1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2012 Giacomo Po <gpo@ucla.edu> 5 // Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@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 12 #ifndef EIGEN_MINRES_H_ 13 #define EIGEN_MINRES_H_ 14 15 16 namespace Eigen { 17 18 namespace internal { 19 20 /** \internal Low-level MINRES algorithm 21 * \param mat The matrix A 22 * \param rhs The right hand side vector b 23 * \param x On input and initial solution, on output the computed solution. 24 * \param precond A right preconditioner being able to efficiently solve for an 25 * approximation of Ax=b (regardless of b) 26 * \param iters On input the max number of iteration, on output the number of performed iterations. 27 * \param tol_error On input the tolerance error, on output an estimation of the relative error. 28 */ 29 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> 30 EIGEN_DONT_INLINE minres(const MatrixType & mat,const Rhs & rhs,Dest & x,const Preconditioner & precond,int & iters,typename Dest::RealScalar & tol_error)31 void minres(const MatrixType& mat, const Rhs& rhs, Dest& x, 32 const Preconditioner& precond, int& iters, 33 typename Dest::RealScalar& tol_error) 34 { 35 using std::sqrt; 36 typedef typename Dest::RealScalar RealScalar; 37 typedef typename Dest::Scalar Scalar; 38 typedef Matrix<Scalar,Dynamic,1> VectorType; 39 40 // initialize 41 const int maxIters(iters); // initialize maxIters to iters 42 const int N(mat.cols()); // the size of the matrix 43 const RealScalar rhsNorm2(rhs.squaredNorm()); 44 const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2) 45 46 // Initialize preconditioned Lanczos 47 // VectorType v_old(N); // will be initialized inside loop 48 VectorType v( VectorType::Zero(N) ); //initialize v 49 VectorType v_new(rhs-mat*x); //initialize v_new 50 RealScalar residualNorm2(v_new.squaredNorm()); 51 // VectorType w(N); // will be initialized inside loop 52 VectorType w_new(precond.solve(v_new)); // initialize w_new 53 // RealScalar beta; // will be initialized inside loop 54 RealScalar beta_new2(v_new.dot(w_new)); 55 eigen_assert(beta_new2 >= 0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 56 RealScalar beta_new(sqrt(beta_new2)); 57 const RealScalar beta_one(beta_new); 58 v_new /= beta_new; 59 w_new /= beta_new; 60 // Initialize other variables 61 RealScalar c(1.0); // the cosine of the Givens rotation 62 RealScalar c_old(1.0); 63 RealScalar s(0.0); // the sine of the Givens rotation 64 RealScalar s_old(0.0); // the sine of the Givens rotation 65 // VectorType p_oold(N); // will be initialized in loop 66 VectorType p_old(VectorType::Zero(N)); // initialize p_old=0 67 VectorType p(p_old); // initialize p=0 68 RealScalar eta(1.0); 69 70 iters = 0; // reset iters 71 while ( iters < maxIters ){ 72 73 // Preconditioned Lanczos 74 /* Note that there are 4 variants on the Lanczos algorithm. These are 75 * described in Paige, C. C. (1972). Computational variants of 76 * the Lanczos method for the eigenproblem. IMA Journal of Applied 77 * Mathematics, 10(3), 373–381. The current implementation corresponds 78 * to the case A(2,7) in the paper. It also corresponds to 79 * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear 80 * Systems, 2003 p.173. For the preconditioned version see 81 * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987). 82 */ 83 const RealScalar beta(beta_new); 84 // v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter 85 const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT 86 v = v_new; // update 87 // w = w_new; // update 88 const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT 89 v_new.noalias() = mat*w - beta*v_old; // compute v_new 90 const RealScalar alpha = v_new.dot(w); 91 v_new -= alpha*v; // overwrite v_new 92 w_new = precond.solve(v_new); // overwrite w_new 93 beta_new2 = v_new.dot(w_new); // compute beta_new 94 eigen_assert(beta_new2 >= 0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 95 beta_new = sqrt(beta_new2); // compute beta_new 96 v_new /= beta_new; // overwrite v_new for next iteration 97 w_new /= beta_new; // overwrite w_new for next iteration 98 99 // Givens rotation 100 const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration 101 const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration 102 const RealScalar r1_hat=c*alpha-c_old*s*beta; 103 const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) ); 104 c_old = c; // store for next iteration 105 s_old = s; // store for next iteration 106 c=r1_hat/r1; // new cosine 107 s=beta_new/r1; // new sine 108 109 // Update solution 110 // p_oold = p_old; 111 const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT 112 p_old = p; 113 p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED? 114 x += beta_one*c*eta*p; 115 residualNorm2 *= s*s; 116 117 if ( residualNorm2 < threshold2){ 118 break; 119 } 120 121 eta=-s*eta; // update eta 122 iters++; // increment iteration number (for output purposes) 123 } 124 tol_error = std::sqrt(residualNorm2 / rhsNorm2); // return error. Note that this is the estimated error. The real error |Ax-b|/|b| may be slightly larger 125 } 126 127 } 128 129 template< typename _MatrixType, int _UpLo=Lower, 130 typename _Preconditioner = IdentityPreconditioner> 131 // typename _Preconditioner = IdentityPreconditioner<typename _MatrixType::Scalar> > // preconditioner must be positive definite 132 class MINRES; 133 134 namespace internal { 135 136 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 137 struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> > 138 { 139 typedef _MatrixType MatrixType; 140 typedef _Preconditioner Preconditioner; 141 }; 142 143 } 144 145 /** \ingroup IterativeLinearSolvers_Module 146 * \brief A minimal residual solver for sparse symmetric problems 147 * 148 * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm 149 * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite). 