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GMRES.h
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 
14 namespace Eigen {
15 
16 namespace internal {
17 
55 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
56 bool 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  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
68 
69  if(rhs.norm() <= considerAsZero)
70  {
71  x.setZero();
72  tol_error = 0;
73  return true;
74  }
75 
76  RealScalar tol = tol_error;
77  const Index maxIters = iters;
78  iters = 0;
79 
80  const Index m = mat.rows();
81 
82  // residual and preconditioned residual
83  VectorType p0 = rhs - mat*x;
84  VectorType r0 = precond.solve(p0);
85 
86  const RealScalar r0Norm = r0.norm();
87 
88  // is initial guess already good enough?
89  if(r0Norm == 0)
90  {
91  tol_error = 0;
92  return true;
93  }
94 
95  // storage for Hessenberg matrix and Householder data
96  FMatrixType H = FMatrixType::Zero(m, restart + 1);
97  VectorType w = VectorType::Zero(restart + 1);
98  VectorType tau = VectorType::Zero(restart + 1);
99 
100  // storage for Jacobi rotations
101  std::vector < JacobiRotation < Scalar > > G(restart);
102 
103  // storage for temporaries
104  VectorType t(m), v(m), workspace(m), x_new(m);
105 
106  // generate first Householder vector
107  Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
108  RealScalar beta;
109  r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
110  w(0) = Scalar(beta);
111 
112  for (Index k = 1; k <= restart; ++k)
113  {
114  ++iters;
115 
116  v = VectorType::Unit(m, k - 1);
117 
118  // apply Householder reflections H_{1} ... H_{k-1} to v
119  // TODO: use a HouseholderSequence
120  for (Index i = k - 1; i >= 0; --i) {
121  v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
122  }
123 
124  // apply matrix M to v: v = mat * v;
125  t.noalias() = mat * v;
126  v = precond.solve(t);
127 
128  // apply Householder reflections H_{k-1} ... H_{1} to v
129  // TODO: use a HouseholderSequence
130  for (Index i = 0; i < k; ++i) {
131  v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
132  }
133 
134  if (v.tail(m - k).norm() != 0.0)
135  {
136  if (k <= restart)
137  {
138  // generate new Householder vector
139  Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
140  v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
141 
142  // apply Householder reflection H_{k} to v
143  v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
144  }
145  }
146 
147  if (k > 1)
148  {
149  for (Index i = 0; i < k - 1; ++i)
150  {
151  // apply old Givens rotations to v
152  v.applyOnTheLeft(i, i + 1, G[i].adjoint());
153  }
154  }
155 
156  if (k<m && v(k) != (Scalar) 0)
157  {
158  // determine next Givens rotation
159  G[k - 1].makeGivens(v(k - 1), v(k));
160 
161  // apply Givens rotation to v and w
162  v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
163  w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
164  }
165 
166  // insert coefficients into upper matrix triangle
167  H.col(k-1).head(k) = v.head(k);
168 
169  tol_error = abs(w(k)) / r0Norm;
170  bool stop = (k==m || tol_error < tol || iters == maxIters);
171 
172  if (stop || k == restart)
173  {
174  // solve upper triangular system
175  Ref<VectorType> y = w.head(k);
176  H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
177 
178  // use Horner-like scheme to calculate solution vector
179  x_new.setZero();
180  for (Index i = k - 1; i >= 0; --i)
181  {
182  x_new(i) += y(i);
183  // apply Householder reflection H_{i} to x_new
184  x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
185  }
186 
187  x += x_new;
188 
189  if(stop)
190  {
191  return true;
192  }
193  else
194  {
195  k=0;
196 
197  // reset data for restart
198  p0.noalias() = rhs - mat*x;
199  r0 = precond.solve(p0);
200 
201  // clear Hessenberg matrix and Householder data
202  H.setZero();
203  w.setZero();
204  tau.setZero();
205 
206  // generate first Householder vector
207  r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
208  w(0) = Scalar(beta);
209  }
210  }
211  }
212 
213  return false;
214 
215 }
216 
217 }
218 
219 template< typename _MatrixType,
220  typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
221 class GMRES;
222 
223 namespace internal {
224 
225 template< typename _MatrixType, typename _Preconditioner>
226 struct traits<GMRES<_MatrixType,_Preconditioner> >
227 {
228  typedef _MatrixType MatrixType;
229  typedef _Preconditioner Preconditioner;
230 };
231 
232 }
233 
268 template< typename _MatrixType, typename _Preconditioner>
269 class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
270 {
272  using Base::matrix;
273  using Base::m_error;
274  using Base::m_iterations;
275  using Base::m_info;
276  using Base::m_isInitialized;
277 
278 private:
279  Index m_restart;
280 
281 public:
282  using Base::_solve_impl;
283  typedef _MatrixType MatrixType;
284  typedef typename MatrixType::Scalar Scalar;
285  typedef typename MatrixType::RealScalar RealScalar;
286  typedef _Preconditioner Preconditioner;
287 
288 public:
289 
291  GMRES() : Base(), m_restart(30) {}
292 
303  template<typename MatrixDerived>
304  explicit GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}
305 
306  ~GMRES() {}
307 
310  Index get_restart() { return m_restart; }
311 
315  void set_restart(const Index restart) { m_restart=restart; }
316 
318  template<typename Rhs,typename Dest>
319  void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
320  {
321  m_iterations = Base::maxIterations();
322  m_error = Base::m_tolerance;
323  bool ret = internal::gmres(matrix(), b, x, Base::m_preconditioner, m_iterations, m_restart, m_error);
324  m_info = (!ret) ? NumericalIssue
325  : m_error <= Base::m_tolerance ? Success
326  : NoConvergence;
327  }
328 
329 protected:
330 
331 };
332 
333 } // end namespace Eigen
334 
335 #endif // EIGEN_GMRES_H
GMRES()
Definition: GMRES.h:291
Index get_restart()
Definition: GMRES.h:310
Namespace containing all symbols from the Eigen library.
NumericalIssue
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
GMRES(const EigenBase< MatrixDerived > &A)
Definition: GMRES.h:304
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
void set_restart(const Index restart)
Definition: GMRES.h:315
A GMRES solver for sparse square problems.
Definition: GMRES.h:221