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@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
conjugate_gradient(const MatrixType & mat,const Rhs & rhs,Dest & x,const Preconditioner & precond,Index & iters,typename Dest::RealScalar & tol_error)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