150 * The vectors x and b can be either dense or sparse. 151 * 152 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 153 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower 154 * or Upper. Default is Lower. 155 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 156 * 157 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 158 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 159 * and NumTraits<Scalar>::epsilon() for the tolerance. 160 * 161 * This class can be used as the direct solver classes. Here is a typical usage example: 162 * \code 163 * int n = 10000; 164 * VectorXd x(n), b(n); 165 * SparseMatrix<double> A(n,n); 166 * // fill A and b 167 * MINRES<SparseMatrix<double> > mr; 168 * mr.compute(A); 169 * x = mr.solve(b); 170 * std::cout << "#iterations: " << mr.iterations() << std::endl; 171 * std::cout << "estimated error: " << mr.error() << std::endl; 172 * // update b, and solve again 173 * x = mr.solve(b); 174 * \endcode 175 * 176 * By default the iterations start with x=0 as an initial guess of the solution. 177 * One can control the start using the solveWithGuess() method. Here is a step by 178 * step execution example starting with a random guess and printing the evolution 179 * of the estimated error: 180 * * \code 181 * x = VectorXd::Random(n); 182 * mr.setMaxIterations(1); 183 * int i = 0; 184 * do { 185 * x = mr.solveWithGuess(b,x); 186 * std::cout << i << " : " << mr.error() << std::endl; 187 * ++i; 188 * } while (mr.info()!=Success && i<100); 189 * \endcode 190 * Note that such a step by step excution is slightly slower. 191 * 192 * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 193 */ 194 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 195 class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> > 196 { 197 198 typedef IterativeSolverBase<MINRES> Base; 199 using Base::mp_matrix; 200 using Base::m_error; 201 using Base::m_iterations; 202 using Base::m_info; 203 using Base::m_isInitialized; 204 public: 205 typedef _MatrixType MatrixType; 206 typedef typename MatrixType::Scalar Scalar; 207 typedef typename MatrixType::Index Index; 208 typedef typename MatrixType::RealScalar RealScalar; 209 typedef _Preconditioner Preconditioner; 210 211 enum {UpLo = _UpLo}; 212 213 public: 214 215 /** Default constructor. */ 216 MINRES() : Base() {} 217 218 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 219 * 220 * This constructor is a shortcut for the default constructor followed 221 * by a call to compute(). 222 * 223 * \warning this class stores a reference to the matrix A as well as some 224 * precomputed values that depend on it. Therefore, if \a A is changed 225 * this class becomes invalid. Call compute() to update it with the new 226 * matrix A, or modify a copy of A. 227 */ 228 MINRES(const MatrixType& A) : Base(A) {} 229 230 /** Destructor. */ 231 ~MINRES(){} 232 233 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 234 * \a x0 as an initial solution. 235 * 236 * \sa compute() 237 */ 238 template<typename Rhs,typename Guess> 239 inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess> 240 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 241 { 242 eigen_assert(m_isInitialized && "MINRES is not initialized."); 243 eigen_assert(Base::rows()==b.rows() 244 && "MINRES::solve(): invalid number of rows of the right hand side matrix b"); 245 return internal::solve_retval_with_guess 246 <MINRES, Rhs, Guess>(*this, b.derived(), x0); 247 } 248 249 /** \internal */ 250 template<typename Rhs,typename Dest> 251 void _solveWithGuess(const Rhs& b, Dest& x) const 252 { 253 m_iterations = Base::maxIterations(); 254 m_error = Base::m_tolerance; 255 256 for(int j=0; j<b.cols(); ++j) 257 { 258 m_iterations = Base::maxIterations(); 259 m_error = Base::m_tolerance; 260 261 typename Dest::ColXpr xj(x,j); 262 internal::minres(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj, 263 Base::m_preconditioner, m_iterations, m_error); 264 } 265 266 m_isInitialized = true; 267 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 268 } 269 270 /** \internal */ 271 template<typename Rhs,typename Dest> 272 void _solve(const Rhs& b, Dest& x) const 273 { 274 x.setZero(); 275 _solveWithGuess(b,x); 276 } 277 278 protected: 279 280 }; 281 282 namespace internal { 283 284 template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs> 285 struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> 286 : solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> 287 { 288 typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec; 289 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 290 291 template<typename Dest> void evalTo(Dest& dst) const 292 { 293 dec()._solve(rhs(),dst); 294 } 295 }; 296 297 } // end namespace internal 298 299 } // end namespace Eigen 300 301 #endif // EIGEN_MINRES_H 302 303