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