source: pacpussensors/trunk/Vislab/lib3dv/eigen/Eigen/src/Eigenvalues/Tridiagonalization.h@ 136

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1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008 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_TRIDIAGONALIZATION_H
12#define EIGEN_TRIDIAGONALIZATION_H
13
14namespace Eigen {
15
16namespace internal {
17
18template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
19template<typename MatrixType>
20struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
21{
22 typedef typename MatrixType::PlainObject ReturnType;
23};
24
25template<typename MatrixType, typename CoeffVectorType>
26void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
27}
28
29/** \eigenvalues_module \ingroup Eigenvalues_Module
30 *
31 *
32 * \class Tridiagonalization
33 *
34 * \brief Tridiagonal decomposition of a selfadjoint matrix
35 *
36 * \tparam _MatrixType the type of the matrix of which we are computing the
37 * tridiagonal decomposition; this is expected to be an instantiation of the
38 * Matrix class template.
39 *
40 * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
41 * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
42 *
43 * A tridiagonal matrix is a matrix which has nonzero elements only on the
44 * main diagonal and the first diagonal below and above it. The Hessenberg
45 * decomposition of a selfadjoint matrix is in fact a tridiagonal
46 * decomposition. This class is used in SelfAdjointEigenSolver to compute the
47 * eigenvalues and eigenvectors of a selfadjoint matrix.
48 *
49 * Call the function compute() to compute the tridiagonal decomposition of a
50 * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
51 * constructor which computes the tridiagonal Schur decomposition at
52 * construction time. Once the decomposition is computed, you can use the
53 * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
54 * decomposition.
55 *
56 * The documentation of Tridiagonalization(const MatrixType&) contains an
57 * example of the typical use of this class.
58 *
59 * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
60 */
61template<typename _MatrixType> class Tridiagonalization
62{
63 public:
64
65 /** \brief Synonym for the template parameter \p _MatrixType. */
66 typedef _MatrixType MatrixType;
67
68 typedef typename MatrixType::Scalar Scalar;
69 typedef typename NumTraits<Scalar>::Real RealScalar;
70 typedef typename MatrixType::Index Index;
71
72 enum {
73 Size = MatrixType::RowsAtCompileTime,
74 SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
75 Options = MatrixType::Options,
76 MaxSize = MatrixType::MaxRowsAtCompileTime,
77 MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
78 };
79
80 typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
81 typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
82 typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
83 typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
84 typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
85
86 typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
87 typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
88 const Diagonal<const MatrixType>
89 >::type DiagonalReturnType;
90
91 typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
92 typename internal::add_const_on_value_type<typename Diagonal<
93 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
94 const Diagonal<
95 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
96 >::type SubDiagonalReturnType;
97
98 /** \brief Return type of matrixQ() */
99 typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
100
101 /** \brief Default constructor.
102 *
103 * \param [in] size Positive integer, size of the matrix whose tridiagonal
104 * decomposition will be computed.
105 *
106 * The default constructor is useful in cases in which the user intends to
107 * perform decompositions via compute(). The \p size parameter is only
108 * used as a hint. It is not an error to give a wrong \p size, but it may
109 * impair performance.
110 *
111 * \sa compute() for an example.
112 */
113 Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
114 : m_matrix(size,size),
115 m_hCoeffs(size > 1 ? size-1 : 1),
116 m_isInitialized(false)
117 {}
118
119 /** \brief Constructor; computes tridiagonal decomposition of given matrix.
120 *
121 * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
122 * is to be computed.
123 *
124 * This constructor calls compute() to compute the tridiagonal decomposition.
125 *
126 * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
127 * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
128 */
129 Tridiagonalization(const MatrixType& matrix)
130 : m_matrix(matrix),
131 m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
132 m_isInitialized(false)
133 {
134 internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
135 m_isInitialized = true;
136 }
137
138 /** \brief Computes tridiagonal decomposition of given matrix.
139 *
140 * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
141 * is to be computed.
142 * \returns Reference to \c *this
143 *
144 * The tridiagonal decomposition is computed by bringing the columns of
145 * the matrix successively in the required form using Householder
146 * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
147 * the size of the given matrix.
148 *
149 * This method reuses of the allocated data in the Tridiagonalization
150 * object, if the size of the matrix does not change.
