1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2012 Dsir Nuentsa-Wakam <desire.nuentsa_wakam (at) inria.fr> 5 // 6 // This Source Code Form is subject to the terms of the Mozilla 7 // Public License v. 2.0. If a copy of the MPL was not distributed 8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9 10 #ifndef EIGEN_INCOMPLETE_LUT_H 11 #define EIGEN_INCOMPLETE_LUT_H 12 13 14 namespace Eigen { 15 16 namespace internal { 17 18 /** \internal 19 * Compute a quick-sort split of a vector 20 * On output, the vector row is permuted such that its elements satisfy 21 * abs(row(i)) >= abs(row(ncut)) if i<ncut 22 * abs(row(i)) <= abs(row(ncut)) if i>ncut 23 * \param row The vector of values 24 * \param ind The array of index for the elements in @p row 25 * \param ncut The number of largest elements to keep 26 **/ 27 template <typename VectorV, typename VectorI, typename Index> 28 Index QuickSplit(VectorV &row, VectorI &ind, Index ncut) 29 { 30 typedef typename VectorV::RealScalar RealScalar; 31 using std::swap; 32 using std::abs; 33 Index mid; 34 Index n = row.size(); /* length of the vector */ 35 Index first, last ; 36 37 ncut--; /* to fit the zero-based indices */ 38 first = 0; 39 last = n-1; 40 if (ncut < first || ncut > last ) return 0; 41 42 do { 43 mid = first; 44 RealScalar abskey = abs(row(mid)); 45 for (Index j = first + 1; j <= last; j++) { 46 if ( abs(row(j)) > abskey) { 47 ++mid; 48 swap(row(mid), row(j)); 49 swap(ind(mid), ind(j)); 50 } 51 } 52 /* Interchange for the pivot element */ 53 swap(row(mid), row(first)); 54 swap(ind(mid), ind(first)); 55 56 if (mid > ncut) last = mid - 1; 57 else if (mid < ncut ) first = mid + 1; 58 } while (mid != ncut ); 59 60 return 0; /* mid is equal to ncut */ 61 } 62 63 }// end namespace internal 64 65 /** \ingroup IterativeLinearSolvers_Module 66 * \class IncompleteLUT 67 * \brief Incomplete LU factorization with dual-threshold strategy 68 * 69 * During the numerical factorization, two dropping rules are used : 70 * 1) any element whose magnitude is less than some tolerance is dropped. 71 * This tolerance is obtained by multiplying the input tolerance @p droptol 72 * by the average magnitude of all the original elements in the current row. 73 * 2) After the elimination of the row, only the @p fill largest elements in 74 * the L part and the @p fill largest elements in the U part are kept 75 * (in addition to the diagonal element ). Note that @p fill is computed from 76 * the input parameter @p fillfactor which is used the ratio to control the fill_in 77 * relatively to the initial number of nonzero elements. 78 * 79 * The two extreme cases are when @p droptol=0 (to keep all the @p fill*2 largest elements) 80 * and when @p fill=n/2 with @p droptol being different to zero. 81 * 82 * References : Yousef Saad, ILUT: A dual threshold incomplete LU factorization, 83 * Numerical Linear Algebra with Applications, 1(4), pp 387-402, 1994. 84 * 85 * NOTE : The following implementation is derived from the ILUT implementation 86 * in the SPARSKIT package, Copyright (C) 2005, the Regents of the University of Minnesota 87 * released under the terms of the GNU LGPL: 88 * http://www-users.cs.umn.edu/~saad/software/SPARSKIT/README 89 * However, Yousef Saad gave us permission to relicense his ILUT code to MPL2. 