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 // Copyright (C) 2015 Gael Guennebaud <gael.guennebaud (at) inria.fr> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 #ifndef EIGEN_INCOMPLETE_CHOlESKY_H 12 #define EIGEN_INCOMPLETE_CHOlESKY_H 13 14 #include <vector> 15 #include <list> 16 17 namespace Eigen { 18 /** 19 * \brief Modified Incomplete Cholesky with dual threshold 20 * 21 * References : C-J. Lin and J. J. Mor, Incomplete Cholesky Factorizations with 22 * Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999 23 * 24 * \tparam Scalar the scalar type of the input matrices 25 * \tparam _UpLo The triangular part that will be used for the computations. It can be Lower 26 * or Upper. Default is Lower. 27 * \tparam _OrderingType The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<int>, 28 * unless EIGEN_MPL2_ONLY is defined, in which case the default is NaturalOrdering<int>. 29 * 30 * \implsparsesolverconcept 31 * 32 * It performs the following incomplete factorization: \f$ S P A P' S \approx L L' \f$ 33 * where L is a lower triangular factor, S is a diagonal scaling matrix, and P is a 34 * fill-in reducing permutation as computed by the ordering method. 35 * 36 * \b Shifting \b strategy: Let \f$ B = S P A P' S \f$ be the scaled matrix on which the factorization is carried out, 37 * and \f$ \beta \f$ be the minimum value of the diagonal. If \f$ \beta > 0 \f$ then, the factorization is directly performed 38 * on the matrix B. Otherwise, the factorization is performed on the shifted matrix \f$ B + (\sigma+|\beta| I \f$ where 39 * \f$ \sigma \f$ is the initial shift value as returned and set by setInitialShift() method. The default value is \f$ \sigma = 10^{-3} \f$. 40 * If the factorization fails, then the shift in doubled until it succeed or a maximum of ten attempts. If it still fails, as returned by 41 * the info() method, then you can either increase the initial shift, or better use another preconditioning technique. 42 * 43 */ 44 template <typename Scalar, int _UpLo = Lower, typename _OrderingType = 45 #ifndef EIGEN_MPL2_ONLY 46 AMDOrdering<int> 47 #else 48 NaturalOrdering<int> 49 #endif 50 > 51 class IncompleteCholesky : public SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> > 52 { 53 protected: 54 typedef SparseSolverBase<IncompleteCholesky<Scalar,_UpLo,_OrderingType> > Base; 55 using Base::m_isInitialized; 56 public: 57 typedef typename NumTraits<Scalar>::Real RealScalar; 58 typedef _OrderingType OrderingType; 59 typedef typename OrderingType::PermutationType PermutationType; 60 typedef typename PermutationType::StorageIndex StorageIndex; 61 typedef SparseMatrix<Scalar,ColMajor,StorageIndex> FactorType; 62 typedef Matrix<Scalar,Dynamic,1> VectorSx; 63 typedef Matrix<RealScalar,Dynamic,1> VectorRx; 64 typedef Matrix<StorageIndex,Dynamic, 1> VectorIx; 65 typedef std::vector<std::list<StorageIndex> > VectorList; 66 enum { UpLo = _UpLo }; 67 enum { 68 ColsAtCompileTime = Dynamic, 69 MaxColsAtCompileTime = Dynamic 70 }; 71 public: 72 73 /** Default constructor leaving the object in a partly non-initialized stage. 74 * 75 * You must call compute() or the pair analyzePattern()/factorize() to make it valid. 76 * 77 * \sa IncompleteCholesky(const MatrixType&) 78 */ 79 IncompleteCholesky() : m_initialShift(1e-3),m_factorizationIsOk(false) {} 80 81 /** Constructor computing the incomplete factorization for the given matrix \a matrix. 