151 *
152 * Example: \include Tridiagonalization_compute.cpp
153 * Output: \verbinclude Tridiagonalization_compute.out
154 */
155 Tridiagonalization& compute(const MatrixType& matrix)
156 {
157 m_matrix = matrix;
158 m_hCoeffs.resize(matrix.rows()-1, 1);
159 internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
160 m_isInitialized = true;
161 return *this;
162 }
163
164 /** \brief Returns the Householder coefficients.
165 *
166 * \returns a const reference to the vector of Householder coefficients
167 *
168 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
169 * the member function compute(const MatrixType&) has been called before
170 * to compute the tridiagonal decomposition of a matrix.
171 *
172 * The Householder coefficients allow the reconstruction of the matrix
173 * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
174 *
175 * Example: \include Tridiagonalization_householderCoefficients.cpp
176 * Output: \verbinclude Tridiagonalization_householderCoefficients.out
177 *
178 * \sa packedMatrix(), \ref Householder_Module "Householder module"
179 */
180 inline CoeffVectorType householderCoefficients() const
181 {
182 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
183 return m_hCoeffs;
184 }
185
186 /** \brief Returns the internal representation of the decomposition
187 *
188 * \returns a const reference to a matrix with the internal representation
189 * of the decomposition.
190 *
191 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
192 * the member function compute(const MatrixType&) has been called before
193 * to compute the tridiagonal decomposition of a matrix.
194 *
195 * The returned matrix contains the following information:
196 * - the strict upper triangular part is equal to the input matrix A.
197 * - the diagonal and lower sub-diagonal represent the real tridiagonal
198 * symmetric matrix T.
199 * - the rest of the lower part contains the Householder vectors that,
200 * combined with Householder coefficients returned by
201 * householderCoefficients(), allows to reconstruct the matrix Q as
202 * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
203 * Here, the matrices \f$ H_i \f$ are the Householder transformations
204 * \f$ H_i = (I - h_i v_i v_i^T) \f$
205 * where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
206 * \f$ v_i \f$ is the Householder vector defined by
207 * \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
208 * with M the matrix returned by this function.
209 *
210 * See LAPACK for further details on this packed storage.
211 *
212 * Example: \include Tridiagonalization_packedMatrix.cpp
213 * Output: \verbinclude Tridiagonalization_packedMatrix.out
214 *
215 * \sa householderCoefficients()
216 */
217 inline const MatrixType& packedMatrix() const
218 {
219 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
220 return m_matrix;
221 }
222
223 /** \brief Returns the unitary matrix Q in the decomposition
224 *
225 * \returns object representing the matrix Q
226 *
227 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
228 * the member function compute(const MatrixType&) has been called before
229 * to compute the tridiagonal decomposition of a matrix.
230 *
231 * This function returns a light-weight object of template class
232 * HouseholderSequence. You can either apply it directly to a matrix or
233 * you can convert it to a matrix of type #MatrixType.
234 *
235 * \sa Tridiagonalization(const MatrixType&) for an example,
236 * matrixT(), class HouseholderSequence
237 */
238 HouseholderSequenceType matrixQ() const
239 {
240 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
241 return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
242 .setLength(m_matrix.rows() - 1)
243 .setShift(1);
244 }
245
246 /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
247 *
248 * \returns expression object representing the matrix T
249 *
250 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
251 * the member function compute(const MatrixType&) has been called before
252 * to compute the tridiagonal decomposition of a matrix.
253 *
254 * Currently, this function can be used to extract the matrix T from internal
255 * data and copy it to a dense matrix object. In most cases, it may be
256 * sufficient to directly use the packed matrix or the vector expressions
257 * returned by diagonal() and subDiagonal() instead of creating a new
258 * dense copy matrix with this function.
259 *
260 * \sa Tridiagonalization(const MatrixType&) for an example,
261 * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
262 */
263 MatrixTReturnType matrixT() const
264 {
265 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
266 return MatrixTReturnType(m_matrix.real());
267 }
268
269 /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
270 *
271 * \returns expression representing the diagonal of T
272 *
273 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
274 * the member function compute(const MatrixType&) has been called before
275 * to compute the tridiagonal decomposition of a matrix.
