MINRES.h (12397B)
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) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr> 6 // Copyright (C) 2018 David Hyde <dabh@stanford.edu> 7 // 8 // This Source Code Form is subject to the terms of the Mozilla 9 // Public License v. 2.0. If a copy of the MPL was not distributed 10 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 11 12 13 #ifndef EIGEN_MINRES_H_ 14 #define EIGEN_MINRES_H_ 15 16 17 namespace Eigen { 18 19 namespace internal { 20 21 /** \internal Low-level MINRES algorithm 22 * \param mat The matrix A 23 * \param rhs The right hand side vector b 24 * \param x On input and initial solution, on output the computed solution. 25 * \param precond A right preconditioner being able to efficiently solve for an 26 * approximation of Ax=b (regardless of b) 27 * \param iters On input the max number of iteration, on output the number of performed iterations. 28 * \param tol_error On input the tolerance error, on output an estimation of the relative error. 29 */ 30 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> 31 EIGEN_DONT_INLINE 32 void minres(const MatrixType& mat, const Rhs& rhs, Dest& x, 33 const Preconditioner& precond, Index& iters, 34 typename Dest::RealScalar& tol_error) 35 { 36 using std::sqrt; 37 typedef typename Dest::RealScalar RealScalar; 38 typedef typename Dest::Scalar Scalar; 39 typedef Matrix<Scalar,Dynamic,1> VectorType; 40 41 // Check for zero rhs 42 const RealScalar rhsNorm2(rhs.squaredNorm()); 43 if(rhsNorm2 == 0) 44 { 45 x.setZero(); 46 iters = 0; 47 tol_error = 0; 48 return; 49 } 50 51 // initialize 52 const Index maxIters(iters); // initialize maxIters to iters 53 const Index N(mat.cols()); // the size of the matrix 54 const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2) 55 56 // Initialize preconditioned Lanczos 57 VectorType v_old(N); // will be initialized inside loop 58 VectorType v( VectorType::Zero(N) ); //initialize v 59 VectorType v_new(rhs-mat*x); //initialize v_new 60 RealScalar residualNorm2(v_new.squaredNorm()); 61 VectorType w(N); // will be initialized inside loop 62 VectorType w_new(precond.solve(v_new)); // initialize w_new 63 // RealScalar beta; // will be initialized inside loop 64 RealScalar beta_new2(v_new.dot(w_new)); 65 eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 66 RealScalar beta_new(sqrt(beta_new2)); 67 const RealScalar beta_one(beta_new); 68 // Initialize other variables 69 RealScalar c(1.0); // the cosine of the Givens rotation 70 RealScalar c_old(1.0); 71 RealScalar s(0.0); // the sine of the Givens rotation 72 RealScalar s_old(0.0); // the sine of the Givens rotation 73 VectorType p_oold(N); // will be initialized in loop 74 VectorType p_old(VectorType::Zero(N)); // initialize p_old=0 75 VectorType p(p_old); // initialize p=0 76 RealScalar eta(1.0); 77 78 iters = 0; // reset iters 79 while ( iters < maxIters ) 80 { 81 // Preconditioned Lanczos 82 /* Note that there are 4 variants on the Lanczos algorithm. These are 83 * described in Paige, C. C. (1972). Computational variants of 84 * the Lanczos method for the eigenproblem. IMA Journal of Applied 85 * Mathematics, 10(3), 373-381. The current implementation corresponds 86 * to the case A(2,7) in the paper. It also corresponds to 87 * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear 88 * Systems, 2003 p.173. For the preconditioned version see 89 * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987). 90 */ 91 const RealScalar beta(beta_new); 92 v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter 93 v_new /= beta_new; // overwrite v_new for next iteration 94 w_new /= beta_new; // overwrite w_new for next iteration 95 v = v_new; // update 96 w = w_new; // update 97 v_new.noalias() = mat*w - beta*v_old; // compute v_new 98 const RealScalar alpha = v_new.dot(w); 99 v_new -= alpha*v; // overwrite v_new 100 w_new = precond.solve(v_new); // overwrite w_new 101 beta_new2 = v_new.dot(w_new); // compute beta_new 102 eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); 103 beta_new = sqrt(beta_new2); // compute beta_new 104 105 // Givens rotation 106 const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration 107 const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration 108 const RealScalar r1_hat=c*alpha-c_old*s*beta; 109 const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) ); 110 c_old = c; // store for next iteration 111 s_old = s; // store for next iteration 112 c=r1_hat/r1; // new cosine 113 s=beta_new/r1; // new sine 114 115 // Update solution 116 p_oold = p_old; 117 p_old = p; 118 p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED? 119 x += beta_one*c*eta*p; 120 121 /* Update the squared residual. Note that this is the estimated residual. 