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: 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,int & iters,const int & restart,typename Dest::RealScalar & tol_error)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
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.
250 *
251 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
252 */
253 template< typename _MatrixType, typename _Preconditioner>
254 class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
255 {
256 typedef IterativeSolverBase<GMRES> Base;
257 using Base::mp_matrix;
258 using Base::m_error;
259 using Base::m_iterations;
260 using Base::m_info;
261 using Base::m_isInitialized;
262
263 private:
264 int m_restart;
265
266 public:
267 typedef _MatrixType MatrixType;
268 typedef typename MatrixType::Scalar Scalar;
269 typedef typename MatrixType::Index Index;
270 typedef typename MatrixType::RealScalar RealScalar;
271 typedef _Preconditioner Preconditioner;
272
273 public:
274
275 /** Default constructor. */
276 GMRES() : Base(), m_restart(30) {}
277
278 /** Initialize the solver with matrix \a A for further \c Ax=b solving.
279 *
280 * This constructor is a shortcut for the default constructor followed
281 * by a call to compute().
282 *
283 * \warning this class stores a reference to the matrix A as well as some
284 * precomputed values that depend on it. Therefore, if \a A is changed
285 * this class becomes invalid. Call compute() to update it with the new
286 * matrix A, or modify a copy of A.
287 */
288 GMRES(const MatrixType& A) : Base(A), m_restart(30) {}
289
290 ~GMRES() {}
291
292 /** Get the number of iterations after that a restart is performed.
293 */
294 int get_restart() { return m_restart; }
295
296 /** Set the number of iterations after that a restart is performed.
297 * \param restart number of iterations for a restarti, default is 30.
298 */
299 void set_restart(const int restart) { m_restart=restart; }
300
301 /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
302 * \a x0 as an initial solution.
303 *
304 * \sa compute()
305 */
306 template<typename Rhs,typename Guess>
307 inline const internal::solve_retval_with_guess<GMRES, Rhs, Guess>
308 solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
309 {
310 eigen_assert(m_isInitialized && "GMRES is not initialized.");
311 eigen_assert(Base::rows()==b.rows()
312 && "GMRES::solve(): invalid number of rows of the right hand side matrix b");
313 return internal::solve_retval_with_guess
314 <GMRES, Rhs, Guess>(*this, b.derived(), x0);
315 }
316
317 /** \internal */
318 template<typename Rhs,typename Dest>
319 void _solveWithGuess(const Rhs& b, Dest& x) const
320 {
321 bool failed = false;
322 for(int j=0; j<b.cols(); ++j)
323 {
324 m_iterations = Base::maxIterations();
325 m_error = Base::m_tolerance;
326
327 typename Dest::ColXpr xj(x,j);
328 if(!internal::gmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_restart, m_error))
329 failed = true;
330 }
331 m_info = failed ? NumericalIssue
332 : m_error <= Base::m_tolerance ? Success
333 : NoConvergence;
334 m_isInitialized = true;
335 }
336
337 /** \internal */
338 template<typename Rhs,typename Dest>
339 void _solve(const Rhs& b, Dest& x) const
340 {
341 x = b;
342 if(x.squaredNorm() == 0) return; // Check Zero right hand side
343 _solveWithGuess(b,x);
344 }
345
346 protected:
347
348 };
349
350
351 namespace internal {
352
353 template<typename _MatrixType, typename _Preconditioner, typename Rhs>
354 struct solve_retval<GMRES<_MatrixType, _Preconditioner>, Rhs>
355 : solve_retval_base<GMRES<_MatrixType, _Preconditioner>, Rhs>
356 {
357 typedef GMRES<_MatrixType, _Preconditioner> Dec;
358 EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
359
360 template<typename Dest> void evalTo(Dest& dst) const
361 {
362 dec()._solve(rhs(),dst);
363 }
364 };
365
366 } // end namespace internal
367
368 } // end namespace Eigen
369
370 #endif // EIGEN_GMRES_H
371