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// Copyright (C) 2012, 2014 Kolja Brix <brix@igpm.rwth-aaachen.de>
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_GMRES_H
12#define EIGEN_GMRES_H
13
14namespace Eigen {
15
16namespace internal {
17
18/**
19* Generalized Minimal Residual Algorithm based on the
20* Arnoldi algorithm implemented with Householder reflections.
21*
22* Parameters:
23*  \param mat       matrix of linear system of equations
24*  \param Rhs       right hand side vector of linear system of equations
25*  \param x         on input: initial guess, on output: solution
26*  \param precond   preconditioner used
27*  \param iters     on input: maximum number of iterations to perform
28*                   on output: number of iterations performed
29*  \param restart   number of iterations for a restart
30*  \param tol_error on input: relative residual tolerance
31*                   on output: residuum achieved
32*
33* \sa IterativeMethods::bicgstab()
34*
35*
36* For references, please see:
37*
38* Saad, Y. and Schultz, M. H.
39* GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
40* SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
41*
42* Saad, Y.
43* Iterative Methods for Sparse Linear Systems.
44* Society for Industrial and Applied Mathematics, Philadelphia, 2003.
45*
46* Walker, H. F.
47* Implementations of the GMRES method.
48* Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
49*
50* Walker, H. F.
51* Implementation of the GMRES Method using Householder Transformations.
52* SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
53*
54*/
55template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
56bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
57    Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) {
58
59  using std::sqrt;
60  using std::abs;
61
62  typedef typename Dest::RealScalar RealScalar;
63  typedef typename Dest::Scalar Scalar;
64  typedef Matrix < Scalar, Dynamic, 1 > VectorType;
65  typedef Matrix < Scalar, Dynamic, Dynamic, ColMajor> FMatrixType;
66
67  RealScalar tol = tol_error;
68  const Index maxIters = iters;
69  iters = 0;
70
71  const Index m = mat.rows();
72
73  // residual and preconditioned residual
74  VectorType p0 = rhs - mat*x;
75  VectorType r0 = precond.solve(p0);
76
77  const RealScalar r0Norm = r0.norm();
78
79  // is initial guess already good enough?
80  if(r0Norm == 0)
81  {
82    tol_error = 0;
83    return true;
84  }
85
86  // storage for Hessenberg matrix and Householder data
87  FMatrixType H   = FMatrixType::Zero(m, restart + 1);
88  VectorType w    = VectorType::Zero(restart + 1);
89  VectorType tau  = VectorType::Zero(restart + 1);
90
91  // storage for Jacobi rotations
92  std::vector < JacobiRotation < Scalar > > G(restart);
93
94  // storage for temporaries
95  VectorType t(m), v(m), workspace(m), x_new(m);
96
97  // generate first Householder vector
98  Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
99  RealScalar beta;
100  r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
101  w(0) = Scalar(beta);
102
103  for (Index k = 1; k <= restart; ++k)
104  {
105    ++iters;
106
107    v = VectorType::Unit(m, k - 1);
108
109    // apply Householder reflections H_{1} ... H_{k-1} to v
110    // TODO: use a HouseholderSequence
111    for (Index i = k - 1; i >= 0; --i) {
112      v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
113    }
114
115    // apply matrix M to v:  v = mat * v;
116    t.noalias() = mat * v;
117    v = precond.solve(t);
118
119    // apply Householder reflections H_{k-1} ... H_{1} to v
120    // TODO: use a HouseholderSequence
121    for (Index i = 0; i < k; ++i) {
122      v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
123    }
124
125    if (v.tail(m - k).norm() != 0.0)
126    {
127      if (k <= restart)
128      {
129        // generate new Householder vector
130        Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
131        v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
132
133        // apply Householder reflection H_{k} to v
134        v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
135      }
136    }
137
138    if (k > 1)
139    {
140      for (Index i = 0; i < k - 1; ++i)
141      {
142        // apply old Givens rotations to v
143        v.applyOnTheLeft(i, i + 1, G[i].adjoint());
144      }
145    }
146
147    if (k<m && v(k) != (Scalar) 0)
148    {
149      // determine next Givens rotation
150      G[k - 1].makeGivens(v(k - 1), v(k));
151
152      // apply Givens rotation to v and w
153      v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
154      w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
155    }
156
157    // insert coefficients into upper matrix triangle
158    H.col(k-1).head(k) = v.head(k);
159
160    tol_error = abs(w(k)) / r0Norm;
161    bool stop = (k==m || tol_error < tol || iters == maxIters);
162
163    if (stop || k == restart)
164    {
165      // solve upper triangular system
166      Ref<VectorType> y = w.head(k);
167      H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
168
169      // use Horner-like scheme to calculate solution vector
170      x_new.setZero();
171      for (Index i = k - 1; i >= 0; --i)
172      {
173        x_new(i) += y(i);
174        // apply Householder reflection H_{i} to x_new
175        x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
176      }
177
178      x += x_new;
179
180      if(stop)
181      {
182        return true;
183      }
184      else
185      {
186        k=0;
187
188        // reset data for restart
189        p0.noalias() = rhs - mat*x;
190        r0 = precond.solve(p0);
191
192        // clear Hessenberg matrix and Householder data
193        H.setZero();
194        w.setZero();
195        tau.setZero();
196
197        // generate first Householder vector
198        r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
199        w(0) = Scalar(beta);
200      }
201    }
202  }
203
204  return false;
205
206}
207
208}
209
210template< typename _MatrixType,
211          typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
212class GMRES;
213
214namespace internal {
215
216template< typename _MatrixType, typename _Preconditioner>
217struct traits<GMRES<_MatrixType,_Preconditioner> >
218{
219  typedef _MatrixType MatrixType;
220  typedef _Preconditioner Preconditioner;
221};
222
223}
224
225/** \ingroup IterativeLinearSolvers_Module
226  * \brief A GMRES solver for sparse square problems
227  *
228  * This class allows to solve for A.x = b sparse linear problems using a generalized minimal
229  * residual method. The vectors x and b can be either dense or sparse.