82 */ 83 template<typename MatrixType> 84 IncompleteCholesky(const MatrixType& matrix) : m_initialShift(1e-3),m_factorizationIsOk(false) 85 { 86 compute(matrix); 87 } 88 89 /** \returns number of rows of the factored matrix */ 90 Index rows() const { return m_L.rows(); } 91 92 /** \returns number of columns of the factored matrix */ 93 Index cols() const { return m_L.cols(); } 94 95 96 /** \brief Reports whether previous computation was successful. 97 * 98 * It triggers an assertion if \c *this has not been initialized through the respective constructor, 99 * or a call to compute() or analyzePattern(). 100 * 101 * \returns \c Success if computation was successful, 102 * \c NumericalIssue if the matrix appears to be negative. 103 */ 104 ComputationInfo info() const 105 { 106 eigen_assert(m_isInitialized && "IncompleteCholesky is not initialized."); 107 return m_info; 108 } 109 110 /** \brief Set the initial shift parameter \f$ \sigma \f$. 111 */ 112 void setInitialShift(RealScalar shift) { m_initialShift = shift; } 113 114 /** \brief Computes the fill reducing permutation vector using the sparsity pattern of \a mat 115 */ 116 template<typename MatrixType> 117 void analyzePattern(const MatrixType& mat) 118 { 119 OrderingType ord; 120 PermutationType pinv; 121 ord(mat.template selfadjointView<UpLo>(), pinv); 122 if(pinv.size()>0) m_perm = pinv.inverse(); 123 else m_perm.resize(0); 124 m_L.resize(mat.rows(), mat.cols()); 125 m_analysisIsOk = true; 126 m_isInitialized = true; 127 m_info = Success; 128 } 129 130 /** \brief Performs the numerical factorization of the input matrix \a mat 131 * 132 * The method analyzePattern() or compute() must have been called beforehand 133 * with a matrix having the same pattern. 134 * 135 * \sa compute(), analyzePattern() 136 */ 137 template<typename MatrixType> 138 void factorize(const MatrixType& mat); 139 140 /** Computes or re-computes the incomplete Cholesky factorization of the input matrix \a mat 141 * 142 * It is a shortcut for a sequential call to the analyzePattern() and factorize() methods. 143 * 144 * \sa analyzePattern(), factorize() 145 */ 146 template<typename MatrixType> 147 void compute(const MatrixType& mat) 148 { 149 analyzePattern(mat); 150 factorize(mat); 151 } 152 153 // internal 154 template<typename Rhs, typename Dest> 155 void _solve_impl(const Rhs& b, Dest& x) const 156 { 157 eigen_assert(m_factorizationIsOk && "factorize() should be called first"); 158 if (m_perm.rows() == b.rows()) x = m_perm * b; 159 else x = b; 160 x = m_scale.asDiagonal() * x; 161 x = m_L.template triangularView<Lower>().solve(x); 162 x = m_L.adjoint().template triangularView<Upper>().solve(x); 163 x = m_scale.asDiagonal() * x; 164 if (m_perm.rows() == b.rows()) 165 x = m_perm.inverse() * x; 166 } 167 168 /** \returns the sparse lower triangular factor L */ 169 const FactorType& matrixL() const { eigen_assert("m_factorizationIsOk"); return m_L; } 170 171 /** \returns a vector representing the scaling factor S */ 172 const VectorRx& scalingS() const { eigen_assert("m_factorizationIsOk"); return m_scale; } 173 174 /** \returns the fill-in reducing permutation P (can be empty for a natural ordering) */ 175 const PermutationType& permutationP() const { eigen_assert("m_analysisIsOk"); return m_perm; } 176 177 protected: 178 FactorType m_L; // The lower part stored in CSC 179 VectorRx m_scale; // The vector for scaling the matrix 180 RealScalar m_initialShift; // The initial shift parameter 181 bool m_analysisIsOk; 182 bool m_factorizationIsOk; 183 ComputationInfo m_info; 184 PermutationType m_perm; 185 186 private: 187 inline void updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol); 188 }; 189 190 // Based on