1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <g.gael@free.fr>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_HESSENBERGDECOMPOSITION_H
#define EIGEN_HESSENBERGDECOMPOSITION_H
/** \eigenvalues_module \ingroup Eigenvalues_Module
* \nonstableyet
*
* \class HessenbergDecomposition
*
* \brief Reduces a squared matrix to an Hessemberg form
*
* \param MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
*
* This class performs an Hessenberg decomposition of a matrix \f$ A \f$ such that:
* \f$ A = Q H Q^* \f$ where \f$ Q \f$ is unitary and \f$ H \f$ a Hessenberg matrix.
*
* \sa class Tridiagonalization, class Qr
*/
template<typename _MatrixType> class HessenbergDecomposition
{
public:
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
enum {
Size = MatrixType::RowsAtCompileTime,
SizeMinusOne = MatrixType::RowsAtCompileTime==Dynamic
? Dynamic
: MatrixType::RowsAtCompileTime-1
};
typedef Matrix<Scalar, SizeMinusOne, 1> CoeffVectorType;
/** This constructor initializes a HessenbergDecomposition object for
* further use with HessenbergDecomposition::compute()
*/
HessenbergDecomposition(int size = Size==Dynamic ? 2 : Size)
: m_matrix(size,size)
{
if(size>1)
m_hCoeffs.resize(size-1);
}
HessenbergDecomposition(const MatrixType& matrix)
: m_matrix(matrix)
{
if(matrix.rows()<2)
return;
m_hCoeffs.resize(matrix.rows()-1,1);
_compute(m_matrix, m_hCoeffs);
}
/** Computes or re-compute the Hessenberg decomposition for the matrix \a matrix.
*
* This method allows to re-use the allocated data.
*/
void compute(const MatrixType& matrix)
{
m_matrix = matrix;
if(matrix.rows()<2)
return;
m_hCoeffs.resize(matrix.rows()-1,1);
_compute(m_matrix, m_hCoeffs);
}
/** \returns a const reference to the householder coefficients allowing to
* reconstruct the matrix Q from the packed data.
*
* \sa packedMatrix()
*/
const CoeffVectorType& householderCoefficients() const { return m_hCoeffs; }
/** \returns a const reference to the internal representation of the decomposition.
*
* The returned matrix contains the following information:
* - the upper part and lower sub-diagonal represent the Hessenberg matrix H
* - the rest of the lower part contains the Householder vectors that, combined with
* Householder coefficients returned by householderCoefficients(),
* allows to reconstruct the matrix Q as follow:
* Q = H_{N-1} ... H_1 H_0
* where the matrices H are the Householder transformation:
* H_i = (I - h_i * v_i * v_i')
* where h_i == householderCoefficients()[i] and v_i is a Householder vector:
* v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]
*
* See LAPACK for further details on this packed storage.
*/
const MatrixType& packedMatrix(void) const { return m_matrix; }
MatrixType matrixQ() const;
MatrixType matrixH() const;
private:
static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
protected:
MatrixType m_matrix;
CoeffVectorType m_hCoeffs;
};
#ifndef EIGEN_HIDE_HEAVY_CODE
/** \internal
* Performs a tridiagonal decomposition of \a matA in place.
*
* \param matA the input selfadjoint matrix
* \param hCoeffs returned Householder coefficients
*
* The result is written in the lower triangular part of \a matA.
*
* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
*
* \sa packedMatrix()
*/
template<typename MatrixType>
void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
{
assert(matA.rows()==matA.cols());
int n = matA.rows();
Matrix<Scalar,1,Dynamic> temp(n);
for (int i = 0; i<n-1; ++i)
{
// let's consider the vector v = i-th column starting at position i+1
int remainingSize = n-i-1;
RealScalar beta;
Scalar h;
matA.col(i).end(remainingSize).makeHouseholderInPlace(h, beta);
matA.col(i).coeffRef(i+1) = beta;
hCoeffs.coeffRef(i) = h;
// Apply similarity transformation to remaining columns,
// i.e., compute A = H A H'
// A = H A
matA.corner(BottomRight, remainingSize, remainingSize)
.applyHouseholderOnTheLeft(matA.col(i).end(remainingSize-1), h, &temp.coeffRef(0));
// A = A H'
matA.corner(BottomRight, n, remainingSize)
.applyHouseholderOnTheRight(matA.col(i).end(remainingSize-1).conjugate(), ei_conj(h), &temp.coeffRef(0));
}
}
/** reconstructs and returns the matrix Q */
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
HessenbergDecomposition<MatrixType>::matrixQ() const
{
int n = m_matrix.rows();
MatrixType matQ = MatrixType::Identity(n,n);
Matrix<Scalar,1,MatrixType::ColsAtCompileTime> temp(n);
for (int i = n-2; i>=0; i--)
{
matQ.corner(BottomRight,n-i-1,n-i-1)
.applyHouseholderOnTheLeft(m_matrix.col(i).end(n-i-2), ei_conj(m_hCoeffs.coeff(i)), &temp.coeffRef(0,0));
}
return matQ;
}
#endif // EIGEN_HIDE_HEAVY_CODE
/** constructs and returns the matrix H.
* Note that the matrix H is equivalent to the upper part of the packed matrix
* (including the lower sub-diagonal). Therefore, it might be often sufficient
* to directly use the packed matrix instead of creating a new one.
*/
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
HessenbergDecomposition<MatrixType>::matrixH() const
{
// FIXME should this function (and other similar) rather take a matrix as argument
// and fill it (to avoid temporaries)
int n = m_matrix.rows();
MatrixType matH = m_matrix;
if (n>2)
matH.corner(BottomLeft,n-2, n-2).template triangularView<LowerTriangular>().setZero();
return matH;
}
#endif // EIGEN_HESSENBERGDECOMPOSITION_H
|
