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 // Check for zero rhs 41 const RealScalar rhsNorm2(rhs.squaredNorm()); 42 if(rhsNorm2 == 0) 43 { 44 x.setZero(); 45 iters = 0; 46 tol_error = 0; 47 return; 48 } 49 50 // initialize 51 const int maxIters(iters); // initialize maxIters to iters 52 const int N(mat.cols()); // the size of the matrix 53 const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2) 54 55 // Initialize preconditioned Lanczos 56 VectorType v_old(N); // will be initialized inside loop 57 VectorType v( VectorType::Zero(N) ); //initialize v 58 VectorType v_new(rhs-mat*x); //initialize v_new 59 RealScalar residualNorm2(v_new.squaredNorm()); 60 VectorType w(N); // will be initialized inside loop 61 VectorType w_new(precond.solve(v_new)); // initialize w_new 62 // RealScalar beta; // will be initialized inside loop 63 RealScalar beta_new2(v_new.dot(w_new)); 64 eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 65 RealScalar beta_new(sqrt(beta_new2)); 66 const RealScalar beta_one(beta_new); 67 v_new /= beta_new; 68 w_new /= beta_new; 69 // Initialize other variables 70 RealScalar c(1.0); // the cosine of the Givens rotation 71 RealScalar c_old(1.0); 72 RealScalar s(0.0); // the sine of the Givens rotation 73 RealScalar s_old(0.0); // the sine of the Givens rotation 74 VectorType p_oold(N); // will be initialized in loop 75 VectorType p_old(VectorType::Zero(N)); // initialize p_old=0 76 VectorType p(p_old); // initialize p=0 77 RealScalar eta(1.0); 78 79 iters = 0; // reset iters 80 while ( iters < maxIters ) 81 { 82 // Preconditioned Lanczos 83 /* Note that there are 4 variants on the Lanczos algorithm. These are 84 * described in Paige, C. C. (1972). Computational variants of 85 * the Lanczos method for the eigenproblem. IMA Journal of Applied 86 * Mathematics, 10(3), 373–381. The current implementation corresponds 87 * to the case A(2,7) in the paper. It also corresponds to 88 * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear 89 * Systems, 2003 p.173. For the preconditioned version see 90 * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987). 91 */ 92 const RealScalar beta(beta_new); 93 v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter 94 // const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT 95 v = v_new; // update 96 w = w_new; // update 97 // const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT 98 v_new.noalias() = mat*w - beta*v_old; // compute v_new 99 const RealScalar alpha = v_new.dot(w); 100 v_new -= alpha*v; // overwrite v_new 101 w_new = precond.solve(v_new); // overwrite w_new 102 beta_new2 = v_new.dot(w_new); // compute beta_new 103 eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 104 beta_new = sqrt(beta_new2); // compute beta_new 105 v_new /= beta_new; // overwrite v_new for next iteration 106 w_new /= beta_new; // overwrite w_new for next iteration 107 108 // Givens rotation 109 const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration 110 const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration 111 const RealScalar r1_hat=c*alpha-c_old*s*beta; 112 const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) ); 113 c_old = c; // store for next iteration 114 s_old = s; // store for next iteration 115 c=r1_hat/r1; // new cosine 116 s=beta_new/r1; // new sine 117 118 // Update solution 119 p_oold = p_old; 120 // const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT 121 p_old = p; 122 p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED? 123 x += beta_one*c*eta*p; 124 125 /* Update the squared residual. Note that this is the estimated residual. 126 The real residual |Ax-b|^2 may be slightly larger */ 127 residualNorm2 *= s*s; 128 129 if ( residualNorm2 < threshold2) 130 { 131 break; 132 } 133 134 eta=-s*eta; // update eta 135 iters++; // increment iteration number (for output purposes) 136 } 137 138 /* Compute error. Note that this is the estimated error. The real 139 error |Ax-b|/|b| may be slightly larger */ 140 tol_error = std::sqrt(residualNorm2 / rhsNorm2); 141 } 142 143 } 144 145 template< typename _MatrixType, int _UpLo=Lower, 146 typename _Preconditioner = IdentityPreconditioner> 147 // typename _Preconditioner = IdentityPreconditioner<typename _MatrixType::Scalar> > // preconditioner must be positive definite 148 class MINRES; 149 150 namespace internal { 151 152 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 153 struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> > 154 { 155 typedef _MatrixType MatrixType; 156 typedef _Preconditioner Preconditioner; 157 }; 158 159 } 160 161 /** \ingroup IterativeLinearSolvers_Module 162 * \brief A minimal residual solver for sparse symmetric problems 163 * 164 * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm 165 * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite). 