Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 1 | // This file is part of Eigen, a lightweight C++ template library |
| 2 | // for linear algebra. |
| 3 | // |
| 4 | // Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@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_DGMRES_H |
| 11 | #define EIGEN_DGMRES_H |
| 12 | |
| 13 | #include <Eigen/Eigenvalues> |
| 14 | |
| 15 | namespace Eigen { |
| 16 | |
| 17 | template< typename _MatrixType, |
| 18 | typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> > |
| 19 | class DGMRES; |
| 20 | |
| 21 | namespace internal { |
| 22 | |
| 23 | template< typename _MatrixType, typename _Preconditioner> |
| 24 | struct traits<DGMRES<_MatrixType,_Preconditioner> > |
| 25 | { |
| 26 | typedef _MatrixType MatrixType; |
| 27 | typedef _Preconditioner Preconditioner; |
| 28 | }; |
| 29 | |
| 30 | /** \brief Computes a permutation vector to have a sorted sequence |
| 31 | * \param vec The vector to reorder. |
| 32 | * \param perm gives the sorted sequence on output. Must be initialized with 0..n-1 |
| 33 | * \param ncut Put the ncut smallest elements at the end of the vector |
| 34 | * WARNING This is an expensive sort, so should be used only |
| 35 | * for small size vectors |
| 36 | * TODO Use modified QuickSplit or std::nth_element to get the smallest values |
| 37 | */ |
| 38 | template <typename VectorType, typename IndexType> |
| 39 | void sortWithPermutation (VectorType& vec, IndexType& perm, typename IndexType::Scalar& ncut) |
| 40 | { |
| 41 | eigen_assert(vec.size() == perm.size()); |
| 42 | typedef typename IndexType::Scalar Index; |
| 43 | typedef typename VectorType::Scalar Scalar; |
| 44 | bool flag; |
| 45 | for (Index k = 0; k < ncut; k++) |
| 46 | { |
| 47 | flag = false; |
| 48 | for (Index j = 0; j < vec.size()-1; j++) |
| 49 | { |
| 50 | if ( vec(perm(j)) < vec(perm(j+1)) ) |
| 51 | { |
| 52 | std::swap(perm(j),perm(j+1)); |
| 53 | flag = true; |
| 54 | } |
| 55 | if (!flag) break; // The vector is in sorted order |
| 56 | } |
| 57 | } |
| 58 | } |
| 59 | |
| 60 | } |
| 61 | /** |
| 62 | * \ingroup IterativeLInearSolvers_Module |
| 63 | * \brief A Restarted GMRES with deflation. |
| 64 | * This class implements a modification of the GMRES solver for |
| 65 | * sparse linear systems. The basis is built with modified |
| 66 | * Gram-Schmidt. At each restart, a few approximated eigenvectors |
| 67 | * corresponding to the smallest eigenvalues are used to build a |
| 68 | * preconditioner for the next cycle. This preconditioner |
| 69 | * for deflation can be combined with any other preconditioner, |
| 70 | * the IncompleteLUT for instance. The preconditioner is applied |
| 71 | * at right of the matrix and the combination is multiplicative. |
| 72 | * |
| 73 | * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. |
| 74 | * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner |
| 75 | * Typical usage : |
| 76 | * \code |
| 77 | * SparseMatrix<double> A; |
| 78 | * VectorXd x, b; |
| 79 | * //Fill A and b ... |
| 80 | * DGMRES<SparseMatrix<double> > solver; |
| 81 | * solver.set_restart(30); // Set restarting value |
| 82 | * solver.setEigenv(1); // Set the number of eigenvalues to deflate |
| 83 | * solver.compute(A); |
| 84 | * x = solver.solve(b); |
| 85 | * \endcode |
| 86 | * |
| 87 | * References : |
| 88 | * [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid |
| 89 | * Algebraic Solvers for Linear Systems Arising from Compressible |
| 90 | * Flows, Computers and Fluids, In Press, |
| 91 | * http://dx.doi.org/10.1016/j.compfluid.2012.03.023 |
