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// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_BICGSTAB_H
12#define EIGEN_BICGSTAB_H
13
14namespace Eigen {
15
16namespace internal {
17
18/** \internal Low-level bi conjugate gradient stabilized algorithm
19  * \param mat The matrix A
20  * \param rhs The right hand side vector b
21  * \param x On input and initial solution, on output the computed solution.
22  * \param precond A preconditioner being able to efficiently solve for an
23  *                approximation of Ax=b (regardless of b)
24  * \param iters On input the max number of iteration, on output the number of performed iterations.
25  * \param tol_error On input the tolerance error, on output an estimation of the relative error.
26  * \return false in the case of numerical issue, for example a break down of BiCGSTAB.
27  */
28template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
29bool bicgstab(const MatrixType& mat, const Rhs& rhs, Dest& x,
30              const Preconditioner& precond, Index& iters,
31              typename Dest::RealScalar& tol_error)
32{
33  using std::sqrt;
34  using std::abs;
35  typedef typename Dest::RealScalar RealScalar;
36  typedef typename Dest::Scalar Scalar;
37  typedef Matrix<Scalar,Dynamic,1> VectorType;
38  RealScalar tol = tol_error;
39  Index maxIters = iters;
40
41  Index n = mat.cols();
42  VectorType r  = rhs - mat * x;
43  VectorType r0 = r;
44
45  RealScalar r0_sqnorm = r0.squaredNorm();
46  RealScalar rhs_sqnorm = rhs.squaredNorm();
47  if(rhs_sqnorm == 0)
48  {
49    x.setZero();
50    return true;
51  }
52  Scalar rho    = 1;
53  Scalar alpha  = 1;
54  Scalar w      = 1;
55
56  VectorType v = VectorType::Zero(n), p = VectorType::Zero(n);
57  VectorType y(n),  z(n);
58  VectorType kt(n), ks(n);
59
60  VectorType s(n), t(n);
61
62  RealScalar tol2 = tol*tol*rhs_sqnorm;
63  RealScalar eps2 = NumTraits<Scalar>::epsilon()*NumTraits<Scalar>::epsilon();
64  Index i = 0;
65  Index restarts = 0;
66
67  while ( r.squaredNorm() > tol2 && i<maxIters )
68  {
69    Scalar rho_old = rho;
70
71    rho = r0.dot(r);
72    if (abs(rho) < eps2*r0_sqnorm)
73    {
74      // The new residual vector became too orthogonal to the arbitrarily chosen direction r0
75      // Let's restart with a new r0:
76      r  = rhs - mat * x;
77      r0 = r;
78      rho = r0_sqnorm = r.squaredNorm();
79      if(restarts++ == 0)
80        i = 0;
81    }
82    Scalar beta = (rho/rho_old) * (alpha / w);
83    p = r + beta * (p - w * v);
84
85    y = precond.solve(p);
86
87    v.noalias() = mat * y;
88
89    alpha = rho / r0.dot(v);
90    s = r - alpha * v;
91
92    z = precond.solve(s);
93    t.noalias() = mat * z;
94
95    RealScalar tmp = t.squaredNorm();
96    if(tmp>RealScalar(0))
97      w = t.dot(s) / tmp;
98    else
99      w = Scalar(0);
100    x += alpha * y + w * z;
101    r = s - w * t;
102    ++i;
103  }
104  tol_error = sqrt(r.squaredNorm()/rhs_sqnorm);
105  iters = i;
106  return true;
107}
108
109}
110
111template< typename _MatrixType,
112          typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
113class BiCGSTAB;
114
115namespace internal {
116
117template< typename _MatrixType, typename _Preconditioner>
118struct traits<BiCGSTAB<_MatrixType,_Preconditioner> >
119{
120  typedef _MatrixType MatrixType;
121  typedef _Preconditioner Preconditioner;
122};
123
124}
125
126/** \ingroup IterativeLinearSolvers_Module
127  * \brief A bi conjugate gradient stabilized solver for sparse square problems
128  *
129  * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient
130  * stabilized algorithm. The vectors x and b can be either dense or sparse.
131  *
132  * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
133  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
134  *
135  * \implsparsesolverconcept
136  *
137  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
138  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
139  * and NumTraits<Scalar>::epsilon() for the tolerance.
140  *
141  * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
142  *
143  * \b Performance: when using sparse matrices, best performance is achied for a row-major sparse matrix format.
144  * Moreover, in this case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
145  * See \ref TopicMultiThreading for details.
146  *
147  * This class can be used as the direct solver classes. Here is a typical usage example:
148  * \include BiCGSTAB_simple.cpp
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  * BiCGSTAB can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
154  *
155  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
156  */
157template< typename _MatrixType, typename _Preconditioner>
158class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType,_Preconditioner> >
159{
160  typedef IterativeSolverBase<BiCGSTAB> 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;
166public:
167  typedef _MatrixType MatrixType;
168  typedef typename MatrixType::Scalar Scalar;
169  typedef typename MatrixType::RealScalar RealScalar;
170  typedef _Preconditioner Preconditioner;
171
172public:
173
174  /** Default constructor. */
175  BiCGSTAB() : Base() {}
176
177  /** Initialize the solver with matrix \a A for further \c Ax=b solving.
178    *
179    * This constructor is a shortcut for the default constructor followed
180    * by a call to compute().
181    *
182    * \warning this class stores a reference to the matrix A as well as some
183    * precomputed values that depend on it. Therefore, if \a A is changed
184    * this class becomes invalid. Call compute() to update it with the new
185    * matrix A, or modify a copy of A.
186    */
187  template<typename MatrixDerived>
188  explicit BiCGSTAB(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
189
190  ~BiCGSTAB() {}
191
192  /** \internal */
193  template<typename Rhs,typename Dest>
194  void _solve_with_guess_impl(const Rhs& b, Dest& x) const
195  {
196    bool failed = false;
197    for(Index j=0; j<b.cols(); ++j)
198    {
199      m_iterations = Base::maxIterations();
200      m_error = Base::m_tolerance;
201
202      typename Dest::ColXpr xj(x,j);
203      if(!internal::bicgstab(matrix(), b.col(j), xj, Base::m_preconditioner, m_iterations, m_error))
204        failed = true;
205    }
206    m_info = failed ? NumericalIssue
207           : m_error <= Base::m_tolerance ? Success
208           : NoConvergence;
209    m_isInitialized = true;
210  }
211
212  /** \internal */
213  using Base::_solve_impl;
214  template<typename Rhs,typename Dest>
215  void _solve_impl(const MatrixBase<Rhs>& b, Dest& x) const
216  {
217    x.resize(this->rows(),b.cols());
218    x.setZero();
219    _solve_with_guess_impl(b,x);
220  }
221
222protected:
223
224};
225
226} // end namespace Eigen
227
228#endif // EIGEN_BICGSTAB_H
229