90 * See the Eigen mailing list archive, thread: ILUT, date: July 8, 2012: 91 * http://listengine.tuxfamily.org/lists.tuxfamily.org/eigen/2012/07/msg00064.html 92 * alternatively, on GMANE: 93 * http://comments.gmane.org/gmane.comp.lib.eigen/3302 94 */ 95 template <typename _Scalar> 96 class IncompleteLUT : internal::noncopyable 97 { 98 typedef _Scalar Scalar; 99 typedef typename NumTraits<Scalar>::Real RealScalar; 100 typedef Matrix<Scalar,Dynamic,1> Vector; 101 typedef SparseMatrix<Scalar,RowMajor> FactorType; 102 typedef SparseMatrix<Scalar,ColMajor> PermutType; 103 typedef typename FactorType::Index Index; 104 105 public: 106 typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType; 107 108 IncompleteLUT() 109 : m_droptol(NumTraits<Scalar>::dummy_precision()), m_fillfactor(10), 110 m_analysisIsOk(false), m_factorizationIsOk(false), m_isInitialized(false) 111 {} 112 113 template<typename MatrixType> 114 IncompleteLUT(const MatrixType& mat, const RealScalar& droptol=NumTraits<Scalar>::dummy_precision(), int fillfactor = 10) 115 : m_droptol(droptol),m_fillfactor(fillfactor), 116 m_analysisIsOk(false),m_factorizationIsOk(false),m_isInitialized(false) 117 { 118 eigen_assert(fillfactor != 0); 119 compute(mat); 120 } 121 122 Index rows() const { return m_lu.rows(); } 123 124 Index cols() const { return m_lu.cols(); } 125 126 /** \brief Reports whether previous computation was successful. 127 * 128 * \returns \c Success if computation was succesful, 129 * \c NumericalIssue if the matrix.appears to be negative. 130 */ 131 ComputationInfo info() const 132 { 133 eigen_assert(m_isInitialized && "IncompleteLUT is not initialized."); 134 return m_info; 135 } 136 137 template<typename MatrixType> 138 void analyzePattern(const MatrixType& amat); 139 140 template<typename MatrixType> 141 void factorize(const MatrixType& amat); 142 143 /** 144 * Compute an incomplete LU factorization with dual threshold on the matrix mat 145 * No pivoting is done in this version 146 * 147 **/ 148 template<typename MatrixType> 149 IncompleteLUT<Scalar>& compute(const MatrixType& amat) 150 { 151 analyzePattern(amat); 152 factorize(amat); 153 m_isInitialized = m_factorizationIsOk; 154 return *this; 155 } 156 157 void setDroptol(const RealScalar& droptol); 158 void setFillfactor(int fillfactor); 159 160 template<typename Rhs, typename Dest> 161 void _solve(const Rhs& b, Dest& x) const 162 { 163 x = m_Pinv * b; 164 x = m_lu.template triangularView<UnitLower>().solve(x); 165 x = m_lu.template triangularView<Upper>().solve(x); 166 x = m_P * x; 167 } 168 169 template<typename Rhs> inline const internal::solve_retval<IncompleteLUT, Rhs> 170 solve(const MatrixBase<Rhs>& b) const 171 { 172 eigen_assert(m_isInitialized && "IncompleteLUT is not initialized."); 173 eigen_assert(cols()==b.rows() 174 && "IncompleteLUT::solve(): invalid number of rows of the right hand side matrix b"); 175 return internal::solve_retval<IncompleteLUT, Rhs>(*this, b.derived()); 176 } 177 178 protected: 179 180 /** keeps off-diagonal entries; drops diagonal entries */ 181 struct keep_diag { 182 inline bool operator() (const Index& row, const Index& col, const Scalar&) const 183 { 184 return row!=col; 185 } 186 }; 187 188 protected: 189 190 FactorType m_lu; 191 RealScalar m_droptol; 192 int m_fillfactor; 193 bool m_analysisIsOk; 194 bool m_factorizationIsOk; 195 bool m_isInitialized; 196 ComputationInfo m_info; 197 PermutationMatrix<Dynamic,Dynamic,Index> m_P; // Fill-reducing permutation 198 PermutationMatrix<Dynamic,Dynamic,Index> m_Pinv; // Inverse permutation 199 }; 200 201 /** 202 * Set control parameter droptol 203 * \param droptol Drop any element whose magnitude is less than this tolerance 204 **/ 205 template<typename Scalar> 206 void IncompleteLUT<Scalar>::setDroptol(const RealScalar& droptol) 207 { 208 this->m_droptol = droptol; 209 } 210 211 /** 212 * Set control parameter fillfactor 213 * \param fillfactor This is used to compute the number @p fill_in of largest elements to keep on each row. 214 **/ 215 template<typename Scalar> 216 void IncompleteLUT<Scalar>::setFillfactor(int fillfactor) 217 { 218 this->m_fillfactor = fillfactor; 219 } 220 221 template <typename Scalar> 222 template<typename _MatrixType> 223 void IncompleteLUT<Scalar>::analyzePattern(const _MatrixType& amat) 224 { 225 // Compute the Fill-reducing permutation 226 SparseMatrix<Scalar,ColMajor, Index> mat1 = amat; 227 SparseMatrix<Scalar,ColMajor, Index> mat2 = amat.transpose(); 228 // Symmetrize the pattern 229 // FIXME for a matrix with nearly symmetric pattern, mat2+mat1 is the appropriate choice. 