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 const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
54 RealScalar threshold = numext::maxi(RealScalar(tol*tol*rhsNorm2),considerAsZero);
55 RealScalar residualNorm2 = residual.squaredNorm();
56 if (residualNorm2 < threshold)
57 {
58 iters = 0;
59 tol_error = sqrt(residualNorm2 / rhsNorm2);
60 return;
61 }
62
63 VectorType p(n);
64 p = precond.solve(residual); // initial search direction
65
66 VectorType z(n), tmp(n);
67 RealScalar absNew = numext::real(residual.dot(p)); // the square of the absolute value of r scaled by invM
68 Index i = 0;
69 while(i < maxIters)
70 {
71 tmp.noalias() = mat * p; // the bottleneck of the algorithm
72
73 Scalar alpha = absNew / p.dot(tmp); // the amount we travel on dir
74 x += alpha * p; // update solution
75 residual -= alpha * tmp; // update residual
76
77 residualNorm2 = residual.squaredNorm();
78 if(residualNorm2 < threshold)
79 break;
80
81 z = precond.solve(residual); // approximately solve for "A z = residual"
82
83 RealScalar absOld = absNew;
84 absNew = numext::real(residual.dot(z)); // update the absolute value of r
85 RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction
86 p = z + beta * p; // update search direction
87 i++;
88 }
89 tol_error = sqrt(residualNorm2 / rhsNorm2);
90 iters = i;
91 }
92
93 }
94
95 template< typename _MatrixType, int _UpLo=Lower,
96 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
97 class ConjugateGradient;
98
99 namespace internal {
100
101 template< typename _MatrixType, int _UpLo, typename _Preconditioner>
102 struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
103 {
104 typedef _MatrixType MatrixType;
105 typedef _Preconditioner Preconditioner;
106 };
107
108 }
109
110 /** \ingroup IterativeLinearSolvers_Module
111 * \brief A conjugate gradient solver for sparse (or dense) self-adjoint problems
112 *
113 * This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm.
114 * The matrix A must be selfadjoint. The matrix A and the vectors x and b can be either dense or sparse.
115 *
116 * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
117 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
118 * \c Upper, or \c Lower|Upper in which the full matrix entries will be considered.
119 * Default is \c Lower, best performance is \c Lower|Upper.
120 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
121 *
122 * \implsparsesolverconcept
123 *
124 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
125 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
126 * and NumTraits<Scalar>::epsilon() for the tolerance.
127 *
128 * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
129 *
130 * \b Performance: Even though the default value of \c _UpLo is \c Lower, significantly higher performance is
131 * achieved when using a complete matrix and \b Lower|Upper as the \a _UpLo template parameter. Moreover, in this
132 * case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
133 * See \ref TopicMultiThreading for details.
134 *
135 * This class can be used as the direct solver classes. Here is a typical usage example:
136 \code
137 int n = 10000;
138 VectorXd x(n), b(n);
139 SparseMatrix<double> A(n,n);
140 // fill A and b
141 ConjugateGradient<SparseMatrix<double>, Lower|Upper> cg;
142 cg.compute(A);
143 x = cg.solve(b);
144 std::cout << "#iterations: " << cg.iterations() << std::endl;
145 std::cout << "estimated error: " << cg.error() << std::endl;
146 // update b, and solve again
147 x = cg.solve(b);
148 \endcode
149 *
150 * By default the iterations start with x=0 as an initial guess of the solution.
151 * One can control the start using the solveWithGuess() method.
152 *
153 * ConjugateGradient can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
154 *
155 * \sa class LeastSquaresConjugateGradient, 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::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::RealScalar RealScalar;
170 typedef _Preconditioner Preconditioner;
171
172 enum {
173 UpLo = _UpLo
174 };
175
176 public:
177
178 /** Default constructor. */
179 ConjugateGradient() : Base() {}
180
181 /** Initialize the solver with matrix \a A for further \c Ax=b solving.
182 *
183 * This constructor is a shortcut for the default constructor followed
184 * by a call to compute().
185 *
186 * \warning this class stores a reference to the matrix A as well as some
187 * precomputed values that depend on it. Therefore, if \a A is changed
188 * this class becomes invalid. Call compute() to update it with the new
189 * matrix A, or modify a copy of A.
190 */
191 template<typename MatrixDerived>
192 explicit ConjugateGradient(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
193
194 ~ConjugateGradient() {}
195
196 /** \internal */
197 template<typename Rhs,typename Dest>
198 void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
199 {
200 typedef typename Base::MatrixWrapper MatrixWrapper;
201 typedef typename Base::ActualMatrixType ActualMatrixType;
202 enum {
203 TransposeInput = (!MatrixWrapper::MatrixFree)
204 && (UpLo==(Lower|Upper))
205 && (!MatrixType::IsRowMajor)
206 && (!NumTraits<Scalar>::IsComplex)
207 };
208 typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper;
209 EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
210 typedef typename internal::conditional<UpLo==(Lower|Upper),
211 RowMajorWrapper,
212 typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type
213 >::type SelfAdjointWrapper;
214
215 m_iterations = Base::maxIterations();
216 m_error = Base::m_tolerance;
217
218 RowMajorWrapper row_mat(matrix());
219 internal::conjugate_gradient(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
220 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
221 }
222
223 protected:
224
225 };
226
227 } // end namespace Eigen
228
229 #endif // EIGEN_CONJUGATE_GRADIENT_H
230