1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
5 // Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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_GENERALIZEDSELFADJOINTEIGENSOLVER_H
12 #define EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H
13
14 #include "./Tridiagonalization.h"
15
16 namespace Eigen {
17
18 /** \eigenvalues_module \ingroup Eigenvalues_Module
19 *
20 *
21 * \class GeneralizedSelfAdjointEigenSolver
22 *
23 * \brief Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem
24 *
25 * \tparam _MatrixType the type of the matrix of which we are computing the
26 * eigendecomposition; this is expected to be an instantiation of the Matrix
27 * class template.
28 *
29 * This class solves the generalized eigenvalue problem
30 * \f$ Av = \lambda Bv \f$. In this case, the matrix \f$ A \f$ should be
31 * selfadjoint and the matrix \f$ B \f$ should be positive definite.
32 *
33 * Only the \b lower \b triangular \b part of the input matrix is referenced.
34 *
35 * Call the function compute() to compute the eigenvalues and eigenvectors of
36 * a given matrix. Alternatively, you can use the
37 * GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
38 * constructor which computes the eigenvalues and eigenvectors at construction time.
39 * Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues()
40 * and eigenvectors() functions.
41 *
42 * The documentation for GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
43 * contains an example of the typical use of this class.
44 *
45 * \sa class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver
46 */
47 template<typename _MatrixType>
48 class GeneralizedSelfAdjointEigenSolver : public SelfAdjointEigenSolver<_MatrixType>
49 {
50 typedef SelfAdjointEigenSolver<_MatrixType> Base;
51 public:
52
53 typedef _MatrixType MatrixType;
54
55 /** \brief Default constructor for fixed-size matrices.
56 *
57 * The default constructor is useful in cases in which the user intends to
58 * perform decompositions via compute(). This constructor
59 * can only be used if \p _MatrixType is a fixed-size matrix; use
60 * GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.
61 */
GeneralizedSelfAdjointEigenSolver()62 GeneralizedSelfAdjointEigenSolver() : Base() {}
63
64 /** \brief Constructor, pre-allocates memory for dynamic-size matrices.
65 *
66 * \param [in] size Positive integer, size of the matrix whose
67 * eigenvalues and eigenvectors will be computed.
68 *
69 * This constructor is useful for dynamic-size matrices, when the user
70 * intends to perform decompositions via compute(). The \p size
71 * parameter is only used as a hint. It is not an error to give a wrong
72 * \p size, but it may impair performance.
73 *
74 * \sa compute() for an example
75 */
GeneralizedSelfAdjointEigenSolver(Index size)76 explicit GeneralizedSelfAdjointEigenSolver(Index size)
77 : Base(size)
78 {}
79
80 /** \brief Constructor; computes generalized eigendecomposition of given matrix pencil.
81 *
82 * \param[in] matA Selfadjoint matrix in matrix pencil.
83 * Only the lower triangular part of the matrix is referenced.
84 * \param[in] matB Positive-definite matrix in matrix pencil.
85 * Only the lower triangular part of the matrix is referenced.
86 * \param[in] options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
87 * Default is #ComputeEigenvectors|#Ax_lBx.
88 *
89 * This constructor calls compute(const MatrixType&, const MatrixType&, int)
90 * to compute the eigenvalues and (if requested) the eigenvectors of the
91 * generalized eigenproblem \f$ Ax = \lambda B x \f$ with \a matA the
92 * selfadjoint matrix \f$ A \f$ and \a matB the positive definite matrix
93 * \f$ B \f$. Each eigenvector \f$ x \f$ satisfies the property
94 * \f$ x^* B x = 1 \f$. The eigenvectors are computed if
95 * \a options contains ComputeEigenvectors.
96 *
97 * In addition, the two following variants can be solved via \p options:
98 * - \c ABx_lx: \f$ ABx = \lambda x \f$
99 * - \c BAx_lx: \f$ BAx = \lambda x \f$
100 *
101 * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.cpp
102 * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.out
103 *
104 * \sa compute(const MatrixType&, const MatrixType&, int)
105 */
106 GeneralizedSelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB,
107 int options = ComputeEigenvectors|Ax_lBx)
108 : Base(matA.cols())
109 {
110 compute(matA, matB, options);
111 }
112
113 /** \brief Computes generalized eigendecomposition of given matrix pencil.
114 *
115 * \param[in] matA Selfadjoint matrix in matrix pencil.
116 * Only the lower triangular part of the matrix is referenced.
117 * \param[in] matB Positive-definite matrix in matrix pencil.
118 * Only the lower triangular part of the matrix is referenced.
119 * \param[in] options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
120 * Default is #ComputeEigenvectors|#Ax_lBx.
