1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2012 Giacomo Po <gpo (at) ucla.edu> 5 // Copyright (C) 2011-2014 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 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 31 void minres(const MatrixType& mat, const Rhs& rhs, Dest& x, 32 const Preconditioner& precond, Index& 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 Index maxIters(iters); // initialize maxIters to iters 52 const Index 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), 373381. 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 class MINRES; 148 149 namespace internal { 150 151 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 152 struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> > 153 { 154 typedef _MatrixType MatrixType; 155 typedef _Preconditioner Preconditioner; 156 }; 157 158 } 159 160 /** \ingroup IterativeLinearSolvers_Module 161 * \brief A minimal residual solver for sparse symmetric problems 162 * 163 * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm 164 * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite). 165 * The vectors x and b can be either dense or sparse. 166 * 167 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 168 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower, 169 * Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower. 170 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 171 * 172 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 173 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 174 * and NumTraits<Scalar>::epsilon() for the tolerance. 175 * 176 * This class can be used as the direct solver classes. Here is a typical usage example: 177 * \code 178 * int n = 10000; 179 * VectorXd x(n), b(n); 180 * SparseMatrix<double> A(n,n); 181 * // fill A and b 182 * MINRES<SparseMatrix<double> > mr; 183 * mr.compute(A); 184 * x = mr.solve(b); 185 * std::cout << "#iterations: " << mr.iterations() << std::endl; 186 * std::cout << "estimated error: " << mr.error() << std::endl; 187 * // update b, and solve again 188 * x = mr.solve(b); 189 * \endcode 190 * 191 * By default the iterations start with x=0 as an initial guess of the solution. 192 * One can control the start using the solveWithGuess() method. 193 * 194 * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink. 195 * 196 * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 197 */ 198 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 199 class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> > 200 { 201 202 typedef IterativeSolverBase<MINRES> Base; 203 using Base::matrix; 204 using Base::m_error; 205 using Base::m_iterations; 206 using Base::m_info; 207 using Base::m_isInitialized; 208 public: 209 using Base::_solve_impl; 210 typedef _MatrixType MatrixType; 211 typedef typename MatrixType::Scalar Scalar; 212 typedef typename MatrixType::RealScalar RealScalar; 213 typedef _Preconditioner Preconditioner; 214 215 enum {UpLo = _UpLo}; 216 217 public: 218 219 /** Default constructor. */ 220 MINRES() : Base() {} 221 222 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 223 * 224 * This constructor is a shortcut for the default constructor followed 225 * by a call to compute(). 226 * 227 * \warning this class stores a reference to the matrix A as well as some 228 * precomputed values that depend on it. Therefore, if \a A is changed 229 * this class becomes invalid. Call compute() to update it with the new 230 * matrix A, or modify a copy of A. 231 */ 232 template<typename MatrixDerived> 233 explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {} 234 235 /** Destructor. */ 236 ~MINRES(){} 237 238 /** \internal */ 239 template<typename Rhs,typename Dest> 240 void _solve_with_guess_impl(const Rhs& b, Dest& x) const 241 { 242 typedef typename Base::MatrixWrapper MatrixWrapper; 243 typedef typename Base::ActualMatrixType ActualMatrixType; 244 enum { 245 TransposeInput = (!MatrixWrapper::MatrixFree) 246 && (UpLo==(Lower|Upper)) 247 && (!MatrixType::IsRowMajor) 248 && (!NumTraits<Scalar>::IsComplex) 249 }; 250 typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper; 251 EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY); 252 typedef typename internal::conditional<UpLo==(Lower|Upper), 253 RowMajorWrapper, 254 typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type 255 >::type SelfAdjointWrapper; 256 257 m_iterations = Base::maxIterations(); 258 m_error = Base::m_tolerance; 259 RowMajorWrapper row_mat(matrix()); 260 for(int j=0; j<b.cols(); ++j) 261 { 262 m_iterations = Base::maxIterations(); 263 m_error = Base::m_tolerance; 264 265 typename Dest::ColXpr xj(x,j); 266 internal::minres(SelfAdjointWrapper(row_mat), b.col(j), xj, 267 Base::m_preconditioner, m_iterations, m_error); 268 } 269 270 m_isInitialized = true; 271 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 272 } 273 274 /** \internal */ 275 template<typename Rhs,typename Dest> 276 void _solve_impl(const Rhs& b, MatrixBase<Dest> &x) const 277 { 278 x.setZero(); 279 _solve_with_guess_impl(b,x.derived()); 280 } 281 282 protected: 283 284 }; 285 286 } // end namespace Eigen 287 288 #endif // EIGEN_MINRES_H 289 290
