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 // 6 // This Source Code Form is subject to the terms of the Mozilla 7 // Public License v. 2.0. If a copy of the MPL was not distributed 8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9 10 #ifndef EIGEN_CONJUGATE_GRADIENT_H 11 #define EIGEN_CONJUGATE_GRADIENT_H 12 13 namespace Eigen { 14 15 namespace internal { 16 17 /** \internal Low-level conjugate gradient algorithm 18 * \param mat The matrix A 19 * \param rhs The right hand side vector b 20 * \param x On input and initial solution, on output the computed solution. 21 * \param precond A preconditioner being able to efficiently solve for an 22 * approximation of Ax=b (regardless of b) 23 * \param iters On input the max number of iteration, on output the number of performed iterations. 24 * \param tol_error On input the tolerance error, on output an estimation of the relative error. 25 */ 26 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> 27 EIGEN_DONT_INLINE 28 void conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x, 29 const Preconditioner& precond, int& iters, 30 typename Dest::RealScalar& tol_error) 31 { 32 using std::sqrt; 33 using std::abs; 34 typedef typename Dest::RealScalar RealScalar; 35 typedef typename Dest::Scalar Scalar; 36 typedef Matrix<Scalar,Dynamic,1> VectorType; 37 38 RealScalar tol = tol_error; 39 int maxIters = iters; 40 41 int n = mat.cols(); 42 43 VectorType residual = rhs - mat * x; //initial residual 44 VectorType p(n); 45 46 p = precond.solve(residual); //initial search direction 47 48 VectorType z(n), tmp(n); 49 RealScalar absNew = internal::real(residual.dot(p)); // the square of the absolute value of r scaled by invM 50 RealScalar rhsNorm2 = rhs.squaredNorm(); 51 RealScalar residualNorm2 = 0; 52 RealScalar threshold = tol*tol*rhsNorm2; 53 int i = 0; 54 while(i < maxIters) 55 { 56 tmp.noalias() = mat * p; // the bottleneck of the algorithm 57 58 Scalar alpha = absNew / p.dot(tmp); // the amount we travel on dir 59 x += alpha * p; // update solution 60 residual -= alpha * tmp; // update residue 61 62 residualNorm2 = residual.squaredNorm(); 63 if(residualNorm2 < threshold) 64 break; 65 66 z = precond.solve(residual); // approximately solve for "A z = residual" 67 68 RealScalar absOld = absNew; 69 absNew = internal::real(residual.dot(z)); // update the absolute value of r 70 RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction 71 p = z + beta * p; // update search direction 72 i++; 73 } 74 tol_error = sqrt(residualNorm2 / rhsNorm2); 75 iters = i; 76 } 77 78 } 79 80 template< typename _MatrixType, int _UpLo=Lower, 81 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> > 82 class ConjugateGradient; 83 84 namespace internal { 85 86 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 87 struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> > 88 { 89 typedef _MatrixType MatrixType; 90 typedef _Preconditioner Preconditioner; 91 }; 92 93 } 94 95 /** \ingroup IterativeLinearSolvers_Module 96 * \brief A conjugate gradient solver for sparse self-adjoint problems 97 * 98 * This class allows to solve for A.x = b sparse linear problems using a conjugate gradient algorithm. 99 * The sparse matrix A must be selfadjoint. The vectors x and b can be either dense or sparse. 100 * 101 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 102 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower 103 * or Upper. Default is Lower. 104 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 105 * 106 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 107 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 108 * and NumTraits<Scalar>::epsilon() for the tolerance. 109 * 110 * This class can be used as the direct solver classes. Here is a typical usage example: 111 * \code 112 * int n = 10000; 113 * VectorXd x(n), b(n); 114 * SparseMatrix<double> A(n,n); 115 * // fill A and b 116 * ConjugateGradient<SparseMatrix<double> > cg; 117 * cg.compute(A); 118 * x = cg.solve(b); 119 * std::cout << "#iterations: " << cg.iterations() << std::endl; 120 * std::cout << "estimated error: " << cg.error() << std::endl; 121 * // update b, and solve again 122 * x = cg.solve(b); 123 * \endcode 124 * 125 * By default the iterations start with x=0 as an initial guess of the solution. 