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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
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// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_TRIDIAGONALIZATION_H
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#define EIGEN_TRIDIAGONALIZATION_H
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namespace Eigen {
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namespace internal {
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template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
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template<typename MatrixType>
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struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
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{
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typedef typename MatrixType::PlainObject ReturnType;
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};
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template<typename MatrixType, typename CoeffVectorType>
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void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
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}
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/** \eigenvalues_module \ingroup Eigenvalues_Module
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*
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*
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* \class Tridiagonalization
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*
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* \brief Tridiagonal decomposition of a selfadjoint matrix
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*
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* \tparam _MatrixType the type of the matrix of which we are computing the
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* tridiagonal decomposition; this is expected to be an instantiation of the
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* Matrix class template.
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*
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* This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
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* \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
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*
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* A tridiagonal matrix is a matrix which has nonzero elements only on the
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* main diagonal and the first diagonal below and above it. The Hessenberg
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* decomposition of a selfadjoint matrix is in fact a tridiagonal
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* decomposition. This class is used in SelfAdjointEigenSolver to compute the
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* eigenvalues and eigenvectors of a selfadjoint matrix.
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*
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* Call the function compute() to compute the tridiagonal decomposition of a
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* given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
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* constructor which computes the tridiagonal Schur decomposition at
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* construction time. Once the decomposition is computed, you can use the
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* matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
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* decomposition.
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*
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* The documentation of Tridiagonalization(const MatrixType&) contains an
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* example of the typical use of this class.
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*
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* \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
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*/
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template<typename _MatrixType> class Tridiagonalization
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{
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public:
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/** \brief Synonym for the template parameter \p _MatrixType. */
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef typename MatrixType::Index Index;
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enum {
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Size = MatrixType::RowsAtCompileTime,
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SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
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Options = MatrixType::Options,
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MaxSize = MatrixType::MaxRowsAtCompileTime,
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MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
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};
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typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
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typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
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typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
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typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
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typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
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typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
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typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
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const Diagonal<const MatrixType>
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>::type DiagonalReturnType;
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typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
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typename internal::add_const_on_value_type<typename Diagonal<
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Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
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const Diagonal<
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Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
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>::type SubDiagonalReturnType;
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/** \brief Return type of matrixQ() */
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typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
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/** \brief Default constructor.
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*
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* \param [in] size Positive integer, size of the matrix whose tridiagonal
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* decomposition will be computed.
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*
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via compute(). The \p size parameter is only
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* used as a hint. It is not an error to give a wrong \p size, but it may
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* impair performance.
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*
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* \sa compute() for an example.
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*/
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Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
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: m_matrix(size,size),
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m_hCoeffs(size > 1 ? size-1 : 1),
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m_isInitialized(false)
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{}
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/** \brief Constructor; computes tridiagonal decomposition of given matrix.
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*
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* \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
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* is to be computed.
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*
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* This constructor calls compute() to compute the tridiagonal decomposition.
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*
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* Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
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* Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
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*/
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Tridiagonalization(const MatrixType& matrix)
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: m_matrix(matrix),
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m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
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m_isInitialized(false)
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{
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internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
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m_isInitialized = true;
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}
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/** \brief Computes tridiagonal decomposition of given matrix.
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*
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* \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
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* is to be computed.
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* \returns Reference to \c *this
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*
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* The tridiagonal decomposition is computed by bringing the columns of
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* the matrix successively in the required form using Householder
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* reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
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* the size of the given matrix.
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*
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* This method reuses of the allocated data in the Tridiagonalization
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* object, if the size of the matrix does not change.
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*
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* Example: \include Tridiagonalization_compute.cpp
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* Output: \verbinclude Tridiagonalization_compute.out
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*/
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Tridiagonalization& compute(const MatrixType& matrix)
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{
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m_matrix = matrix;
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m_hCoeffs.resize(matrix.rows()-1, 1);
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internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
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m_isInitialized = true;
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return *this;
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}
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/** \brief Returns the Householder coefficients.
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*
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* \returns a const reference to the vector of Householder coefficients
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*
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* \pre Either the constructor Tridiagonalization(const MatrixType&) or
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* the member function compute(const MatrixType&) has been called before
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* to compute the tridiagonal decomposition of a matrix.
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*
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* The Householder coefficients allow the reconstruction of the matrix
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* \f$ Q \f$ in the tridiagonal decomposition from the packed data.
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*
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* Example: \include Tridiagonalization_householderCoefficients.cpp
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* Output: \verbinclude Tridiagonalization_householderCoefficients.out
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*
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* \sa packedMatrix(), \ref Householder_Module "Householder module"
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*/
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inline CoeffVectorType householderCoefficients() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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return m_hCoeffs;
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}
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/** \brief Returns the internal representation of the decomposition
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*
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* \returns a const reference to a matrix with the internal representation
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* of the decomposition.
