1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2011-2014 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, Index& 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 Index maxIters = iters; 40 41 Index 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 Index 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 residual 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 (or dense) self-adjoint problems 111 * 112 * This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm. 113 * The matrix A must be selfadjoint. The matrix A and the vectors x and b can be either dense or sparse. 114 * 115 * \tparam _MatrixType the type of the 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 * \c Upper, or \c Lower|Upper in which the full matrix entries will be considered. 118 * Default is \c Lower, best performance is \c Lower|Upper. 119 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner 120 * 121 * \implsparsesolverconcept 122 * 123 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() 124 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations 125 * and NumTraits<Scalar>::epsilon() for the tolerance. 126 * 127 * The tolerance corresponds to the relative residual error: |Ax-b|/|b| 128 * 129 * \b Performance: Even though the default value of \c _UpLo is \c Lower, significantly higher performance is 130 * achieved when using a complete matrix and \b Lower|Upper as the \a _UpLo template parameter. Moreover, in this 131 * case multi-threading can be exploited if the user code is compiled with OpenMP enabled. 132 * See \ref TopicMultiThreading for details. 133 * 134 * This class can be used as the direct solver classes. Here is a typical usage example: 135 \code 136 int n = 10000; 137 VectorXd x(n), b(n); 138 SparseMatrix<double> A(n,n); 139 // fill A and b 140 ConjugateGradient<SparseMatrix<double>, Lower|Upper> cg; 141 cg.compute(A); 142 x = cg.solve(b); 143 std::cout << "#iterations: " << cg.iterations() << std::endl; 144 std::cout << "estimated error: " << cg.error() << std::endl; 145 // update b, and solve again 146 x = cg.solve(b); 147 \endcode 148 * 149 * By default the iterations start with x=0 as an initial guess of the solution. 150 * One can control the start using the solveWithGuess() method. 151 * 152 * ConjugateGradient can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink. 153 * 154 * \sa class LeastSquaresConjugateGradient, class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner 155 */ 156 template< typename _MatrixType, int _UpLo, typename _Preconditioner> 157 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> > 158 { 159 typedef IterativeSolverBase<ConjugateGradient> Base; 160 using Base::matrix; 161 using Base::m_error; 162 using Base::m_iterations; 163 using Base::m_info; 164 using Base::m_isInitialized; 165 public: 166 typedef _MatrixType MatrixType; 167 typedef typename MatrixType::Scalar Scalar; 168 typedef typename MatrixType::RealScalar RealScalar; 169 typedef _Preconditioner Preconditioner; 170 171 enum { 172 UpLo = _UpLo 173 }; 174 175 public: 176 177 /** Default constructor. */ 178 ConjugateGradient() : Base() {} 179 180 /** Initialize the solver with matrix \a A for further \c Ax=b solving. 181 * 182 * This constructor is a shortcut for the default constructor followed 183 * by a call to compute(). 184 * 185 * \warning this class stores a reference to the matrix A as well as some 186 * precomputed values that depend on it. Therefore, if \a A is changed 187 * this class becomes invalid. Call compute() to update it with the new 188 * matrix A, or modify a copy of A. 189 */ 190 template<typename MatrixDerived> 191 explicit ConjugateGradient(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {} 192 193 ~ConjugateGradient() {} 194 195 /** \internal */ 196 template<typename Rhs,typename Dest> 197 void _solve_with_guess_impl(const Rhs& b, Dest& x) const 198 { 199 typedef typename Base::MatrixWrapper MatrixWrapper; 200 typedef typename Base::ActualMatrixType ActualMatrixType; 201 enum { 202 TransposeInput = (!MatrixWrapper::MatrixFree) 203 && (UpLo==(Lower|Upper)) 204 && (!MatrixType::IsRowMajor) 205 && (!NumTraits<Scalar>::IsComplex) 206 }; 207 typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper; 208 EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY); 209 typedef typename internal::conditional<UpLo==(Lower|Upper), 210 RowMajorWrapper, 211 typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type 212 >::type SelfAdjointWrapper; 213 m_iterations = Base::maxIterations(); 214 m_error = Base::m_tolerance; 215 216 for(Index j=0; j<b.cols(); ++j) 217 { 218 m_iterations = Base::maxIterations(); 219 m_error = Base::m_tolerance; 220 221 typename Dest::ColXpr xj(x,j); 222 RowMajorWrapper row_mat(matrix()); 223 internal::conjugate_gradient(SelfAdjointWrapper(row_mat), b.col(j), xj, Base::m_preconditioner, m_iterations, m_error); 224 } 225 226 m_isInitialized = true; 227 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; 228 } 229 230 /** \internal */ 231 using Base::_solve_impl; 232 template<typename Rhs,typename Dest> 233 void _solve_impl(const MatrixBase<Rhs>& b, Dest& x) const 234 { 235 x.setZero(); 236 _solve_with_guess_impl(b.derived(),x); 237 } 238 239 protected: 240 241 }; 242 243 } // end namespace Eigen 244 245 #endif // EIGEN_CONJUGATE_GRADIENT_H 246