166 * The vectors x and b can be either dense or sparse. 167 * 168 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 169 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower 170 * or Upper. Default is Lower. 171 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 172 * 173 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 174 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 175 * and NumTraits<Scalar>::epsilon() for the tolerance. 176 * 177 * This class can be used as the direct solver classes. Here is a typical usage example: 178 * \code 179 * int n = 10000; 180 * VectorXd x(n), b(n); 181 * SparseMatrix<double> A(n,n); 182 * // fill A and b 183 * MINRES<SparseMatrix<double> > mr; 184 * mr.compute(A); 185 * x = mr.solve(b); 186 * std::cout << "#iterations: " << mr.iterations() << std::endl; 187 * std::cout << "estimated error: " << mr.error() << std::endl; 188 * // update b, and solve again 189 * x = mr.solve(b); 190 * \endcode 191 * 192 * By default the iterations start with x=0 as an initial guess of the solution. 193 * One can control the start using the solveWithGuess() method. 194 * 195 * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 196 */ 197 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 198 class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> > 199 { 200 201 typedef IterativeSolverBase<MINRES> Base; 202 using Base::mp_matrix; 203 using Base::m_error; 204 using Base::m_iterations; 205 using Base::m_info; 206 using Base::m_isInitialized; 207 public: 208 typedef _MatrixType MatrixType; 209 typedef typename MatrixType::Scalar Scalar; 210 typedef typename MatrixType::Index Index; 211 typedef typename MatrixType::RealScalar RealScalar; 212 typedef _Preconditioner Preconditioner; 213 214 enum {UpLo = _UpLo}; 215 216 public: 217 218 /** Default constructor. */ 219 MINRES() : Base() {} 220 221 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 222 * 223 * This constructor is a shortcut for the default constructor followed 224 * by a call to compute(). 225 * 226 * \warning this class stores a reference to the matrix A as well as some 227 * precomputed values that depend on it. Therefore, if \a A is changed 228 * this class becomes invalid. Call compute() to update it with the new 229 * matrix A, or modify a copy of A. 230 */ 231 MINRES(const MatrixType& A) : Base(A) {} 232 233 /** Destructor. */ 234 ~MINRES(){} 235 236 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 237 * \a x0 as an initial solution. 238 * 239 * \sa compute() 240 */ 241 template<typename Rhs,typename Guess> 242 inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess> 243 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 244 { 245 eigen_assert(m_isInitialized && "MINRES is not initialized."); 246 eigen_assert(Base::rows()==b.rows() 247 && "MINRES::solve(): invalid number of rows of the right hand side matrix b"); 248 return internal::solve_retval_with_guess 249 <MINRES, Rhs, Guess>(*this, b.derived(), x0); 250 } 251 252 /** \internal */ 253 template<typename Rhs,typename Dest> 254 void _solveWithGuess(const Rhs& b, Dest& x) const 255 { 256 typedef typename internal::conditional<UpLo==(Lower|Upper), 257 const MatrixType&, 258 SparseSelfAdjointView<const MatrixType, UpLo> 259 >::type MatrixWrapperType; 260 261 m_iterations = Base::maxIterations(); 262 m_error = Base::m_tolerance; 263 264 for(int j=0; j<b.cols(); ++j) 265 { 266 m_iterations = Base::maxIterations(); 267 m_error = Base::m_tolerance; 268 269 typename Dest::ColXpr xj(x,j); 270 internal::minres(MatrixWrapperType(*mp_matrix), b.col(j), xj, 271 Base::m_preconditioner, m_iterations, m_error); 272 } 273 274 m_isInitialized = true; 275 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 276 } 277 278 /** \internal */ 279 template<typename Rhs,typename Dest> 280 void _solve(const Rhs& b, Dest& x) const 281 { 282 x.setZero(); 283 _solveWithGuess(b,x); 284 } 285 286 protected: 287 288 }; 289 290 namespace internal { 291 292 template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs> 293 struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> 294 : solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> 295 { 296 typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec; 297 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 298 299 template<typename Dest> void evalTo(Dest& dst) const 300 { 301 dec()._solve(rhs(),dst); 302 } 303 }; 304 305 } // end namespace internal 306 307 } // end namespace Eigen 308 309 #endif // EIGEN_MINRES_H 310 311