the following paper: 191 // C-J. Lin and J. J. Mor, Incomplete Cholesky Factorizations with 192 // Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999 193 // http://ftp.mcs.anl.gov/pub/tech_reports/reports/P682.pdf 194 template<typename Scalar, int _UpLo, typename OrderingType> 195 template<typename _MatrixType> 196 void IncompleteCholesky<Scalar,_UpLo, OrderingType>::factorize(const _MatrixType& mat) 197 { 198 using std::sqrt; 199 eigen_assert(m_analysisIsOk && "analyzePattern() should be called first"); 200 201 // Dropping strategy : Keep only the p largest elements per column, where p is the number of elements in the column of the original matrix. Other strategies will be added 202 203 // Apply the fill-reducing permutation computed in analyzePattern() 204 if (m_perm.rows() == mat.rows() ) // To detect the null permutation 205 { 206 // The temporary is needed to make sure that the diagonal entry is properly sorted 207 FactorType tmp(mat.rows(), mat.cols()); 208 tmp = mat.template selfadjointView<_UpLo>().twistedBy(m_perm); 209 m_L.template selfadjointView<Lower>() = tmp.template selfadjointView<Lower>(); 210 } 211 else 212 { 213 m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>(); 214 } 215 216 Index n = m_L.cols(); 217 Index nnz = m_L.nonZeros(); 218 Map<VectorSx> vals(m_L.valuePtr(), nnz); //values 219 Map<VectorIx> rowIdx(m_L.innerIndexPtr(), nnz); //Row indices 220 Map<VectorIx> colPtr( m_L.outerIndexPtr(), n+1); // Pointer to the beginning of each row 221 VectorIx firstElt(n-1); // for each j, points to the next entry in vals that will be used in the factorization 222 VectorList listCol(n); // listCol(j) is a linked list of columns to update column j 223 VectorSx col_vals(n); // Store a nonzero values in each column 224 VectorIx col_irow(n); // Row indices of nonzero elements in each column 225 VectorIx col_pattern(n); 226 col_pattern.fill(-1); 227 StorageIndex col_nnz; 228 229 230 // Computes the scaling factors 231 m_scale.resize(n); 232 m_scale.setZero(); 233 for (Index j = 0; j < n; j++) 234 for (Index k = colPtr[j]; k < colPtr[j+1]; k++) 235 { 236 m_scale(j) += numext::abs2(vals(k)); 237 if(rowIdx[k]!=j) 238 m_scale(rowIdx[k]) += numext::abs2(vals(k)); 239 } 240 241 m_scale = m_scale.cwiseSqrt().cwiseSqrt(); 242 243 for (Index j = 0; j < n; ++j) 244 if(m_scale(j)>(std::numeric_limits<RealScalar>::min)()) 245 m_scale(j) = RealScalar(1)/m_scale(j); 246 else 247 m_scale(j) = 1; 248 249 // TODO disable scaling if not needed, i.e., if it is roughly uniform? (this will make solve() faster) 250 251 // Scale and compute the shift for the matrix 252 RealScalar mindiag = NumTraits<RealScalar>::highest(); 253 for (Index j = 0; j < n; j++) 254 { 255 for (Index k = colPtr[j]; k < colPtr[j+1]; k++) 256 vals[k] *= (m_scale(j)*m_scale(rowIdx[k])); 257 eigen_internal_assert(rowIdx[colPtr[j]]==j && "IncompleteCholesky: only the lower triangular part must be stored"); 258 mindiag = numext::mini(numext::real(vals[colPtr[j]]), mindiag); 259 } 260 261 FactorType L_save = m_L; 262 263 RealScalar shift = 0; 264 if(mindiag <= RealScalar(0.)) 