| 92 | * [2] K. Burrage and J. Erhel, On the performance of various |
| 93 | * adaptive preconditioned GMRES strategies, 5(1998), 101-121. |
| 94 | * [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES |
| 95 | * preconditioned by deflation,J. Computational and Applied |
| 96 | * Mathematics, 69(1996), 303-318. |
| 97 | |
| 98 | * |
| 99 | */ |
| 100 | template< typename _MatrixType, typename _Preconditioner> |
| 101 | class DGMRES : public IterativeSolverBase<DGMRES<_MatrixType,_Preconditioner> > |
| 102 | { |
| 103 | typedef IterativeSolverBase<DGMRES> Base; |
| 104 | using Base::mp_matrix; |
| 105 | using Base::m_error; |
| 106 | using Base::m_iterations; |
| 107 | using Base::m_info; |
| 108 | using Base::m_isInitialized; |
| 109 | using Base::m_tolerance; |
| 110 | public: |
| 111 | typedef _MatrixType MatrixType; |
| 112 | typedef typename MatrixType::Scalar Scalar; |
| 113 | typedef typename MatrixType::Index Index; |
| 114 | typedef typename MatrixType::RealScalar RealScalar; |
| 115 | typedef _Preconditioner Preconditioner; |
| 116 | typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix; |
| 117 | typedef Matrix<RealScalar,Dynamic,Dynamic> DenseRealMatrix; |
| 118 | typedef Matrix<Scalar,Dynamic,1> DenseVector; |
| 119 | typedef Matrix<RealScalar,Dynamic,1> DenseRealVector; |
| 120 | typedef Matrix<std::complex<RealScalar>, Dynamic, 1> ComplexVector; |
| 121 | |
| 122 | |
| 123 | /** Default constructor. */ |
| 124 | DGMRES() : Base(),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {} |
| 125 | |
| 126 | /** Initialize the solver with matrix \a A for further \c Ax=b solving. |
| 127 | * |
| 128 | * This constructor is a shortcut for the default constructor followed |
| 129 | * by a call to compute(). |
| 130 | * |
| 131 | * \warning this class stores a reference to the matrix A as well as some |
| 132 | * precomputed values that depend on it. Therefore, if \a A is changed |
| 133 | * this class becomes invalid. Call compute() to update it with the new |
| 134 | * matrix A, or modify a copy of A. |
| 135 | */ |
| 136 | DGMRES(const MatrixType& A) : Base(A),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) |
| 137 | {} |
| 138 | |
| 139 | ~DGMRES() {} |
| 140 | |
| 141 | /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A |
| 142 | * \a x0 as an initial solution. |
| 143 | * |
| 144 | * \sa compute() |
| 145 | */ |
| 146 | template<typename Rhs,typename Guess> |
| 147 | inline const internal::solve_retval_with_guess<DGMRES, Rhs, Guess> |
| 148 | solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const |
| 149 | { |
| 150 | eigen_assert(m_isInitialized && "DGMRES is not initialized."); |
| 151 | eigen_assert(Base::rows()==b.rows() |
| 152 | && "DGMRES::solve(): invalid number of rows of the right hand side matrix b"); |
| 153 | return internal::solve_retval_with_guess |
| 154 | <DGMRES, Rhs, Guess>(*this, b.derived(), x0); |
| 155 | } |
| 156 | |
| 157 | /** \internal */ |
| 158 | template<typename Rhs,typename Dest> |
| 159 | void _solveWithGuess(const Rhs& b, Dest& x) const |
| 160 | { |
| 161 | bool failed = false; |
| 162 | for(int j=0; j<b.cols(); ++j) |
| 163 | { |
| 164 | m_iterations = Base::maxIterations(); |
| 165 | m_error = Base::m_tolerance; |
| 166 | |
| 167 | typename Dest::ColXpr xj(x,j); |
| 168 | dgmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner); |
| 169 | } |
| 170 | m_info = failed ? NumericalIssue |
| 171 | : m_error <= Base::m_tolerance ? Success |
| 172 | : NoConvergence; |
| 173 | m_isInitialized = true; |
| 174 | } |
| 175 | |
| 176 | /** \internal */ |