276 *
277 * Example: \include Tridiagonalization_diagonal.cpp
278 * Output: \verbinclude Tridiagonalization_diagonal.out
279 *
280 * \sa matrixT(), subDiagonal()
281 */
282 DiagonalReturnType diagonal() const;
283
284 /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
285 *
286 * \returns expression representing the subdiagonal of T
287 *
288 * \pre Either the constructor Tridiagonalization(const MatrixType&) or
289 * the member function compute(const MatrixType&) has been called before
290 * to compute the tridiagonal decomposition of a matrix.
291 *
292 * \sa diagonal() for an example, matrixT()
293 */
294 SubDiagonalReturnType subDiagonal() const;
295
296 protected:
297
298 MatrixType m_matrix;
299 CoeffVectorType m_hCoeffs;
300 bool m_isInitialized;
301};
302
303template<typename MatrixType>
304typename Tridiagonalization<MatrixType>::DiagonalReturnType
305Tridiagonalization<MatrixType>::diagonal() const
306{
307 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
308 return m_matrix.diagonal();
309}
310
311template<typename MatrixType>
312typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
313Tridiagonalization<MatrixType>::subDiagonal() const
314{
315 eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
316 Index n = m_matrix.rows();
317 return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
318}
319
320namespace internal {
321
322/** \internal
323 * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
324 *
325 * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
326 * On output, the strict upper part is left unchanged, and the lower triangular part
327 * represents the T and Q matrices in packed format has detailed below.
328 * \param[out] hCoeffs returned Householder coefficients (see below)
329 *
330 * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
331 * and lower sub-diagonal of the matrix \a matA.
332 * The unitary matrix Q is represented in a compact way as a product of
333 * Householder reflectors \f$ H_i \f$ such that:
334 * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
335 * The Householder reflectors are defined as
336 * \f$ H_i = (I - h_i v_i v_i^T) \f$
337 * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
338 * \f$ v_i \f$ is the Householder vector defined by
339 * \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
340 *
341 * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
342 *
343 * \sa Tridiagonalization::packedMatrix()
344 */
345template<typename MatrixType, typename CoeffVectorType>
346void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
347{
348 using numext::conj;
349 typedef typename MatrixType::Index Index;
350 typedef typename MatrixType::Scalar Scalar;
351 typedef typename MatrixType::RealScalar RealScalar;
352 Index n = matA.rows();
353 eigen_assert(n==matA.cols());
354 eigen_assert(n==hCoeffs.size()+1 || n==1);
355
356 for (Index i = 0; i<n-1; ++i)
357 {
358 Index remainingSize = n-i-1;
359 RealScalar beta;
360 Scalar h;
361 matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
362
363 // Apply similarity transformation to remaining columns,
364 // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
365 matA.col(i).coeffRef(i+1) = 1;
366
367 hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
368 * (conj(h) * matA.col(i).tail(remainingSize)));
369
370 hCoeffs.tail(n-i-1) += (conj(h)*RealScalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
371
372 matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
373 .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
374
375 matA.col(i).coeffRef(i+1) = beta;
376 hCoeffs.coeffRef(i) = h;
377 }
378}
379
380// forward declaration, implementation at the end of this file
381template<typename MatrixType,
382 int Size=MatrixType::ColsAtCompileTime,
383 bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
384struct tridiagonalization_inplace_selector;
385
386/** \brief Performs a full tridiagonalization in place
387 *
388 * \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
389 * decomposition is to be computed. Only the lower triangular part referenced.
390 * The rest is left unchanged. On output, the orthogonal matrix Q
391 * in the decomposition if \p extractQ is true.
392 * \param[out] diag The diagonal of the tridiagonal matrix T in the
393 * decomposition.
394 * \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
395 * the decomposition.
396 * \param[in] extractQ If true, the orthogonal matrix Q in the
397 * decomposition is computed and stored in \p mat.
398 *
399 * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
400 * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
401 * symmetric tridiagonal matrix.
402 *
403 * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
404 * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
405 * part of the matrix \p mat is destroyed.
406 *
407 * The vectors \p diag and \p subdiag are not resized. The function
408 * assumes that they are already of the correct size. The length of the
409 * vector \p diag should equal the number of rows in \p mat, and the
410 * length of the vector \p subdiag should be one left.
