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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  int &iters, const int &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 > FMatrixType;
66 
67  RealScalar tol = tol_error;
68  const int maxIters = iters;
69  iters = 0;
70 
71  const int m = mat.rows();
72 
73  VectorType p0 = rhs - mat*x;
74  VectorType r0 = precond.solve(p0);
75 
76  // is initial guess already good enough?
77  if(abs(r0.norm()) < tol) {
78  return true;
79  }
80 
81  VectorType w = VectorType::Zero(restart + 1);
82 
83  FMatrixType H = FMatrixType::Zero(m, restart + 1); // Hessenberg matrix
84  VectorType tau = VectorType::Zero(restart + 1);
85  std::vector < JacobiRotation < Scalar > > G(restart);
86 
87  // generate first Householder vector
88  VectorType e(m-1);
89  RealScalar beta;
90  r0.makeHouseholder(e, tau.coeffRef(0), beta);
91  w(0)=(Scalar) beta;
92  H.bottomLeftCorner(m - 1, 1) = e;
93 
94  for (int k = 1; k <= restart; ++k) {
95 
96  ++iters;
97 
98  VectorType v = VectorType::Unit(m, k - 1), workspace(m);
99 
100  // apply Householder reflections H_{1} ... H_{k-1} to v
101  for (int i = k - 1; i >= 0; --i) {
102  v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
103  }
104 
105  // apply matrix M to v: v = mat * v;
106  VectorType t=mat*v;
107  v=precond.solve(t);
108 
109  // apply Householder reflections H_{k-1} ... H_{1} to v
110  for (int i = 0; i < k; ++i) {
111  v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
112  }
113 
114  if (v.tail(m - k).norm() != 0.0) {
115 
116  if (k <= restart) {
117 
118  // generate new Householder vector
119  VectorType e(m - k - 1);
120  RealScalar beta;
121  v.tail(m - k).makeHouseholder(e, tau.coeffRef(k), beta);
122  H.col(k).tail(m - k - 1) = e;
123 
124  // apply Householder reflection H_{k} to v
125  v.tail(m - k).applyHouseholderOnTheLeft(H.col(k).tail(m - k - 1), tau.coeffRef(k), workspace.data());
126 
127  }
128  }
129 
130  if (k > 1) {
131  for (int i = 0; i < k - 1; ++i) {
132  // apply old Givens rotations to v
133  v.applyOnTheLeft(i, i + 1, G[i].adjoint());
134  }
135  }
136 
137  if (k<m && v(k) != (Scalar) 0) {
138  // determine next Givens rotation
139  G[k - 1].makeGivens(v(k - 1), v(k));
140 
141  // apply Givens rotation to v and w
142  v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
143  w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
144 
145  }
146 
147  // insert coefficients into upper matrix triangle
148  H.col(k - 1).head(k) = v.head(k);
149 
150  bool stop=(k==m || abs(w(k)) < tol || iters == maxIters);
151 
152  if (stop || k == restart) {
153 
154  // solve upper triangular system
155  VectorType y = w.head(k);
156  H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y);
157 
158  // use Horner-like scheme to calculate solution vector
159  VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
160 
161  // apply Householder reflection H_{k} to x_new
162  x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
163 
164  for (int i = k - 2; i >= 0; --i) {
165  x_new += y(i) * VectorType::Unit(m, i);
166  // apply Householder reflection H_{i} to x_new
167  x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
168  }
169 
170  x += x_new;
171 
172  if (stop) {
173  return true;
174  } else {
175  k=0;
176 
177  // reset data for a restart r0 = rhs - mat * x;
178  VectorType p0=mat*x;
179  VectorType p1=precond.solve(p0);
180  r0 = rhs - p1;
181 // r0_sqnorm = r0.squaredNorm();
182  w = VectorType::Zero(restart + 1);
183  H = FMatrixType::Zero(m, restart + 1);
184  tau = VectorType::Zero(restart + 1);
185 
186  // generate first Householder vector
187  RealScalar beta;
188  r0.makeHouseholder(e, tau.coeffRef(0), beta);