| 177 | template<typename Rhs,typename Dest> |
| 178 | void _solve(const Rhs& b, Dest& x) const |
| 179 | { |
| 180 | x = b; |
| 181 | _solveWithGuess(b,x); |
| 182 | } |
| 183 | /** |
| 184 | * Get the restart value |
| 185 | */ |
| 186 | int restart() { return m_restart; } |
| 187 | |
| 188 | /** |
| 189 | * Set the restart value (default is 30) |
| 190 | */ |
| 191 | void set_restart(const int restart) { m_restart=restart; } |
| 192 | |
| 193 | /** |
| 194 | * Set the number of eigenvalues to deflate at each restart |
| 195 | */ |
| 196 | void setEigenv(const int neig) |
| 197 | { |
| 198 | m_neig = neig; |
| 199 | if (neig+1 > m_maxNeig) m_maxNeig = neig+1; // To allow for complex conjugates |
| 200 | } |
| 201 | |
| 202 | /** |
| 203 | * Get the size of the deflation subspace size |
| 204 | */ |
| 205 | int deflSize() {return m_r; } |
| 206 | |
| 207 | /** |
| 208 | * Set the maximum size of the deflation subspace |
| 209 | */ |
| 210 | void setMaxEigenv(const int maxNeig) { m_maxNeig = maxNeig; } |
| 211 | |
| 212 | protected: |
| 213 | // DGMRES algorithm |
| 214 | template<typename Rhs, typename Dest> |
| 215 | void dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x, const Preconditioner& precond) const; |
| 216 | // Perform one cycle of GMRES |
| 217 | template<typename Dest> |
| 218 | int dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const; |
| 219 | // Compute data to use for deflation |
| 220 | int dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const; |
| 221 | // Apply deflation to a vector |
| 222 | template<typename RhsType, typename DestType> |
| 223 | int dgmresApplyDeflation(const RhsType& In, DestType& Out) const; |
| 224 | ComplexVector schurValues(const ComplexSchur<DenseMatrix>& schurofH) const; |
| 225 | ComplexVector schurValues(const RealSchur<DenseMatrix>& schurofH) const; |
| 226 | // Init data for deflation |
| 227 | void dgmresInitDeflation(Index& rows) const; |
| 228 | mutable DenseMatrix m_V; // Krylov basis vectors |
| 229 | mutable DenseMatrix m_H; // Hessenberg matrix |
| 230 | mutable DenseMatrix m_Hes; // Initial hessenberg matrix wihout Givens rotations applied |
| 231 | mutable Index m_restart; // Maximum size of the Krylov subspace |
| 232 | mutable DenseMatrix m_U; // Vectors that form the basis of the invariant subspace |
| 233 | mutable DenseMatrix m_MU; // matrix operator applied to m_U (for next cycles) |
| 234 | mutable DenseMatrix m_T; /* T=U^T*M^{-1}*A*U */ |
| 235 | mutable PartialPivLU<DenseMatrix> m_luT; // LU factorization of m_T |
| 236 | mutable int m_neig; //Number of eigenvalues to extract at each restart |
| 237 | mutable int m_r; // Current number of deflated eigenvalues, size of m_U |
| 238 | mutable int m_maxNeig; // Maximum number of eigenvalues to deflate |
| 239 | mutable RealScalar m_lambdaN; //Modulus of the largest eigenvalue of A |
| 240 | mutable bool m_isDeflAllocated; |
| 241 | mutable bool m_isDeflInitialized; |
| 242 | |
| 243 | //Adaptive strategy |
| 244 | mutable RealScalar m_smv; // Smaller multiple of the remaining number of steps allowed |
| 245 | mutable bool m_force; // Force the use of deflation at each restart |
| 246 | |
| 247 | }; |
| 248 | /** |
| 249 | * \brief Perform several cycles of restarted GMRES with modified Gram Schmidt, |
| 250 | * |
| 251 | * A right preconditioner is used combined with deflation. |
| 252 | * |
| 253 | */ |
| 254 | template< typename _MatrixType, typename _Preconditioner> |
| 255 | template<typename Rhs, typename Dest> |
| 256 | void DGMRES<_MatrixType, _Preconditioner>::dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x, |
