1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2011 Gael Guennebaud <gael.guennebaud (at) inria.fr> 5 // Copyright (C) 2012 Dsir Nuentsa-Wakam <desire.nuentsa_wakam (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 #ifndef EIGEN_BICGSTAB_H 12 #define EIGEN_BICGSTAB_H 13 14 namespace Eigen { 15 16 namespace internal { 17 18 /** \internal Low-level bi conjugate gradient stabilized algorithm 19 * \param mat The matrix A 20 * \param rhs The right hand side vector b 21 * \param x On input and initial solution, on output the computed solution. 22 * \param precond A preconditioner being able to efficiently solve for an 23 * approximation of Ax=b (regardless of b) 24 * \param iters On input the max number of iteration, on output the number of performed iterations. 25 * \param tol_error On input the tolerance error, on output an estimation of the relative error. 26 * \return false in the case of numerical issue, for example a break down of BiCGSTAB. 27 */ 28 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> 29 bool bicgstab(const MatrixType& mat, const Rhs& rhs, Dest& x, 30 const Preconditioner& precond, int& iters, 31 typename Dest::RealScalar& tol_error) 32 { 33 using std::sqrt; 34 using std::abs; 35 typedef typename Dest::RealScalar RealScalar; 36 typedef typename Dest::Scalar Scalar; 37 typedef Matrix<Scalar,Dynamic,1> VectorType; 38 RealScalar tol = tol_error; 39 int maxIters = iters; 40 41 int n = mat.cols(); 42 VectorType r = rhs - mat * x; 43 VectorType r0 = r; 44 45 RealScalar r0_sqnorm = r0.squaredNorm(); 46 RealScalar rhs_sqnorm = rhs.squaredNorm(); 47 if(rhs_sqnorm == 0) 48 { 49 x.setZero(); 50 return true; 51 } 52 Scalar rho = 1; 53 Scalar alpha = 1; 54 Scalar w = 1; 55 56 VectorType v = VectorType::Zero(n), p = VectorType::Zero(n); 57 VectorType y(n), z(n); 58 VectorType kt(n), ks(n); 59 60 VectorType s(n), t(n); 61 62 RealScalar tol2 = tol*tol; 63 RealScalar eps2 = NumTraits<Scalar>::epsilon()*NumTraits<Scalar>::epsilon(); 64 int i = 0; 65 int restarts = 0; 66 67 while ( r.squaredNorm()/rhs_sqnorm > tol2 && i<maxIters ) 68 { 69 Scalar rho_old = rho; 70 71 rho = r0.dot(r); 72 if (abs(rho) < eps2*r0_sqnorm) 73 { 74 // The new residual vector became too orthogonal to the arbitrarily choosen direction r0 75 // Let's restart with a new r0: 76 r0 = r; 77 rho = r0_sqnorm = r.squaredNorm(); 78 if(restarts++ == 0) 79 i = 0; 80 } 81 Scalar beta = (rho/rho_old) * (alpha / w); 82 p = r + beta * (p - w * v); 83 84 y = precond.solve(p); 85 86 v.noalias() = mat * y; 87 88 alpha = rho / r0.dot(v); 89 s = r - alpha * v; 90 91 z = precond.solve(s); 92 t.noalias() = mat * z; 93 94 RealScalar tmp = t.squaredNorm(); 95 if(tmp>RealScalar(0)) 96 w = t.dot(s) / tmp; 97 else 98 w = Scalar(0); 99 x += alpha * y + w * z; 100 r = s - w * t; 101 ++i; 102 } 103 tol_error = sqrt(r.squaredNorm()/rhs_sqnorm); 104 iters = i; 105 return true; 106 } 107 108 } 109 110 template< typename _MatrixType, 111 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> > 112 class BiCGSTAB; 113 114 namespace internal { 115 116 template< typename _MatrixType, typename _Preconditioner> 117 struct traits<BiCGSTAB<_MatrixType,_Preconditioner> > 118 { 119 typedef _MatrixType MatrixType; 120 typedef _Preconditioner Preconditioner; 121 }; 122 123 } 124 125 /** \ingroup IterativeLinearSolvers_Module 126 * \brief A bi conjugate gradient stabilized solver for sparse square problems 127 * 128 * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient 129 * stabilized algorithm. The vectors x and b can be either dense or sparse. 130 * 131 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 132 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 133 * 134 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 135 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 136 * and NumTraits<Scalar>::epsilon() for the tolerance. 