
for (ip=0; ip<n; ip++) { |
| 287 |
for (iq=0; iq<n; iq++) { |
| 288 |
v(ip, iq) = 0.0; |
| 289 |
} |
| 290 |
v(ip, ip) = 1.0; |
| 291 |
} |
| 292 |
for (ip=0; ip<n; ip++) { |
| 293 |
b[ip] = w[ip] = a(ip, ip); |
| 294 |
z[ip] = 0.0; |
| 295 |
} |
| 296 |
|
| 297 |
// begin rotation sequence |
| 298 |
for (i=0; i<VTK_MAX_ROTATIONS; i++) { |
| 299 |
sm = 0.0; |
| 300 |
for (ip=0; ip<n-1; ip++) { |
| 301 |
for (iq=ip+1; iq<n; iq++) { |
| 302 |
sm += fabs(a(ip, iq)); |
| 303 |
} |
| 304 |
} |
| 305 |
if (sm == 0.0) { |
| 306 |
break; |
| 307 |
} |
| 308 |
|
| 309 |
if (i < 3) { // first 3 sweeps |
| 310 |
tresh = 0.2*sm/(n*n); |
| 311 |
} else { |
| 312 |
tresh = 0.0; |
| 313 |
} |
| 314 |
|
| 315 |
for (ip=0; ip<n-1; ip++) { |
| 316 |
for (iq=ip+1; iq<n; iq++) { |
| 317 |
g = 100.0*fabs(a(ip, iq)); |
| 318 |
|
| 319 |
// after 4 sweeps |
| 320 |
if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip]) |
| 321 |
&& (fabs(w[iq])+g) == fabs(w[iq])) { |
| 322 |
a(ip, iq) = 0.0; |
| 323 |
} else if (fabs(a(ip, iq)) > tresh) { |
| 324 |
h = w[iq] - w[ip]; |
| 325 |
if ( (fabs(h)+g) == fabs(h)) { |
| 326 |
t = (a(ip, iq)) / h; |
| 327 |
} else { |
| 328 |
theta = 0.5*h / (a(ip, iq)); |
| 329 |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
| 330 |
if (theta < 0.0) { |
| 331 |
t = -t; |
| 332 |
} |
| 333 |
} |
| 334 |
c = 1.0 / sqrt(1+t*t); |
| 335 |
s = t*c; |
| 336 |
tau = s/(1.0+c); |
| 337 |
h = t*a(ip, iq); |
| 338 |
z[ip] -= h; |
| 339 |
z[iq] += h; |
| 340 |
w[ip] -= h; |
| 341 |
w[iq] += h; |
| 342 |
a(ip, iq)=0.0; |
| 343 |
|
| 344 |
// ip already shifted left by 1 unit |
| 345 |
for (j = 0;j <= ip-1;j++) { |
| 346 |
VTK_ROTATE(a,j,ip,j,iq); |
| 347 |
} |
| 348 |
// ip and iq already shifted left by 1 unit |
| 349 |
for (j = ip+1;j <= iq-1;j++) { |
| 350 |
VTK_ROTATE(a,ip,j,j,iq); |
| 351 |
} |
| 352 |
// iq already shifted left by 1 unit |
| 353 |
for (j=iq+1; j<n; j++) { |
| 354 |
VTK_ROTATE(a,ip,j,iq,j); |
| 355 |
} |
| 356 |
for (j=0; j<n; j++) { |
| 357 |
VTK_ROTATE(v,j,ip,j,iq); |
| 358 |
} |
| 359 |
} |
| 360 |
} |
| 361 |
} |
| 362 |
|
| 363 |
for (ip=0; ip<n; ip++) { |
| 364 |
b[ip] += z[ip]; |
| 365 |
w[ip] = b[ip]; |
| 366 |
z[ip] = 0.0; |
| 367 |
} |
| 368 |
} |
| 369 |
|
| 370 |
//// this is NEVER called |
| 371 |
if ( i >= VTK_MAX_ROTATIONS ) { |
| 372 |
std::cout << "vtkMath::Jacobi: Error extracting eigenfunctions" << std::endl; |
| 373 |
return 0; |
| 374 |
} |
| 375 |
|
| 376 |
// sort eigenfunctions these changes do not affect accuracy |
| 377 |
for (j=0; j<n-1; j++) { // boundary incorrect |
| 378 |
k = j; |
| 379 |
tmp = w[k]; |
| 380 |
for (i=j+1; i<n; i++) { // boundary incorrect, shifted already |
| 381 |
if (w[i] >= tmp) { // why exchage if same? |
| 382 |
k = i; |
| 383 |
tmp = w[k]; |
| 384 |
} |
| 385 |
} |
| 386 |
if (k != j) { |
| 387 |
w[k] = w[j]; |
| 388 |
w[j] = tmp; |
| 389 |
for (i=0; i<n; i++) { |
| 390 |
tmp = v(i, j); |
| 391 |
v(i, j) = v(i, k); |
| 392 |
v(i, k) = tmp; |
| 393 |
} |
| 394 |
} |
| 395 |
} |
| 396 |
// insure eigenvector consistency (i.e., Jacobi can compute vectors that |
| 397 |
// are negative of one another (.707,.707,0) and (-.707,-.707,0). This can |
| 398 |
// reek havoc in hyperstreamline/other stuff. We will select the most |
| 399 |
// positive eigenvector. |
| 400 |
int ceil_half_n = (n >> 1) + (n & 1); |
| 401 |
for (j=0; j<n; j++) { |
| 402 |
for (numPos=0, i=0; i<n; i++) { |
| 403 |
if ( v(i, j) >= 0.0 ) { |
| 404 |
numPos++; |
| 405 |
} |
| 406 |
} |
| 407 |
// if ( numPos < ceil(RealType(n)/RealType(2.0)) ) |
| 408 |
if ( numPos < ceil_half_n) { |
| 409 |
for (i=0; i<n; i++) { |
| 410 |
v(i, j) *= -1.0; |
| 411 |
} |
| 412 |
} |
| 413 |
} |
| 414 |
|
| 415 |
if (n > 4) { |
| 416 |
delete [] b; |
| 417 |
delete [] z; |
| 418 |
} |
| 419 |
return 1; |
| 420 |
} |
| 421 |
|
| 422 |
|
| 423 |
typedef SquareMatrix<RealType, 6> Mat6x6d; |
| 424 |
} |
| 425 |
#endif //MATH_SQUAREMATRIX_HPP |
| 426 |
|