| 257 | const Preconditioner& precond) const |
| 258 | { |
| 259 | //Initialization |
| 260 | int n = mat.rows(); |
| 261 | DenseVector r0(n); |
| 262 | int nbIts = 0; |
| 263 | m_H.resize(m_restart+1, m_restart); |
| 264 | m_Hes.resize(m_restart, m_restart); |
| 265 | m_V.resize(n,m_restart+1); |
| 266 | //Initial residual vector and intial norm |
| 267 | x = precond.solve(x); |
| 268 | r0 = rhs - mat * x; |
| 269 | RealScalar beta = r0.norm(); |
| 270 | RealScalar normRhs = rhs.norm(); |
| 271 | m_error = beta/normRhs; |
| 272 | if(m_error < m_tolerance) |
| 273 | m_info = Success; |
| 274 | else |
| 275 | m_info = NoConvergence; |
| 276 | |
| 277 | // Iterative process |
| 278 | while (nbIts < m_iterations && m_info == NoConvergence) |
| 279 | { |
| 280 | dgmresCycle(mat, precond, x, r0, beta, normRhs, nbIts); |
| 281 | |
| 282 | // Compute the new residual vector for the restart |
| 283 | if (nbIts < m_iterations && m_info == NoConvergence) |
| 284 | r0 = rhs - mat * x; |
| 285 | } |
| 286 | } |
| 287 | |
| 288 | /** |
| 289 | * \brief Perform one restart cycle of DGMRES |
| 290 | * \param mat The coefficient matrix |
| 291 | * \param precond The preconditioner |
| 292 | * \param x the new approximated solution |
| 293 | * \param r0 The initial residual vector |
| 294 | * \param beta The norm of the residual computed so far |
| 295 | * \param normRhs The norm of the right hand side vector |
| 296 | * \param nbIts The number of iterations |
| 297 | */ |
| 298 | template< typename _MatrixType, typename _Preconditioner> |
| 299 | template<typename Dest> |
| 300 | int DGMRES<_MatrixType, _Preconditioner>::dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const |
| 301 | { |
| 302 | //Initialization |
| 303 | DenseVector g(m_restart+1); // Right hand side of the least square problem |
| 304 | g.setZero(); |
| 305 | g(0) = Scalar(beta); |
| 306 | m_V.col(0) = r0/beta; |
| 307 | m_info = NoConvergence; |
| 308 | std::vector<JacobiRotation<Scalar> >gr(m_restart); // Givens rotations |
| 309 | int it = 0; // Number of inner iterations |
| 310 | int n = mat.rows(); |
| 311 | DenseVector tv1(n), tv2(n); //Temporary vectors |
| 312 | while (m_info == NoConvergence && it < m_restart && nbIts < m_iterations) |
| 313 | { |
| 314 | // Apply preconditioner(s) at right |
| 315 | if (m_isDeflInitialized ) |
| 316 | { |
| 317 | dgmresApplyDeflation(m_V.col(it), tv1); // Deflation |
| 318 | tv2 = precond.solve(tv1); |
| 319 | } |
| 320 | else |
| 321 | { |
| 322 | tv2 = precond.solve(m_V.col(it)); // User's selected preconditioner |
| 323 | } |
| 324 | tv1 = mat * tv2; |
| 325 | |
| 326 | // Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt |
| 327 | Scalar coef; |
| 328 | for (int i = 0; i <= it; ++i) |
| 329 | { |
| 330 | coef = tv1.dot(m_V.col(i)); |
| 331 | tv1 = tv1 - coef * m_V.col(i); |
| 332 | m_H(i,it) = coef; |
| 333 | m_Hes(i,it) = coef; |
| 334 | } |
| 335 | // Normalize the vector |
| 336 | coef = tv1.norm(); |
| 337 | m_V.col(it+1) = tv1/coef; |
| 338 | m_H(it+1, it) = coef; |
| 339 | // m_Hes(it+1,it) = coef; |
| 340 | |
| 341 | // FIXME Check for happy breakdown |
| 342 | |
| 343 | // Update Hessenberg matrix with Givens rotations |
| 344 | for (int i = 1; i <= it; ++i) |
| 345 | { |
| 346 | m_H.col(it).applyOnTheLeft(i-1,i,gr[i-1].adjoint()); |
| 347 | } |
| 348 | // Compute the new plane rotation |
| 349 | gr[it].makeGivens(m_H(it, it), m_H(it+1,it)); |
| 350 | // Apply the new rotation |
| 351 | m_H.col(it).applyOnTheLeft(it,it+1,gr[it].adjoint()); |
| 352 | g.applyOnTheLeft(it,it+1, gr[it].adjoint()); |