230 // on the other hand for a really non-symmetric pattern, mat2*mat1 should be prefered... 231 SparseMatrix<Scalar,ColMajor, Index> AtA = mat2 + mat1; 232 AtA.prune(keep_diag()); 233 internal::minimum_degree_ordering<Scalar, Index>(AtA, m_P); // Then compute the AMD ordering... 234 235 m_Pinv = m_P.inverse(); // ... and the inverse permutation 236 237 m_analysisIsOk = true; 238 } 239 240 template <typename Scalar> 241 template<typename _MatrixType> 242 void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat) 243 { 244 using std::sqrt; 245 using std::swap; 246 using std::abs; 247 248 eigen_assert((amat.rows() == amat.cols()) && "The factorization should be done on a square matrix"); 249 Index n = amat.cols(); // Size of the matrix 250 m_lu.resize(n,n); 251 // Declare Working vectors and variables 252 Vector u(n) ; // real values of the row -- maximum size is n -- 253 VectorXi ju(n); // column position of the values in u -- maximum size is n 254 VectorXi jr(n); // Indicate the position of the nonzero elements in the vector u -- A zero location is indicated by -1 255 256 // Apply the fill-reducing permutation 257 eigen_assert(m_analysisIsOk && "You must first call analyzePattern()"); 258 SparseMatrix<Scalar,RowMajor, Index> mat; 259 mat = amat.twistedBy(m_Pinv); 260 261 // Initialization 262 jr.fill(-1); 263 ju.fill(0); 264 u.fill(0); 265 266 // number of largest elements to keep in each row: 267 Index fill_in = static_cast<Index> (amat.nonZeros()*m_fillfactor)/n+1; 268 if (fill_in > n) fill_in = n; 269 270 // number of largest nonzero elements to keep in the L and the U part of the current row: 271 Index nnzL = fill_in/2; 272 Index nnzU = nnzL; 273 m_lu.reserve(n * (nnzL + nnzU + 1)); 274 275 // global loop over the rows of the sparse matrix 276 for (Index ii = 0; ii < n; ii++) 277 { 278 // 1 - copy the lower and the upper part of the row i of mat in the working vector u 279 280 Index sizeu = 1; // number of nonzero elements in the upper part of the current row 281 Index sizel = 0; // number of nonzero elements in the lower part of the current row 282 ju(ii) = ii; 283 u(ii) = 0; 284 jr(ii) = ii; 285 RealScalar rownorm = 0; 286 287 typename FactorType::InnerIterator j_it(mat, ii); // Iterate through the current row ii 288 for (; j_it; ++j_it) 289 { 290 Index k = j_it.index(); 291 if (k < ii) 292 { 293 // copy the lower part 294 ju(sizel) = k; 295 u(sizel) = j_it.value(); 296 jr(k) = sizel; 297 ++sizel; 298 } 299 else if (k == ii) 300 { 301 u(ii) = j_it.value(); 302 } 303 else 304 { 305 // copy the upper part 306 Index jpos = ii + sizeu; 307 ju(jpos) = k; 308 u(jpos) = j_it.value(); 309 jr(k) = jpos; 310 ++sizeu; 311 } 312 rownorm += numext::abs2(j_it.value()); 313 } 314 315 // 2 - detect possible zero row 316 if(rownorm==0) 317 { 318 m_info = NumericalIssue; 319 return; 320 } 321 // Take the 2-norm of the current row as a relative tolerance 322 rownorm = sqrt(rownorm); 323 324 // 3 - eliminate the previous nonzero rows 325 Index jj = 0; 326 Index len = 0; 327 while (jj < sizel) 328 { 329 // In order to eliminate in the correct order, 330 // we must select first the smallest column index among ju(jj:sizel) 331 