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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
14namespace Eigen {
15
16namespace internal {
17
55template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
56bool 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
210template< typename _MatrixType,
211 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
212class GMRES;
213
214namespace internal {
215
216template< typename _MatrixType, typename _Preconditioner>
217struct traits<GMRES<_MatrixType,_Preconditioner> >
218{
219 typedef _MatrixType MatrixType;
220 typedef _Preconditioner Preconditioner;
221};
222
223}
224
259template< typename _MatrixType, typename _Preconditioner>
260class 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
269private:
270 Index m_restart;
271
272public:
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
279public:
280
282 GMRES() : Base(), m_restart(30) {}
283
294 template<typename MatrixDerived>
295 explicit GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}
296
297 ~GMRES() {}
298
301 Index get_restart() { return m_restart; }
302
306 void set_restart(const Index restart) { m_restart=restart; }
307
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
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
337protected:
338
339};
340
341} // end namespace Eigen
342
343#endif // EIGEN_GMRES_H
A GMRES solver for sparse square problems.
Definition: GMRES.h:261
GMRES()
Definition: GMRES.h:282
GMRES(const EigenBase< MatrixDerived > &A)
Definition: GMRES.h:295
void set_restart(const Index restart)
Definition: GMRES.h:306
Index get_restart()
Definition: GMRES.h:301
bool gmres(const MatrixType &mat, const Rhs &rhs, Dest &x, const Preconditioner &precond, Index &iters, const Index &restart, typename Dest::RealScalar &tol_error)
Definition: GMRES.h:56
Namespace containing all symbols from the Eigen library.
Definition: AdolcForward:45