| 353 | |
| 354 | beta = std::abs(g(it+1)); |
| 355 | m_error = beta/normRhs; |
| 356 | std::cerr << nbIts << " Relative Residual Norm " << m_error << std::endl; |
| 357 | it++; nbIts++; |
| 358 | |
| 359 | if (m_error < m_tolerance) |
| 360 | { |
| 361 | // The method has converged |
| 362 | m_info = Success; |
| 363 | break; |
| 364 | } |
| 365 | } |
| 366 | |
| 367 | // Compute the new coefficients by solving the least square problem |
| 368 | // it++; |
| 369 | //FIXME Check first if the matrix is singular ... zero diagonal |
| 370 | DenseVector nrs(m_restart); |
| 371 | nrs = m_H.topLeftCorner(it,it).template triangularView<Upper>().solve(g.head(it)); |
| 372 | |
| 373 | // Form the new solution |
| 374 | if (m_isDeflInitialized) |
| 375 | { |
| 376 | tv1 = m_V.leftCols(it) * nrs; |
| 377 | dgmresApplyDeflation(tv1, tv2); |
| 378 | x = x + precond.solve(tv2); |
| 379 | } |
| 380 | else |
| 381 | x = x + precond.solve(m_V.leftCols(it) * nrs); |
| 382 | |
| 383 | // Go for a new cycle and compute data for deflation |
| 384 | if(nbIts < m_iterations && m_info == NoConvergence && m_neig > 0 && (m_r+m_neig) < m_maxNeig) |
| 385 | dgmresComputeDeflationData(mat, precond, it, m_neig); |
| 386 | return 0; |
| 387 | |
| 388 | } |
| 389 | |
| 390 | |
| 391 | template< typename _MatrixType, typename _Preconditioner> |
| 392 | void DGMRES<_MatrixType, _Preconditioner>::dgmresInitDeflation(Index& rows) const |
| 393 | { |
| 394 | m_U.resize(rows, m_maxNeig); |
| 395 | m_MU.resize(rows, m_maxNeig); |
| 396 | m_T.resize(m_maxNeig, m_maxNeig); |
| 397 | m_lambdaN = 0.0; |
| 398 | m_isDeflAllocated = true; |
| 399 | } |
| 400 | |
| 401 | template< typename _MatrixType, typename _Preconditioner> |
| 402 | inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const ComplexSchur<DenseMatrix>& schurofH) const |
| 403 | { |
| 404 | return schurofH.matrixT().diagonal(); |
| 405 | } |
| 406 | |
| 407 | template< typename _MatrixType, typename _Preconditioner> |
| 408 | inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const RealSchur<DenseMatrix>& schurofH) const |
| 409 | { |
| 410 | typedef typename MatrixType::Index Index; |
| 411 | const DenseMatrix& T = schurofH.matrixT(); |
| 412 | Index it = T.rows(); |
| 413 | ComplexVector eig(it); |
| 414 | Index j = 0; |
| 415 | while (j < it-1) |
| 416 | { |
| 417 | if (T(j+1,j) ==Scalar(0)) |
| 418 | { |
| 419 | eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0)); |
| 420 | j++; |
| 421 | } |
| 422 | else |
| 423 | { |
| 424 | eig(j) = std::complex<RealScalar>(T(j,j),T(j+1,j)); |
| 425 | eig(j+1) = std::complex<RealScalar>(T(j,j+1),T(j+1,j+1)); |
| 426 | j++; |
| 427 | } |
| 428 | } |
| 429 | if (j < it-1) eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0)); |
| 430 | return eig; |
| 431 | } |
| 432 | |
| 433 | template< typename _MatrixType, typename _Preconditioner> |
| 434 | int DGMRES<_MatrixType, _Preconditioner>::dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const |
| 435 | { |
| 436 | // First, find the Schur form of the Hessenberg matrix H |
| 437 | typename internal::conditional<NumTraits<Scalar>::IsComplex, ComplexSchur<DenseMatrix>, RealSchur<DenseMatrix> >::type schurofH; |
| 438 | bool computeU = true; |
| 439 | DenseMatrix matrixQ(it,it); |
| 440 | matrixQ.setIdentity(); |
| 441 | schurofH.computeFromHessenberg(m_Hes.topLeftCorner(it,it), matrixQ, computeU); |
| 442 | |
| 443 | ComplexVector eig(it); |
| 444 | Matrix<Index,Dynamic,1>perm(it); |
| 445 | eig = this->schurValues(schurofH); |
| 446 | |
| 447 | // Reorder the absolute values of Schur values |