189  w(0)=(Scalar) beta;
190  H.bottomLeftCorner(m - 1, 1) = e;
191 
192  }
193 
194  }
195 
196 
197 
198  }
199 
200  return false;
201 
202 }
203 
204 }
205 
206 template< typename _MatrixType,
207  typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
208 class GMRES;
209 
210 namespace internal {
211 
212 template< typename _MatrixType, typename _Preconditioner>
213 struct traits<GMRES<_MatrixType,_Preconditioner> >
214 {
215  typedef _MatrixType MatrixType;
216  typedef _Preconditioner Preconditioner;
217 };
218 
219 }
220 
266 template< typename _MatrixType, typename _Preconditioner>
267 class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
268 {
269  typedef IterativeSolverBase<GMRES> Base;
270  using Base::mp_matrix;
271  using Base::m_error;
272  using Base::m_iterations;
273  using Base::m_info;
274  using Base::m_isInitialized;
275 
276 private:
277  int m_restart;
278 
279 public:
280  typedef _MatrixType MatrixType;
281  typedef typename MatrixType::Scalar Scalar;
282  typedef typename MatrixType::Index Index;
283  typedef typename MatrixType::RealScalar RealScalar;
284  typedef _Preconditioner Preconditioner;
285 
286 public:
287 
289  GMRES() : Base(), m_restart(30) {}
290 
301  GMRES(const MatrixType& A) : Base(A), m_restart(30) {}
302 
303  ~GMRES() {}
304 
307  int get_restart() { return m_restart; }
308 
312  void set_restart(const int restart) { m_restart=restart; }
313 
319  template<typename Rhs,typename Guess>
320  inline const internal::solve_retval_with_guess<GMRES, Rhs, Guess>
321  solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
322  {
323  eigen_assert(m_isInitialized && "GMRES is not initialized.");
324  eigen_assert(Base::rows()==b.rows()
325  && "GMRES::solve(): invalid number of rows of the right hand side matrix b");
326  return internal::solve_retval_with_guess
327  <GMRES, Rhs, Guess>(*this, b.derived(), x0);
328  }
329 
331  template<typename Rhs,typename Dest>
332  void _solveWithGuess(const Rhs& b, Dest& x) const
333  {
334  bool failed = false;
335  for(int j=0; j<b.cols(); ++j)
336  {
337  m_iterations = Base::maxIterations();
338  m_error = Base::m_tolerance;
339 
340  typename Dest::ColXpr xj(x,j);
341  if(!internal::gmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_restart, m_error))
342  failed = true;
343  }
344  m_info = failed ? NumericalIssue
345  : m_error <= Base::m_tolerance ? Success
346  : NoConvergence;
347  m_isInitialized = true;
348  }
349 
351  template<typename Rhs,typename Dest>
352  void _solve(const Rhs& b, Dest& x) const
353  {
354  x = b;
355  if(x.squaredNorm() == 0) return; // Check Zero right hand side
356  _solveWithGuess(b,x);
357  }
358 
359 protected:
360 
361 };
362 
363 
364 namespace internal {
365 
366  template<typename _MatrixType, typename _Preconditioner, typename Rhs>
367 struct solve_retval<GMRES<_MatrixType, _Preconditioner>, Rhs>
368  : solve_retval_base<GMRES<_MatrixType, _Preconditioner>, Rhs>
369 {
370  typedef GMRES<_MatrixType, _Preconditioner> Dec;
371  EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
372 
373  template<typename Dest> void evalTo(Dest& dst) const
374  {
375  dec()._solve(rhs(),dst);
376  }
377 };
378 
379 } // end namespace internal
380 
381 } // end namespace Eigen
382 
383 #endif // EIGEN_GMRES_H
const internal::solve_retval_with_guess< GMRES, Rhs, Guess > solveWithGuess(const MatrixBase< Rhs > &b, const Guess &x0) const
Definition: GMRES.h:321
GMRES()
Definition: GMRES.h:289
void set_restart(const int restart)
Definition: GMRES.h:312
GMRES(const MatrixType &A)
Definition: GMRES.h:301
A GMRES solver for sparse square problems.
Definition: GMRES.h:208
int get_restart()
Definition: GMRES.h:307