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 45 RealScalar rhsNorm2 = rhs.squaredNorm(); 46 if(rhsNorm2 == 0) 47 { 48 x.setZero(); 49 iters = 0; 50 tol_error = 0; 51 return; 52 } 53 RealScalar threshold = tol*tol*rhsNorm2; 54 RealScalar residualNorm2 = residual.squaredNorm(); 55 if (residualNorm2 < threshold) 56 { 57 iters = 0; 58 tol_error = sqrt(residualNorm2 / rhsNorm2); 59 return; 60 } 61 62 VectorType p(n); 63 p = precond.solve(residual); //initial search direction 64 65 VectorType z(n), tmp(n); 66 RealScalar absNew = numext::real(residual.dot(p)); // the square of the absolute value of r scaled by invM 67 int i = 0; 68 while(i < maxIters) 69 { 70 tmp.noalias() = mat * p; // the bottleneck of the algorithm 71 72 Scalar alpha = absNew / p.dot(tmp); // the amount we travel on dir 73 x += alpha * p; // update solution 74 residual -= alpha * tmp; // update residue 75 76 residualNorm2 = residual.squaredNorm(); 77 if(residualNorm2 < threshold) 78 break; 79 80 z = precond.solve(residual); // approximately solve for "A z = residual" 81 82 RealScalar absOld = absNew; 83 absNew = numext::real(residual.dot(z)); // update the absolute value of r 84 RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction 85 p = z + beta * p; // update search direction 86 i++; 87 } 88 tol_error = sqrt(residualNorm2 / rhsNorm2); 89 iters = i; 90 } 91 92 } 93 94 template< typename _MatrixType, int _UpLo=Lower, 95 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> > 96 class ConjugateGradient; 97 98 namespace internal { 99 100 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 101 struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> > 102 { 103 typedef _MatrixType MatrixType; 104 typedef _Preconditioner Preconditioner; 105 }; 106 107 } 108 109 /** \ingroup IterativeLinearSolvers_Module 110 * \brief A conjugate gradient solver for sparse self-adjoint problems 111 * 112 * This class allows to solve for A.x = b sparse linear problems using a conjugate gradient algorithm. 113 * The sparse matrix A must be selfadjoint. The vectors x and b can be either dense or sparse. 114 * 115 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. 116 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower 117 * or Upper. Default is Lower. 118 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 119 * 120 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 121 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 122 * and NumTraits<Scalar>::epsilon() for the tolerance. 123 * 124 * This class can be used as the direct solver classes. Here is a typical usage example: 125 * \code 126 * int n = 10000; 127 * VectorXd x(n), b(n); 128 * SparseMatrix<double> A(n,n); 129 * // fill A and b 130 * ConjugateGradient<SparseMatrix<double> > cg; 131 * cg.compute(A); 132 * x = cg.solve(b); 133 * std::cout << "#iterations: " << cg.iterations() << std::endl; 134 * std::cout << "estimated error: " << cg.error() << std::endl; 135 * // update b, and solve again 136 * x = cg.solve(b); 137 * \endcode 138 * 139 * By default the iterations start with x=0 as an initial guess of the solution. 140 * One can control the start using the solveWithGuess() method. Here is a step by 141 * step execution example starting with a random guess and printing the evolution 142 * of the estimated error: 143 * * \code 144 * x = VectorXd::Random(n); 145 * cg.setMaxIterations(1); 146 * int i = 0; 147 * do { 148 * x = cg.solveWithGuess(b,x); 149 * std::cout << i << " : " << cg.error() << std::endl; 150 * ++i; 151 * } while (cg.info()!=Success && i<100); 152 * \endcode 153 * Note that such a step by step excution is slightly slower. 154 * 155 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 156 */ 157 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 158 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> > 159 { 160 typedef IterativeSolverBase<ConjugateGradient> Base; 161 using Base::mp_matrix; 162 using Base::m_error; 163 using Base::m_iterations; 164 using Base::m_info; 165 using Base::m_isInitialized; 166 public: 167 typedef _MatrixType MatrixType; 168 typedef typename MatrixType::Scalar Scalar; 169 typedef typename MatrixType::Index Index; 170 typedef typename MatrixType::RealScalar RealScalar; 171 typedef _Preconditioner Preconditioner; 172 173 enum { 174 UpLo = _UpLo 175 }; 176 177 public: 178 179 /** Default constructor. */ 180 ConjugateGradient() : Base() {} 181 182 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 183 * 184 * This constructor is a shortcut for the default constructor followed 185 * by a call to compute(). 186 * 187 * \warning this class stores a reference to the matrix A as well as some 188 * precomputed values that depend on it. Therefore, if \a A is changed 189 * this class becomes invalid. Call compute() to update it with the new 190 * matrix A, or modify a copy of A. 191 */ 192 ConjugateGradient(const MatrixType& A) : Base(A) {} 193 194 ~ConjugateGradient() {} 195 196 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A 197 * \a x0 as an initial solution. 198 * 199 * \sa compute() 200 */ 201 template<typename Rhs,typename Guess> 202 inline const internal::solve_retval_with_guess<ConjugateGradient, Rhs, Guess> 203 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const 204 { 205 eigen_assert(m_isInitialized && "ConjugateGradient is not initialized."); 206 eigen_assert(Base::rows()==b.rows() 207 && "ConjugateGradient::solve(): invalid number of rows of the right hand side matrix b"); 208 return internal::solve_retval_with_guess 209 <ConjugateGradient, Rhs, Guess>(*this, b.derived(), x0); 210 } 211 212 /** \internal */ 213 template<typename Rhs,typename Dest> 214 void _solveWithGuess(const Rhs& b, Dest& x) const 215 { 216 m_iterations = Base::maxIterations(); 217 m_error = Base::m_tolerance; 218 219 for(int j=0; j<b.cols(); ++j) 220 { 221 m_iterations = Base::maxIterations(); 222 m_error = Base::m_tolerance; 223 224 typename Dest::ColXpr xj(x,j); 225 internal::conjugate_gradient(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj, 226 Base::m_preconditioner, m_iterations, m_error); 227 } 228 229 m_isInitialized = true; 230 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 231 } 232 233 /** \internal */ 234 template<typename Rhs,typename Dest> 235 void _solve(const Rhs& b, Dest& x) const 236 { 237 x.setOnes(); 238 _solveWithGuess(b,x); 239 } 240 241 protected: 242 243 }; 244 245 246 namespace internal { 247 248 template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs> 249 struct solve_retval<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs> 250 : solve_retval_base<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs> 251 { 252 typedef ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> Dec; 253 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) 254 255 template<typename Dest> void evalTo(Dest& dst) const 256 { 257 dec()._solve(rhs(),dst); 258 } 259 }; 260 261 } // end namespace internal 262 263 } // end namespace Eigen 264 265 #endif // EIGEN_CONJUGATE_GRADIENT_H 266