137 * 138 * This class can be used as the direct solver classes. Here is a typical usage example: 139 * \code 140 * int n = 10000; 141 * VectorXd x(n), b(n); 142 * SparseMatrix<double> A(n,n); 143 * // fill A and b 144 * BiCGSTAB<SparseMatrix<double> > solver; 145 * solver.compute(A); 146 * x = solver.solve(b); 147 * std::cout << "#iterations: " << solver.iterations() << std::endl; 148 * std::cout << "estimated error: " << solver.error() << std::endl; 149 * // update b, and solve again 150 * x = solver.solve(b); 151 * \endcode 152 * 153 * By default the iterations start with x=0 as an initial guess of the solution. 154 * One can control the start using the solveWithGuess() method. 155 * 156 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 157 */ 158 template< typename _MatrixType, typename _Preconditioner> 159 class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType,_Preconditioner> > 160 { 161 typedef IterativeSolverBase<BiCGSTAB> Base; 162 using Base::mp_matrix; 163 using Base::m_error; 164 using Base::m_iterations; 165 using Base::m_info; 166 using Base::m_isInitialized; 167 public: 168 typedef _MatrixType MatrixType; 169 typedef typename MatrixType::Scalar Scalar; 170 typedef typename MatrixType::Index Index; 171 typedef typename MatrixType::RealScalar RealScalar; 172 typedef _Preconditioner Preconditioner; 173 174 public: 175 176 /** Default constructor. */ 177 BiCGSTAB() : Base() {} 178 179 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 180 * 181 * This constructor is a shortcut for the default constructor followed 182 * by a call to compute(). 183 * 184 * \warning this class stores a reference to the matrix A as well as some 185 * precomputed values that depend on it. Therefore, if \a A is changed 186 * this class becomes invalid. Call compute() to update it with the new 187 * matrix A, or modify a copy of A. 188 */ 189 BiCGSTAB(const MatrixType& A) : Base(A) {} 190 191 ~BiCGSTAB() {} 192 193 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 194 * \a x0 as an initial solution. 195 * 196 * \sa compute() 197 */ 198 template<typename Rhs,typename Guess> 199 inline const internal::solve_retval_with_guess<BiCGSTAB, Rhs, Guess> 200 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 201 { 202 eigen_assert(m_isInitialized && "BiCGSTAB is not initialized."); 203 eigen_assert(Base::rows()==b.rows() 204 && "BiCGSTAB::solve(): invalid number of rows of the right hand side matrix b"); 205 return internal::solve_retval_with_guess 206 <BiCGSTAB, Rhs, Guess>(*this, b.derived(), x0); 207 } 208 209 /** \internal */ 210 template<typename Rhs,typename Dest> 211 void _solveWithGuess(const Rhs& b, Dest& x) const 212 { 213 bool failed = false; 214 for(int j=0; j<b.cols(); ++j) 215 { 216 m_iterations = Base::maxIterations(); 217 m_error = Base::m_tolerance; 218 219 typename Dest::ColXpr xj(x,j); 220 if(!internal::bicgstab(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_error)) 221 failed = true; 222 } 223 m_info = failed ? NumericalIssue 224 : m_error <= Base::m_tolerance ? Success 225 : NoConvergence; 226 m_isInitialized = true; 227 } 228 229 /** \internal */ 230 template<typename Rhs,typename Dest> 231 void _solve(const Rhs& b, Dest& x) const 232 { 233 // x.setZero(); 234 x = b; 235 _solveWithGuess(b,x); 236 } 237 238 protected: 239 240 }; 241 242 243 namespace internal { 244 245 template<typename _MatrixType, typename _Preconditioner, typename Rhs> 246 struct solve_retval<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs> 247 : solve_retval_base<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs> 248 { 249 typedef BiCGSTAB<_MatrixType, _Preconditioner> Dec; 250 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 251 252 template<typename Dest> void evalTo(Dest& dst) const 253 { 254 dec()._solve(rhs(),dst); 255 } 256 }; 257 258 } // end namespace internal 259 260 } // end namespace Eigen 261 262 #endif // EIGEN_BICGSTAB_H 263