1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2015 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_LEAST_SQUARE_CONJUGATE_GRADIENT_H
11 #define EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
12
13 namespace Eigen {
14
15 namespace internal {
16
17 /** \internal Low-level conjugate gradient algorithm for least-square problems
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 A'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
least_square_conjugate_gradient(const MatrixType & mat,const Rhs & rhs,Dest & x,const Preconditioner & precond,Index & iters,typename Dest::RealScalar & tol_error)28 void least_square_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 m = mat.rows(), n = mat.cols();
42
43 VectorType residual = rhs - mat * x;
44 VectorType normal_residual = mat.adjoint() * residual;
45
46 RealScalar rhsNorm2 = (mat.adjoint()*rhs).squaredNorm();
47 if(rhsNorm2 == 0)
48 {
49 x.setZero();
50 iters = 0;
51 tol_error = 0;
52 return;
53 }
54 RealScalar threshold = tol*tol*rhsNorm2;
55 RealScalar residualNorm2 = normal_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(normal_residual); // initial search direction
65
66 VectorType z(n), tmp(m);
67 RealScalar absNew = numext::real(normal_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;
72
73 Scalar alpha = absNew / tmp.squaredNorm(); // the amount we travel on dir
74 x += alpha * p; // update solution
75 residual -= alpha * tmp; // update residual
76 normal_residual = mat.adjoint() * residual; // update residual of the normal equation
77
78 residualNorm2 = normal_residual.squaredNorm();
79 if(residualNorm2 < threshold)
80 break;
81
82 z = precond.solve(normal_residual); // approximately solve for "A'A z = normal_residual"
83
84 RealScalar absOld = absNew;
85 absNew = numext::real(normal_residual.dot(z)); // update the absolute value of r
86 RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction
87 p = z + beta * p; // update search direction
88 i++;
89 }
90 tol_error = sqrt(residualNorm2 / rhsNorm2);
91 iters = i;
92 }
93
94 }
95
96 template< typename _MatrixType,
97 typename _Preconditioner = LeastSquareDiagonalPreconditioner<typename _MatrixType::Scalar> >
98 class LeastSquaresConjugateGradient;
99
100 namespace internal {
101
102 template< typename _MatrixType, typename _Preconditioner>
103 struct traits<LeastSquaresConjugateGradient<_MatrixType,_Preconditioner> >
104 {
105 typedef _MatrixType MatrixType;
106 typedef _Preconditioner Preconditioner;
107 };
108
109 }
110
111 /** \ingroup IterativeLinearSolvers_Module
112 * \brief A conjugate gradient solver for sparse (or dense) least-square problems
113 *
114 * This class allows to solve for A x = b linear problems using an iterative conjugate gradient algorithm.
115 * The matrix A can be non symmetric and rectangular, but the matrix A' A should be positive-definite to guaranty stability.
116 * Otherwise, the SparseLU or SparseQR classes might be preferable.
117 * The matrix A and the vectors x and b can be either dense or sparse.
118 *
119 * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
120 * \tparam _Preconditioner the type of the preconditioner. Default is LeastSquareDiagonalPreconditioner
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 * This class can be used as the direct solver classes. Here is a typical usage example:
129 \code
130 int m=1000000, n = 10000;
131 VectorXd x(n), b(m);
132 SparseMatrix<double> A(m,n);
133 // fill A and b
134 LeastSquaresConjugateGradient<SparseMatrix<double> > lscg;
135 lscg.compute(A);
136 x = lscg.solve(b);
137 std::cout << "#iterations: " << lscg.iterations() << std::endl;
138 std::cout << "estimated error: " << lscg.error() << std::endl;
139 // update b, and solve again
140 x = lscg.solve(b);
141 \endcode
142 *
143 * By default the iterations start with x=0 as an initial guess of the solution.
144 * One can control the start using the solveWithGuess() method.
145 *
146 * \sa class ConjugateGradient, SparseLU, SparseQR
147 */
148 template< typename _MatrixType, typename _Preconditioner>
149 class LeastSquaresConjugateGradient : public IterativeSolverBase<LeastSquaresConjugateGradient<_MatrixType,_Preconditioner> >
150 {
151 typedef IterativeSolverBase<LeastSquaresConjugateGradient> Base;
152 using Base::matrix;
153 using Base::m_error;
154 using Base::m_iterations;
155 using Base::m_info;
156 using Base::m_isInitialized;
157 public:
158 typedef _MatrixType MatrixType;
159 typedef typename MatrixType::Scalar Scalar;
160 typedef typename MatrixType::RealScalar RealScalar;
161 typedef _Preconditioner Preconditioner;
162
163 public:
164
165 /** Default constructor. */
166 LeastSquaresConjugateGradient() : 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 template<typename MatrixDerived>
179 explicit LeastSquaresConjugateGradient(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
180
181 ~LeastSquaresConjugateGradient() {}
182
183 /** \internal */
184 template<typename Rhs,typename Dest>
185 void _solve_with_guess_impl(const Rhs& b, Dest& x) const
186 {
187 m_iterations = Base::maxIterations();
188 m_error = Base::m_tolerance;
189
190 for(Index j=0; j<b.cols(); ++j)
191 {
192 m_iterations = Base::maxIterations();
193 m_error = Base::m_tolerance;
194
195 typename Dest::ColXpr xj(x,j);
196 internal::least_square_conjugate_gradient(matrix(), b.col(j), xj, Base::m_preconditioner, m_iterations, m_error);
197 }
198
199 m_isInitialized = true;
200 m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
201 }
202
203 /** \internal */
204 using Base::_solve_impl;
205 template<typename Rhs,typename Dest>
206 void _solve_impl(const MatrixBase<Rhs>& b, Dest& x) const
207 {
208 x.setZero();
209 _solve_with_guess_impl(b.derived(),x);
210 }
211
212 };
213
214 } // end namespace Eigen
215
216 #endif // EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
217