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
13namespace Eigen {
14
15namespace 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  */
26template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
27EIGEN_DONT_INLINE
28void 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
96template< typename _MatrixType,
97          typename _Preconditioner = LeastSquareDiagonalPreconditioner<typename _MatrixType::Scalar> >
98class LeastSquaresConjugateGradient;
99
100namespace internal {
101
102template< typename _MatrixType, typename _Preconditioner>
103struct 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  */
148template< typename _MatrixType, typename _Preconditioner>
149class 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;
157public:
158  typedef _MatrixType MatrixType;
159  typedef typename MatrixType::Scalar Scalar;
160  typedef typename MatrixType::RealScalar RealScalar;
161  typedef _Preconditioner Preconditioner;
162
163public:
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