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 x = precond.solve(x); 43 VectorType r = rhs - mat * x; 44 VectorType r0 = r; 45 46 RealScalar r0_sqnorm = r0.squaredNorm(); 47 RealScalar rhs_sqnorm = rhs.squaredNorm(); 48 if(rhs_sqnorm == 0) 49 { 50 x.setZero(); 51 return true; 52 } 53 Scalar rho = 1; 54 Scalar alpha = 1; 55 Scalar w = 1; 56 57 VectorType v = VectorType::Zero(n), p = VectorType::Zero(n); 58 VectorType y(n), z(n); 59 VectorType kt(n), ks(n); 60 61 VectorType s(n), t(n); 62 63 RealScalar tol2 = tol*tol; 64 RealScalar eps2 = NumTraits<Scalar>::epsilon()*NumTraits<Scalar>::epsilon(); 65 int i = 0; 66 int restarts = 0; 67 68 while ( r.squaredNorm()/rhs_sqnorm > tol2 && i<maxIters ) 69 { 70 Scalar rho_old = rho; 71 72 rho = r0.dot(r); 73 if (abs(rho) < eps2*r0_sqnorm) 74 { 75 // The new residual vector became too orthogonal to the arbitrarily choosen direction r0 76 // Let's restart with a new r0: 77 r0 = r; 78 rho = r0_sqnorm = r.squaredNorm(); 79 if(restarts++ == 0) 80 i = 0; 81 } 82 Scalar beta = (rho/rho_old) * (alpha / w); 83 p = r + beta * (p - w * v); 84 85 y = precond.solve(p); 86 87 v.noalias() = mat * y; 88 89 alpha = rho / r0.dot(v); 90 s = r - alpha * v; 91 92 z = precond.solve(s); 93 t.noalias() = mat * z; 94 95 RealScalar tmp = t.squaredNorm(); 96 if(tmp>RealScalar(0)) 97 w = t.dot(s) / tmp; 98 else 99 w = Scalar(0); 100 x += alpha * y + w * z; 101 r = s - w * t; 102 ++i; 103 } 104 tol_error = sqrt(r.squaredNorm()/rhs_sqnorm); 105 iters = i; 106 return true; 107 } 108 109 } 110 111 template< typename _MatrixType, 112 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> > 113 class BiCGSTAB; 114 115 namespace internal { 116 117 template< typename _MatrixType, typename _Preconditioner> 118 struct traits<BiCGSTAB<_MatrixType,_Preconditioner> > 119 { 120 typedef _MatrixType MatrixType; 121 typedef _Preconditioner Preconditioner; 122 }; 123 124 } 125 126 /** \ingroup IterativeLinearSolvers_Module 127 * \brief A bi conjugate gradient stabilized solver for sparse square problems 128 * 129 * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient 130 * stabilized algorithm. The vectors x and b can be either dense or sparse. 131 * 132 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 133 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 134 * 135 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 136 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 137 * and NumTraits<Scalar>::epsilon() for the tolerance. 138 * 139 * This class can be used as the direct solver classes. Here is a typical usage example: 140 * \code 141 * int n = 10000; 142 * VectorXd x(n), b(n); 143 * SparseMatrix<double> A(n,n); 144 * // fill A and b 145 * BiCGSTAB<SparseMatrix<double> > solver; 146 * solver(A); 147 * x = solver.solve(b); 148 * std::cout << "#iterations: " << solver.iterations() << std::endl; 149 * std::cout << "estimated error: " << solver.error() << std::endl; 150 * // update b, and solve again 151 * x = solver.solve(b); 152 * \endcode 153 * 154 * By default the iterations start with x=0 as an initial guess of the solution. 155 * One can control the start using the solveWithGuess() method. Here is a step by 156 * step execution example starting with a random guess and printing the evolution 157 * of the estimated error: 158 * * \code 159 * x = VectorXd::Random(n); 160 * solver.setMaxIterations(1); 161 * int i = 0; 162 * do { 163 * x = solver.solveWithGuess(b,x); 164 * std::cout << i << " : " << solver.error() << std::endl; 165 * ++i; 166 * } while (solver.info()!=Success && i<100); 167 * \endcode 168 * Note that such a step by step excution is slightly slower. 169 * 170 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 171 */ 172 template< typename _MatrixType, typename _Preconditioner> 173 class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType,_Preconditioner> > 174 { 175 typedef IterativeSolverBase<BiCGSTAB> Base; 176 using Base::mp_matrix; 177 using Base::m_error; 178 using Base::m_iterations; 179 using Base::m_info; 180 using Base::m_isInitialized; 181 public: 182 typedef _MatrixType MatrixType; 183 typedef typename MatrixType::Scalar Scalar; 184 typedef typename MatrixType::Index Index; 185 typedef typename MatrixType::RealScalar RealScalar; 186 typedef _Preconditioner Preconditioner; 187 188 public: 189 190 /** Default constructor. */ 191 BiCGSTAB() : Base() {} 192 193 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 194 * 195 * This constructor is a shortcut for the default constructor followed 196 * by a call to compute(). 197 * 198 * \warning this class stores a reference to the matrix A as well as some 199 * precomputed values that depend on it. Therefore, if \a A is changed 200 * this class becomes invalid. Call compute() to update it with the new 201 * matrix A, or modify a copy of A. 202 */ 203 BiCGSTAB(const MatrixType& A) : Base(A) {} 204 205 ~BiCGSTAB() {} 206 207 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 208 * \a x0 as an initial solution. 209 * 210 * \sa compute() 211 */ 212 template<typename Rhs,typename Guess> 213 inline const internal::solve_retval_with_guess<BiCGSTAB, Rhs, Guess> 214 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 215 { 216 eigen_assert(m_isInitialized && "BiCGSTAB is not initialized."); 217 eigen_assert(Base::rows()==b.rows() 218 && "BiCGSTAB::solve(): invalid number of rows of the right hand side matrix b"); 219 return internal::solve_retval_with_guess 220 <BiCGSTAB, Rhs, Guess>(*this, b.derived(), x0); 221 } 222 223 /** \internal */ 224 template<typename Rhs,typename Dest> 225 void _solveWithGuess(const Rhs& b, Dest& x) const 226 { 227 bool failed = false; 228 for(int j=0; j<b.cols(); ++j) 229 { 230 m_iterations = Base::maxIterations(); 231 m_error = Base::m_tolerance; 232 233 typename Dest::ColXpr xj(x,j); 234 if(!internal::bicgstab(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_error)) 235 failed = true; 236 } 237 m_info = failed ? NumericalIssue 238 : m_error <= Base::m_tolerance ? Success 239 : NoConvergence; 240 m_isInitialized = true; 241 } 242 243 /** \internal */ 244 template<typename Rhs,typename Dest> 245 void _solve(const Rhs& b, Dest& x) const 246 { 247 // x.setZero(); 248 x = b; 249 _solveWithGuess(b,x); 250 } 251 252 protected: 253 254 }; 255 256 257 namespace internal { 258 259 template<typename _MatrixType, typename _Preconditioner, typename Rhs> 260 struct solve_retval<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs> 261 : solve_retval_base<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs> 262 { 263 typedef BiCGSTAB<_MatrixType, _Preconditioner> Dec; 264 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 265 266 template<typename Dest> void evalTo(Dest& dst) const 267 { 268 dec()._solve(rhs(),dst); 269 } 270 }; 271 272 } // end namespace internal 273 274 } // end namespace Eigen 275 276 #endif // EIGEN_BICGSTAB_H 277