121 *
122 * \returns Reference to \c *this
123 *
124 * Accoring to \p options, this function computes eigenvalues and (if requested)
125 * the eigenvectors of one of the following three generalized eigenproblems:
126 * - \c Ax_lBx: \f$ Ax = \lambda B x \f$
127 * - \c ABx_lx: \f$ ABx = \lambda x \f$
128 * - \c BAx_lx: \f$ BAx = \lambda x \f$
129 * with \a matA the selfadjoint matrix \f$ A \f$ and \a matB the positive definite
130 * matrix \f$ B \f$.
131 * In addition, each eigenvector \f$ x \f$ satisfies the property \f$ x^* B x = 1 \f$.
132 *
133 * The eigenvalues() function can be used to retrieve
134 * the eigenvalues. If \p options contains ComputeEigenvectors, then the
135 * eigenvectors are also computed and can be retrieved by calling
136 * eigenvectors().
137 *
138 * The implementation uses LLT to compute the Cholesky decomposition
139 * \f$ B = LL^* \f$ and computes the classical eigendecomposition
140 * of the selfadjoint matrix \f$ L^{-1} A (L^*)^{-1} \f$ if \p options contains Ax_lBx
141 * and of \f$ L^{*} A L \f$ otherwise. This solves the
142 * generalized eigenproblem, because any solution of the generalized
143 * eigenproblem \f$ Ax = \lambda B x \f$ corresponds to a solution
144 * \f$ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) \f$ of the
145 * eigenproblem for \f$ L^{-1} A (L^*)^{-1} \f$. Similar statements
146 * can be made for the two other variants.
147 *
148 * Example: \include SelfAdjointEigenSolver_compute_MatrixType2.cpp
149 * Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType2.out
150 *
151 * \sa GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
152 */
153 GeneralizedSelfAdjointEigenSolver& compute(const MatrixType& matA, const MatrixType& matB,
154 int options = ComputeEigenvectors|Ax_lBx);
155
156 protected:
157
158 };
159
160
161 template<typename MatrixType>
162 GeneralizedSelfAdjointEigenSolver<MatrixType>& GeneralizedSelfAdjointEigenSolver<MatrixType>::
compute(const MatrixType & matA,const MatrixType & matB,int options)163 compute(const MatrixType& matA, const MatrixType& matB, int options)
164 {
165 eigen_assert(matA.cols()==matA.rows() && matB.rows()==matA.rows() && matB.cols()==matB.rows());
166 eigen_assert((options&~(EigVecMask|GenEigMask))==0
167 && (options&EigVecMask)!=EigVecMask
168 && ((options&GenEigMask)==0 || (options&GenEigMask)==Ax_lBx
169 || (options&GenEigMask)==ABx_lx || (options&GenEigMask)==BAx_lx)
170 && "invalid option parameter");
171
172 bool computeEigVecs = ((options&EigVecMask)==0) || ((options&EigVecMask)==ComputeEigenvectors);
173
174 // Compute the cholesky decomposition of matB = L L' = U'U
175 LLT<MatrixType> cholB(matB);
176
177 int type = (options&GenEigMask);
178 if(type==0)
179 type = Ax_lBx;
180
181 if(type==Ax_lBx)
182 {
183 // compute C = inv(L) A inv(L')
184 MatrixType matC = matA.template selfadjointView<Lower>();
185 cholB.matrixL().template solveInPlace<OnTheLeft>(matC);
186 cholB.matrixU().template solveInPlace<OnTheRight>(matC);
187
188 Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly );
189
190 // transform back the eigen vectors: evecs = inv(U) * evecs
191 if(computeEigVecs)
192 cholB.matrixU().solveInPlace(Base::m_eivec);
193 }
194 else if(type==ABx_lx)
195 {
196 // compute C = L' A L
197 MatrixType matC = matA.template selfadjointView<Lower>();
198 matC = matC * cholB.matrixL();
199 matC = cholB.matrixU() * matC;
200
201 Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
202
203 // transform back the eigen vectors: evecs = inv(U) * evecs
204 if(computeEigVecs)
205 cholB.matrixU().solveInPlace(Base::m_eivec);
206 }
207 else if(type==BAx_lx)
208 {
209 // compute C = L' A L
210 MatrixType matC = matA.template selfadjointView<Lower>();
211 matC = matC * cholB.matrixL();
212 matC = cholB.matrixU() * matC;
213
214 Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
215
216 // transform back the eigen vectors: evecs = L * evecs
217 if(computeEigVecs)
218 Base::m_eivec = cholB.matrixL() * Base::m_eivec;
219 }
220
221 return *this;
222 }
223
224 } // end namespace Eigen
225
226 #endif // EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H
227