230  *
231  * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
232  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
233  *
234  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
235  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
236  * and NumTraits<Scalar>::epsilon() for the tolerance.
237  *
238  * This class can be used as the direct solver classes. Here is a typical usage example:
239  * \code
240  * int n = 10000;
241  * VectorXd x(n), b(n);
242  * SparseMatrix<double> A(n,n);
243  * // fill A and b
244  * GMRES<SparseMatrix<double> > solver(A);
245  * x = solver.solve(b);
246  * std::cout << "#iterations:     " << solver.iterations() << std::endl;
247  * std::cout << "estimated error: " << solver.error()      << std::endl;
248  * // update b, and solve again
249  * x = solver.solve(b);
250  * \endcode
251  *
252  * By default the iterations start with x=0 as an initial guess of the solution.
253  * One can control the start using the solveWithGuess() method.
254  *
255  * GMRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
256  *
257  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
258  */
259template< typename _MatrixType, typename _Preconditioner>
260class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
261{
262  typedef IterativeSolverBase<GMRES> Base;
263  using Base::matrix;
264  using Base::m_error;
265  using Base::m_iterations;
266  using Base::m_info;
267  using Base::m_isInitialized;
268
269private:
270  Index m_restart;
271
272public:
273  using Base::_solve_impl;
274  typedef _MatrixType MatrixType;
275  typedef typename MatrixType::Scalar Scalar;
276  typedef typename MatrixType::RealScalar RealScalar;
277  typedef _Preconditioner Preconditioner;
278
279public:
280
281  /** Default constructor. */
282  GMRES() : Base(), m_restart(30) {}
283
284  /** Initialize the solver with matrix \a A for further \c Ax=b solving.
285    *
286    * This constructor is a shortcut for the default constructor followed
287    * by a call to compute().
288    *
289    * \warning this class stores a reference to the matrix A as well as some
290    * precomputed values that depend on it. Therefore, if \a A is changed
291    * this class becomes invalid. Call compute() to update it with the new
292    * matrix A, or modify a copy of A.
293    */
294  template<typename MatrixDerived>
295  explicit GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}
296
297  ~GMRES() {}
298
299  /** Get the number of iterations after that a restart is performed.
300    */
301  Index get_restart() { return m_restart; }
302
303  /** Set the number of iterations after that a restart is performed.
304    *  \param restart   number of iterations for a restarti, default is 30.
305    */
306  void set_restart(const Index restart) { m_restart=restart; }
307
308  /** \internal */
309  template<typename Rhs,typename Dest>
310  void _solve_with_guess_impl(const Rhs& b, Dest& x) const
311  {
312    bool failed = false;
313    for(Index j=0; j<b.cols(); ++j)
314    {
315      m_iterations = Base::maxIterations();
316      m_error = Base::m_tolerance;
317
318      typename Dest::ColXpr xj(x,j);
319      if(!internal::gmres(matrix(), b.col(j), xj, Base::m_preconditioner, m_iterations, m_restart, m_error))
320        failed = true;
321    }
322    m_info = failed ? NumericalIssue
323          : m_error <= Base::m_tolerance ? Success
324          : NoConvergence;
325    m_isInitialized = true;
326  }
327
328  /** \internal */
329  template<typename Rhs,typename Dest>
330  void _solve_impl(const Rhs& b, MatrixBase<Dest> &x) const
331  {
332    x = b;
333    if(x.squaredNorm() == 0) return; // Check Zero right hand side
334    _solve_with_guess_impl(b,x.derived());
335  }
336
337protected:
338
339};
340
341} // end namespace Eigen
342
343#endif // EIGEN_GMRES_H
344