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