| 448 | DenseRealVector modulEig(it); |
| 449 | for (int j=0; j<it; ++j) modulEig(j) = std::abs(eig(j)); |
| 450 | perm.setLinSpaced(it,0,it-1); |
| 451 | internal::sortWithPermutation(modulEig, perm, neig); |
| 452 | |
| 453 | if (!m_lambdaN) |
| 454 | { |
| 455 | m_lambdaN = (std::max)(modulEig.maxCoeff(), m_lambdaN); |
| 456 | } |
| 457 | //Count the real number of extracted eigenvalues (with complex conjugates) |
| 458 | int nbrEig = 0; |
| 459 | while (nbrEig < neig) |
| 460 | { |
| 461 | if(eig(perm(it-nbrEig-1)).imag() == RealScalar(0)) nbrEig++; |
| 462 | else nbrEig += 2; |
| 463 | } |
| 464 | // Extract the Schur vectors corresponding to the smallest Ritz values |
| 465 | DenseMatrix Sr(it, nbrEig); |
| 466 | Sr.setZero(); |
| 467 | for (int j = 0; j < nbrEig; j++) |
| 468 | { |
| 469 | Sr.col(j) = schurofH.matrixU().col(perm(it-j-1)); |
| 470 | } |
| 471 | |
| 472 | // Form the Schur vectors of the initial matrix using the Krylov basis |
| 473 | DenseMatrix X; |
| 474 | X = m_V.leftCols(it) * Sr; |
| 475 | if (m_r) |
| 476 | { |
| 477 | // Orthogonalize X against m_U using modified Gram-Schmidt |
| 478 | for (int j = 0; j < nbrEig; j++) |
| 479 | for (int k =0; k < m_r; k++) |
| 480 | X.col(j) = X.col(j) - (m_U.col(k).dot(X.col(j)))*m_U.col(k); |
| 481 | } |
| 482 | |
| 483 | // Compute m_MX = A * M^-1 * X |
| 484 | Index m = m_V.rows(); |
| 485 | if (!m_isDeflAllocated) |
| 486 | dgmresInitDeflation(m); |
| 487 | DenseMatrix MX(m, nbrEig); |
| 488 | DenseVector tv1(m); |
| 489 | for (int j = 0; j < nbrEig; j++) |
| 490 | { |
| 491 | tv1 = mat * X.col(j); |
| 492 | MX.col(j) = precond.solve(tv1); |
| 493 | } |
| 494 | |
| 495 | //Update m_T = [U'MU U'MX; X'MU X'MX] |
| 496 | m_T.block(m_r, m_r, nbrEig, nbrEig) = X.transpose() * MX; |
| 497 | if(m_r) |
| 498 | { |
| 499 | m_T.block(0, m_r, m_r, nbrEig) = m_U.leftCols(m_r).transpose() * MX; |
| 500 | m_T.block(m_r, 0, nbrEig, m_r) = X.transpose() * m_MU.leftCols(m_r); |
| 501 | } |
| 502 | |
| 503 | // Save X into m_U and m_MX in m_MU |
| 504 | for (int j = 0; j < nbrEig; j++) m_U.col(m_r+j) = X.col(j); |
| 505 | for (int j = 0; j < nbrEig; j++) m_MU.col(m_r+j) = MX.col(j); |
| 506 | // Increase the size of the invariant subspace |
| 507 | m_r += nbrEig; |
| 508 | |
| 509 | // Factorize m_T into m_luT |
| 510 | m_luT.compute(m_T.topLeftCorner(m_r, m_r)); |
| 511 | |
| 512 | //FIXME CHeck if the factorization was correctly done (nonsingular matrix) |
| 513 | m_isDeflInitialized = true; |
| 514 | return 0; |
| 515 | } |
| 516 | template<typename _MatrixType, typename _Preconditioner> |
| 517 | template<typename RhsType, typename DestType> |
| 518 | int DGMRES<_MatrixType, _Preconditioner>::dgmresApplyDeflation(const RhsType &x, DestType &y) const |
| 519 | { |
| 520 | DenseVector x1 = m_U.leftCols(m_r).transpose() * x; |
| 521 | y = x + m_U.leftCols(m_r) * ( m_lambdaN * m_luT.solve(x1) - x1); |
| 522 | return 0; |
| 523 | } |
| 524 | |
| 525 | namespace internal { |
| 526 | |
| 527 | template<typename _MatrixType, typename _Preconditioner, typename Rhs> |
| 528 | struct solve_retval<DGMRES<_MatrixType, _Preconditioner>, Rhs> |
| 529 | : solve_retval_base<DGMRES<_MatrixType, _Preconditioner>, Rhs> |
| 530 | { |
| 531 | typedef DGMRES<_MatrixType, _Preconditioner> Dec; |
| 532 | EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) |
| 533 | |
| 534 | template<typename Dest> void evalTo(Dest& dst) const |
| 535 | { |
| 536 | dec()._solve(rhs(),dst); |
| 537 | } |
| 538 | }; |
| 539 | } // end namespace internal |
| 540 | |
| 541 | } // end namespace Eigen |
| 542 | #endif |