Index k; 332 Index minrow = ju.segment(jj,sizel-jj).minCoeff(&k); // k is relative to the segment 333 k += jj; 334 if (minrow != ju(jj)) 335 { 336 // swap the two locations 337 Index j = ju(jj); 338 swap(ju(jj), ju(k)); 339 jr(minrow) = jj; jr(j) = k; 340 swap(u(jj), u(k)); 341 } 342 // Reset this location 343 jr(minrow) = -1; 344 345 // Start elimination 346 typename FactorType::InnerIterator ki_it(m_lu, minrow); 347 while (ki_it && ki_it.index() < minrow) ++ki_it; 348 eigen_internal_assert(ki_it && ki_it.col()==minrow); 349 Scalar fact = u(jj) / ki_it.value(); 350 351 // drop too small elements 352 if(abs(fact) <= m_droptol) 353 { 354 jj++; 355 continue; 356 } 357 358 // linear combination of the current row ii and the row minrow 359 ++ki_it; 360 for (; ki_it; ++ki_it) 361 { 362 Scalar prod = fact * ki_it.value(); 363 Index j = ki_it.index(); 364 Index jpos = jr(j); 365 if (jpos == -1) // fill-in element 366 { 367 Index newpos; 368 if (j >= ii) // dealing with the upper part 369 { 370 newpos = ii + sizeu; 371 sizeu++; 372 eigen_internal_assert(sizeu<=n); 373 } 374 else // dealing with the lower part 375 { 376 newpos = sizel; 377 sizel++; 378 eigen_internal_assert(sizel<=ii); 379 } 380 ju(newpos) = j; 381 u(newpos) = -prod; 382 jr(j) = newpos; 383 } 384 else 385 u(jpos) -= prod; 386 } 387 // store the pivot element 388 u(len) = fact; 389 ju(len) = minrow; 390 ++len; 391 392 jj++; 393 } // end of the elimination on the row ii 394 395 // reset the upper part of the pointer jr to zero 396 for(Index k = 0; k <sizeu; k++) jr(ju(ii+k)) = -1; 397 398 // 4 - partially sort and insert the elements in the m_lu matrix 399 400 // sort the L-part of the row 401 sizel = len; 402 len = (std::min)(sizel, nnzL); 403 typename Vector::SegmentReturnType ul(u.segment(0, sizel)); 404 typename VectorXi::SegmentReturnType jul(ju.segment(0, sizel)); 405 internal::QuickSplit(ul, jul, len); 406 407 // store the largest m_fill elements of the L part 408 m_lu.startVec(ii); 409 for(Index k = 0; k < len; k++) 410 m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k); 411 412 // store the diagonal element 413 // apply a shifting rule to avoid zero pivots (we are doing an incomplete factorization) 414 if (u(ii) == Scalar(0)) 415 u(ii) = sqrt(m_droptol) * rownorm; 416 m_lu.insertBackByOuterInnerUnordered(ii, ii) = u(ii); 417 418 // sort the U-part of the row 419 // apply the dropping rule first 420 len = 0; 421 for(Index k = 1; k < sizeu; k++) 422 { 423 if(abs(u(ii+k)) > m_droptol * rownorm ) 424 { 425 ++len; 426 u(ii + len) = u(ii + k); 427 ju(ii + len) = ju(ii + k); 428 } 429 } 430 sizeu = len + 1; // +1 to take into account the diagonal element 431 len = (std::min)(sizeu, nnzU); 432 typename Vector::SegmentReturnType uu(u.segment(ii+1, sizeu-1)); 433 typename VectorXi::SegmentReturnType juu(ju.segment(ii+1, sizeu-1)); 434 internal::QuickSplit(uu, juu, len); 435 436 // store the largest elements of the U part 437 for(Index k = ii + 1; k < ii + len; k++) 438 m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k); 439 } 440 441 m_lu.finalize(); 442 m_lu.makeCompressed(); 443 444 m_factorizationIsOk = true; 445 m_info = Success; 446 } 447 448 namespace internal { 449 450 template<typename _MatrixType, typename Rhs> 451 struct solve_retval<IncompleteLUT<_MatrixType>, Rhs> 452 : solve_retval_base<IncompleteLUT<_MatrixType>, Rhs> 453 { 454 typedef IncompleteLUT<_MatrixType> Dec; 455 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 456 457 template<typename Dest> void evalTo(Dest& dst) const 458 { 459 dec()._solve(rhs(),dst); 460 } 461 }; 462 463 } // end namespace internal 464 465 } // end namespace Eigen 466 467 #endif // EIGEN_INCOMPLETE_LUT_H 468