411 *
412 * This implementation contains an optimized path for 3-by-3 matrices
413 * which is especially useful for plane fitting.
414 *
415 * \note Currently, it requires two temporary vectors to hold the intermediate
416 * Householder coefficients, and to reconstruct the matrix Q from the Householder
417 * reflectors.
418 *
419 * Example (this uses the same matrix as the example in
420 * Tridiagonalization::Tridiagonalization(const MatrixType&)):
421 * \include Tridiagonalization_decomposeInPlace.cpp
422 * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
423 *
424 * \sa class Tridiagonalization
425 */
426template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
427void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
428{
429 eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
430 tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
431}
432
433/** \internal
434 * General full tridiagonalization
435 */
436template<typename MatrixType, int Size, bool IsComplex>
437struct tridiagonalization_inplace_selector
438{
439 typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
440 typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
441 typedef typename MatrixType::Index Index;
442 template<typename DiagonalType, typename SubDiagonalType>
443 static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
444 {
445 CoeffVectorType hCoeffs(mat.cols()-1);
446 tridiagonalization_inplace(mat,hCoeffs);
447 diag = mat.diagonal().real();
448 subdiag = mat.template diagonal<-1>().real();
449 if(extractQ)
450 mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
451 .setLength(mat.rows() - 1)
452 .setShift(1);
453 }
454};
455
456/** \internal
457 * Specialization for 3x3 real matrices.
458 * Especially useful for plane fitting.
459 */
460template<typename MatrixType>
461struct tridiagonalization_inplace_selector<MatrixType,3,false>
462{
463 typedef typename MatrixType::Scalar Scalar;
464 typedef typename MatrixType::RealScalar RealScalar;
465
466 template<typename DiagonalType, typename SubDiagonalType>
467 static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
468 {
469 using std::sqrt;
470 diag[0] = mat(0,0);
471 RealScalar v1norm2 = numext::abs2(mat(2,0));
472 if(v1norm2 == RealScalar(0))
473 {
474 diag[1] = mat(1,1);
475 diag[2] = mat(2,2);
476 subdiag[0] = mat(1,0);
477 subdiag[1] = mat(2,1);
478 if (extractQ)
479 mat.setIdentity();
480 }
481 else
482 {
483 RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
484 RealScalar invBeta = RealScalar(1)/beta;
485 Scalar m01 = mat(1,0) * invBeta;
486 Scalar m02 = mat(2,0) * invBeta;
487 Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
488 diag[1] = mat(1,1) + m02*q;
489 diag[2] = mat(2,2) - m02*q;
490 subdiag[0] = beta;
491 subdiag[1] = mat(2,1) - m01 * q;
492 if (extractQ)
493 {
494 mat << 1, 0, 0,
495 0, m01, m02,
496 0, m02, -m01;
497 }
498 }
499 }
500};
501
502/** \internal
503 * Trivial specialization for 1x1 matrices
504 */
505template<typename MatrixType, bool IsComplex>
506struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
507{
508 typedef typename MatrixType::Scalar Scalar;
509
510 template<typename DiagonalType, typename SubDiagonalType>
511 static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
512 {
513 diag(0,0) = numext::real(mat(0,0));
514 if(extractQ)
515 mat(0,0) = Scalar(1);
516 }
517};
518
519/** \internal
520 * \eigenvalues_module \ingroup Eigenvalues_Module
521 *
522 * \brief Expression type for return value of Tridiagonalization::matrixT()
523 *
524 * \tparam MatrixType type of underlying dense matrix
525 */
526template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
527: public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
528{
529 typedef typename MatrixType::Index Index;
530 public:
531 /** \brief Constructor.
532 *
533 * \param[in] mat The underlying dense matrix
534 */
535 TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
536
537 template <typename ResultType>
538 inline void evalTo(ResultType& result) const
539 {
540 result.setZero();
541 result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
542 result.diagonal() = m_matrix.diagonal();
543 result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
544 }
545
546 Index rows() const { return m_matrix.rows(); }
547 Index cols() const { return m_matrix.cols(); }
548
549 protected:
550 typename MatrixType::Nested m_matrix;
551};
552
553} // end namespace internal
554
555} // end namespace Eigen
556
557#endif // EIGEN_TRIDIAGONALIZATION_H
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