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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 
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 */
55 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
gmres(const MatrixType & mat,const Rhs & rhs,Dest & x,const Preconditioner & precond,Index & iters,const Index & restart,typename Dest::RealScalar & tol_error)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   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 
210 template< typename _MatrixType,
211           typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
212 class GMRES;
213 
214 namespace internal {
215 
216 template< typename _MatrixType, typename _Preconditioner>
217 struct 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   */
259 template< typename _MatrixType, typename _Preconditioner>
260 class 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 
269 private:
270   Index m_restart;
271 
272 public:
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 
279 public:
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 
337 protected:
338 
339 };
340 
341 } // end namespace Eigen
342 
343 #endif // EIGEN_GMRES_H
344