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: 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		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
206template< typename _MatrixType,
207          typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
208class GMRES;
209
210namespace internal {
211
212template< typename _MatrixType, typename _Preconditioner>
213struct traits<GMRES<_MatrixType,_Preconditioner> >
214{
215  typedef _MatrixType MatrixType;
216  typedef _Preconditioner Preconditioner;
217};
218
219}
220
221/** \ingroup IterativeLinearSolvers_Module
222  * \brief A GMRES solver for sparse square problems
223  *
224  * This class allows to solve for A.x = b sparse linear problems using a generalized minimal
225  * residual method. The vectors x and b can be either dense or sparse.
226  *
227  * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
228  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
229  *
230  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
231  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
232  * and NumTraits<Scalar>::epsilon() for the tolerance.
233  *
234  * This class can be used as the direct solver classes. Here is a typical usage example:
235  * \code
236  * int n = 10000;
237  * VectorXd x(n), b(n);
238  * SparseMatrix<double> A(n,n);
239  * // fill A and b
240  * GMRES<SparseMatrix<double> > solver(A);
241  * x = solver.solve(b);
242  * std::cout << "#iterations:     " << solver.iterations() << std::endl;
243  * std::cout << "estimated error: " << solver.error()      << std::endl;
244  * // update b, and solve again
245  * x = solver.solve(b);
246  * \endcode
247  *
248  * By default the iterations start with x=0 as an initial guess of the solution.
249  * One can control the start using the solveWithGuess() method. Here is a step by
250  * step execution example starting with a random guess and printing the evolution
251  * of the estimated error:
252  * * \code
253  * x = VectorXd::Random(n);
254  * solver.setMaxIterations(1);
255  * int i = 0;
256  * do {
257  *   x = solver.solveWithGuess(b,x);
258  *   std::cout << i << " : " << solver.error() << std::endl;
259  *   ++i;
260  * } while (solver.info()!=Success && i<100);
261  * \endcode
262  * Note that such a step by step excution is slightly slower.
263  *
264  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
265  */
266template< typename _MatrixType, typename _Preconditioner>
267class 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
276private:
277  int m_restart;
278
279public:
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
286public:
287
288  /** Default constructor. */
289  GMRES() : Base(), m_restart(30) {}
290
291  /** Initialize the solver with matrix \a A for further \c Ax=b solving.
292    *
293    * This constructor is a shortcut for the default constructor followed
294    * by a call to compute().
295    *
296    * \warning this class stores a reference to the matrix A as well as some
297    * precomputed values that depend on it. Therefore, if \a A is changed
298    * this class becomes invalid. Call compute() to update it with the new
299    * matrix A, or modify a copy of A.
300    */
301  GMRES(const MatrixType& A) : Base(A), m_restart(30) {}
302
303  ~GMRES() {}
304
305  /** Get the number of iterations after that a restart is performed.
306    */
307  int get_restart() { return m_restart; }
308
309  /** Set the number of iterations after that a restart is performed.
310    *  \param restart   number of iterations for a restarti, default is 30.
311    */
312  void set_restart(const int restart) { m_restart=restart; }
313
314  /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
315    * \a x0 as an initial solution.
316    *
317    * \sa compute()
318    */
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
330  /** \internal */
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
350  /** \internal */
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
359protected:
360
361};
362
363
364namespace internal {
365
366  template<typename _MatrixType, typename _Preconditioner, typename Rhs>
367struct 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
384