126 * One can control the start using the solveWithGuess() method. Here is a step by 127 * step execution example starting with a random guess and printing the evolution 128 * of the estimated error: 129 * * \code 130 * x = VectorXd::Random(n); 131 * cg.setMaxIterations(1); 132 * int i = 0; 133 * do { 134 * x = cg.solveWithGuess(b,x); 135 * std::cout << i << " : " << cg.error() << std::endl; 136 * ++i; 137 * } while (cg.info()!=Success && i<100); 138 * \endcode 139 * Note that such a step by step excution is slightly slower. 140 * 141 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 142 */ 143 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 144 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> > 145 { 146 typedef IterativeSolverBase<ConjugateGradient> Base; 147 using Base::mp_matrix; 148 using Base::m_error; 149 using Base::m_iterations; 150 using Base::m_info; 151 using Base::m_isInitialized; 152 public: 153 typedef _MatrixType MatrixType; 154 typedef typename MatrixType::Scalar Scalar; 155 typedef typename MatrixType::Index Index; 156 typedef typename MatrixType::RealScalar RealScalar; 157 typedef _Preconditioner Preconditioner; 158 159 enum { 160 UpLo = _UpLo 161 }; 162 163 public: 164 165 /** Default constructor. */ 166 ConjugateGradient() : Base() {} 167 168 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 169 * 170 * This constructor is a shortcut for the default constructor followed 171 * by a call to compute(). 172 * 173 * \warning this class stores a reference to the matrix A as well as some 174 * precomputed values that depend on it. Therefore, if \a A is changed 175 * this class becomes invalid. Call compute() to update it with the new 176 * matrix A, or modify a copy of A. 177 */ 178 ConjugateGradient(const MatrixType& A) : Base(A) {} 179 180 ~ConjugateGradient() {} 181 182 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 183 * \a x0 as an initial solution. 184 * 185 * \sa compute() 186 */ 187 template<typename Rhs,typename Guess> 188 inline const internal::solve_retval_with_guess<ConjugateGradient, Rhs, Guess> 189 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 190 { 191 eigen_assert(m_isInitialized && "ConjugateGradient is not initialized."); 192 eigen_assert(Base::rows()==b.rows() 193 && "ConjugateGradient::solve(): invalid number of rows of the right hand side matrix b"); 194 return internal::solve_retval_with_guess 195 <ConjugateGradient, Rhs, Guess>(*this, b.derived(), x0); 196 } 197 198 /** \internal */ 199 template<typename Rhs,typename Dest> 200 void _solveWithGuess(const Rhs& b, Dest& x) const 201 { 202 m_iterations = Base::maxIterations(); 203 m_error = Base::m_tolerance; 204 205 for(int j=0; j<b.cols(); ++j) 206 { 207 m_iterations = Base::maxIterations(); 208 m_error = Base::m_tolerance; 209 210 typename Dest::ColXpr xj(x,j); 211 internal::conjugate_gradient(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj, 212 Base::m_preconditioner, m_iterations, m_error); 213 } 214 215 m_isInitialized = true; 216 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 217 } 218 219 /** \internal */ 220 template<typename Rhs,typename Dest> 221 void _solve(const Rhs& b, Dest& x) const 222 { 223 x.setOnes(); 224 _solveWithGuess(b,x); 225 } 226 227 protected: 228 229 }; 230 231 232 namespace internal { 233 234 template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs> 235 struct solve_retval<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs> 236 : solve_retval_base<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs> 237 { 238 typedef ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> Dec; 239 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 240 241 template<typename Dest> void evalTo(Dest& dst) const 242 { 243 dec()._solve(rhs(),dst); 244 } 245 }; 246 247 } // end namespace internal 248 249 } // end namespace Eigen 250 251 #endif // EIGEN_CONJUGATE_GRADIENT_H 252