122 The real residual |Ax-b|^2 may be slightly larger */ 123 residualNorm2 *= s*s; 124 125 if ( residualNorm2 < threshold2) 126 { 127 break; 128 } 129 130 eta=-s*eta; // update eta 131 iters++; // increment iteration number (for output purposes) 132 } 133 134 /* Compute error. Note that this is the estimated error. The real 135 error |Ax-b|/|b| may be slightly larger */ 136 tol_error = std::sqrt(residualNorm2 / rhsNorm2); 137 } 138 139 } 140 141 template< typename _MatrixType, int _UpLo=Lower, 142 typename _Preconditioner = IdentityPreconditioner> 143 class MINRES; 144 145 namespace internal { 146 147 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 148 struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> > 149 { 150 typedef _MatrixType MatrixType; 151 typedef _Preconditioner Preconditioner; 152 }; 153 154 } 155 156 /** \ingroup IterativeLinearSolvers_Module 157 * \brief A minimal residual solver for sparse symmetric problems 158 * 159 * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm 160 * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite). 161 * The vectors x and b can be either dense or sparse. 162 * 163 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 164 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower, 165 * Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower. 166 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 167 * 168 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 169 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 170 * and NumTraits<Scalar>::epsilon() for the tolerance. 171 * 172 * This class can be used as the direct solver classes. Here is a typical usage example: 173 * \code 174 * int n = 10000; 175 * VectorXd x(n), b(n); 176 * SparseMatrix<double> A(n,n); 177 * // fill A and b 178 * MINRES<SparseMatrix<double> > mr; 179 * mr.compute(A); 180 * x = mr.solve(b); 181 * std::cout << "#iterations: " << mr.iterations() << std::endl; 182 * std::cout << "estimated error: " << mr.error() << std::endl; 183 * // update b, and solve again 184 * x = mr.solve(b); 185 * \endcode 186 * 187 * By default the iterations start with x=0 as an initial guess of the solution. 188 * One can control the start using the solveWithGuess() method. 189 * 190 * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink. 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::matrix; 200 using Base::m_error; 201 using Base::m_iterations; 202 using Base::m_info; 203 using Base::m_isInitialized; 204 public: 205 using Base::_solve_impl; 206 typedef _MatrixType MatrixType; 207 typedef typename MatrixType::Scalar Scalar; 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 template<typename MatrixDerived> 229 explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {} 230 231 /** Destructor. */ 232 ~MINRES(){} 233 234 /** \internal */ 235 template<typename Rhs,typename Dest> 236 void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const 237 { 238 typedef typename Base::MatrixWrapper MatrixWrapper; 239 typedef typename Base::ActualMatrixType ActualMatrixType; 240 enum { 241 TransposeInput = (!MatrixWrapper::MatrixFree) 242 && (UpLo==(Lower|Upper)) 243 && (!MatrixType::IsRowMajor) 244 && (!NumTraits<Scalar>::IsComplex) 245 }; 246 typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper; 247 EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY); 248 typedef typename internal::conditional<UpLo==(Lower|Upper), 249 RowMajorWrapper, 250 typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type 251 >::type SelfAdjointWrapper; 252 253 m_iterations = Base::maxIterations(); 254 m_error = Base::m_tolerance; 255 RowMajorWrapper row_mat(matrix()); 256 internal::minres(SelfAdjointWrapper(row_mat), b, x, 257 Base::m_preconditioner, m_iterations, m_error); 258 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 259 } 260 261 protected: 262 263 }; 264 265 } // end namespace Eigen 266 267 #endif // EIGEN_MINRES_H