265 shift = m_initialShift - mindiag; 266 267 m_info = NumericalIssue; 268 269 // Try to perform the incomplete factorization using the current shift 270 int iter = 0; 271 do 272 { 273 // Apply the shift to the diagonal elements of the matrix 274 for (Index j = 0; j < n; j++) 275 vals[colPtr[j]] += shift; 276 277 // jki version of the Cholesky factorization 278 Index j=0; 279 for (; j < n; ++j) 280 { 281 // Left-looking factorization of the j-th column 282 // First, load the j-th column into col_vals 283 Scalar diag = vals[colPtr[j]]; // It is assumed that only the lower part is stored 284 col_nnz = 0; 285 for (Index i = colPtr[j] + 1; i < colPtr[j+1]; i++) 286 { 287 StorageIndex l = rowIdx[i]; 288 col_vals(col_nnz) = vals[i]; 289 col_irow(col_nnz) = l; 290 col_pattern(l) = col_nnz; 291 col_nnz++; 292 } 293 { 294 typename std::list<StorageIndex>::iterator k; 295 // Browse all previous columns that will update column j 296 for(k = listCol[j].begin(); k != listCol[j].end(); k++) 297 { 298 Index jk = firstElt(*k); // First element to use in the column 299 eigen_internal_assert(rowIdx[jk]==j); 300 Scalar v_j_jk = numext::conj(vals[jk]); 301 302 jk += 1; 303 for (Index i = jk; i < colPtr[*k+1]; i++) 304 { 305 StorageIndex l = rowIdx[i]; 306 if(col_pattern[l]<0) 307 { 308 col_vals(col_nnz) = vals[i] * v_j_jk; 309 col_irow[col_nnz] = l; 310 col_pattern(l) = col_nnz; 311 col_nnz++; 312 } 313 else 314 col_vals(col_pattern[l]) -= vals[i] * v_j_jk; 315 } 316 updateList(colPtr,rowIdx,vals, *k, jk, firstElt, listCol); 317 } 318 } 319 320 // Scale the current column 321 if(numext::real(diag) <= 0) 322 { 323 if(++iter>=10) 324 return; 325 326 // increase shift 327 shift = numext::maxi(m_initialShift,RealScalar(2)*shift); 328 // restore m_L, col_pattern, and listCol 329 vals = Map<const VectorSx>(L_save.valuePtr(), nnz); 330 rowIdx = Map<const VectorIx>(L_save.innerIndexPtr(), nnz); 331 colPtr = Map<const VectorIx>(L_save.outerIndexPtr(), n+1); 332 col_pattern.fill(-1); 333 for(Index i=0; i<n; ++i) 334 listCol[i].clear(); 335 336 break; 337 } 338 339 RealScalar rdiag = sqrt(numext::real(diag)); 340 vals[colPtr[j]] = rdiag; 341 for (Index k = 0; k<col_nnz; ++k) 342 { 343 Index i = col_irow[k]; 344 //Scale 345 col_vals(k) /= rdiag; 346 //Update the remaining diagonals with col_vals 347 vals[colPtr[i]] -= numext::abs2(col_vals(k)); 348 } 349 // Select the largest p elements 350 // p is the original number of elements in the column (without the diagonal) 351 Index p = colPtr[j+1] - colPtr[j] - 1 ; 352 Ref<VectorSx> cvals = col_vals.head(col_nnz); 353 Ref<VectorIx> cirow = col_irow.head(col_nnz); 354 internal::QuickSplit(cvals,cirow, p); 355 // Insert the largest p elements in the matrix 356 Index cpt = 0; 357 for (Index i = colPtr[j]+1; i < colPtr[j+1]; i++) 358 { 359 vals[i] = col_vals(cpt); 360 rowIdx[i] = col_irow(cpt); 361 // restore col_pattern: 362 col_pattern(col_irow(cpt)) = -1; 363 cpt++; 364 } 365 // Get the first smallest row index and put it after the diagonal element 366 Index jk = colPtr(j)+1; 367 updateList(colPtr,rowIdx,vals,j,jk,firstElt,listCol); 368 } 369 370 if(j==n) 371 { 372 m_factorizationIsOk = true; 373 m_info = Success; 374 } 375 } while(m_info!=Success); 376 } 377 378 template<typename Scalar, int _UpLo, typename OrderingType> 379 inline void IncompleteCholesky<Scalar,_UpLo, OrderingType>::updateList(Ref<const VectorIx> colPtr, Ref<VectorIx> rowIdx, Ref<VectorSx> vals, const Index& col, const Index& jk, VectorIx& firstElt, VectorList& listCol) 380 { 381 if (jk < colPtr(col+1) ) 382 { 383 Index p = colPtr(col+1) - jk; 384 Index minpos; 385 rowIdx.segment(jk,p).minCoeff(&minpos); 386 minpos += jk; 387 if (rowIdx(minpos) != rowIdx(jk)) 388 { 389 //Swap 390 std::swap(rowIdx(jk),rowIdx(minpos)); 391 std::swap(vals(jk),vals(minpos)); 392 } 393 firstElt(col) = internal::convert_index<StorageIndex,Index>(jk); 394 listCol[rowIdx(jk)].push_back(internal::convert_index<StorageIndex,Index>(col)); 395 } 396 } 397 398 } // end namespace Eigen 399 400 #endif 401
