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@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,int & iters,typename Dest::RealScalar & tol_error)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