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*
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* \pre Either the constructor Tridiagonalization(const MatrixType&) or
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* the member function compute(const MatrixType&) has been called before
|
| ykuroda |
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193
|
* to compute the tridiagonal decomposition of a matrix.
|
| ykuroda |
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194
|
*
|
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195
|
* The returned matrix contains the following information:
|
| ykuroda |
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|
* - the strict upper triangular part is equal to the input matrix A.
|
| ykuroda |
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|
197
|
* - the diagonal and lower sub-diagonal represent the real tridiagonal
|
| ykuroda |
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198
|
* symmetric matrix T.
|
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|
* - the rest of the lower part contains the Householder vectors that,
|
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|
* combined with Householder coefficients returned by
|
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201
|
* householderCoefficients(), allows to reconstruct the matrix Q as
|
| ykuroda |
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|
202
|
* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
|
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|
203
|
* Here, the matrices \f$ H_i \f$ are the Householder transformations
|
| ykuroda |
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204
|
* \f$ H_i = (I - h_i v_i v_i^T) \f$
|
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205
|
* where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
|
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|
* \f$ v_i \f$ is the Householder vector defined by
|
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|
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|
* \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
|
| ykuroda |
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208
|
* with M the matrix returned by this function.
|
| ykuroda |
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209
|
*
|
| ykuroda |
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|
210
|
* See LAPACK for further details on this packed storage.
|
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211
|
*
|
| ykuroda |
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|
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|
* Example: \include Tridiagonalization_packedMatrix.cpp
|
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|
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|
* Output: \verbinclude Tridiagonalization_packedMatrix.out
|
| ykuroda |
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|
214
|
*
|
| ykuroda |
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|
215
|
* \sa householderCoefficients()
|
| ykuroda |
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|
216
|
*/
|
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|
inline const MatrixType& packedMatrix() const
|
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|
218
|
{
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| ykuroda |
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|
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|
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
|
| ykuroda |
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|
220
|
return m_matrix;
|
| ykuroda |
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|
221
|
}
|
| ykuroda |
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|
222
|
|
| ykuroda |
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|
223
|
/** \brief Returns the unitary matrix Q in the decomposition
|
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|
224
|
*
|
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|
* \returns object representing the matrix Q
|
| ykuroda |
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|
226
|
*
|
| ykuroda |
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|
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|
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
|
| ykuroda |
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|
228
|
* the member function compute(const MatrixType&) has been called before
|
| ykuroda |
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|
229
|
* to compute the tridiagonal decomposition of a matrix.
|
| ykuroda |
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|
230
|
*
|
| ykuroda |
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|
231
|
* This function returns a light-weight object of template class
|
| ykuroda |
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|
232
|
* HouseholderSequence. You can either apply it directly to a matrix or
|
| ykuroda |
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|
233
|
* you can convert it to a matrix of type #MatrixType.
|
| ykuroda |
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|
234
|
*
|
| ykuroda |
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|
235
|
* \sa Tridiagonalization(const MatrixType&) for an example,
|
| ykuroda |
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|
236
|
* matrixT(), class HouseholderSequence
|
| ykuroda |
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|
237
|
*/
|
| ykuroda |
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|
238
|
HouseholderSequenceType matrixQ() const
|
| ykuroda |
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|
239
|
{
|
| ykuroda |
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|
240
|
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
|
| ykuroda |
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|
241
|
return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
|
| ykuroda |
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|
242
|
.setLength(m_matrix.rows() - 1)
|
| ykuroda |
0:13a5d365ba16
|
243
|
.setShift(1);
|
| ykuroda |
0:13a5d365ba16
|
244
|
}
|
| ykuroda |
0:13a5d365ba16
|
245
|
|
| ykuroda |
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|
246
|
/** \brief Returns an expression of the tridiagonal matrix T in the decomposition
|
| ykuroda |
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|
247
|
*
|
| ykuroda |
0:13a5d365ba16
|
248
|
* \returns expression object representing the matrix T
|
| ykuroda |
0:13a5d365ba16
|
249
|
*
|
| ykuroda |
0:13a5d365ba16
|
250
|
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
|
| ykuroda |
0:13a5d365ba16
|
251
|
* the member function compute(const MatrixType&) has been called before
|
| ykuroda |
0:13a5d365ba16
|
252
|
* to compute the tridiagonal decomposition of a matrix.
|
| ykuroda |
0:13a5d365ba16
|
253
|
*
|
| ykuroda |
0:13a5d365ba16
|
254
|
* Currently, this function can be used to extract the matrix T from internal
|
| ykuroda |
0:13a5d365ba16
|
255
|
* data and copy it to a dense matrix object. In most cases, it may be
|
| ykuroda |
0:13a5d365ba16
|
256
|
* sufficient to directly use the packed matrix or the vector expressions
|
| ykuroda |
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|
257
|
* returned by diagonal() and subDiagonal() instead of creating a new
|
| ykuroda |
0:13a5d365ba16
|
258
|
* dense copy matrix with this function.
|
| ykuroda |
0:13a5d365ba16
|
259
|
*
|
| ykuroda |
0:13a5d365ba16
|
260
|
* \sa Tridiagonalization(const MatrixType&) for an example,
|
| ykuroda |
0:13a5d365ba16
|
261
|
* matrixQ(), packedMatrix(), diagonal(), subDiagonal()
|
| ykuroda |
0:13a5d365ba16
|
262
|
*/
|
| ykuroda |
0:13a5d365ba16
|
263
|
MatrixTReturnType matrixT() const
|
| ykuroda |
0:13a5d365ba16
|
264
|
{
|
| ykuroda |
0:13a5d365ba16
|
265
|
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
|
| ykuroda |
0:13a5d365ba16
|
266
|
return MatrixTReturnType(m_matrix.real());
|
| ykuroda |
0:13a5d365ba16
|
267
|
}
|
| ykuroda |
0:13a5d365ba16
|
268
|
|
| ykuroda |
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|
269
|
/** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
|
| ykuroda |
0:13a5d365ba16
|
270
|
*
|
| ykuroda |
0:13a5d365ba16
|
271
|
* \returns expression representing the diagonal of T
|
| ykuroda |
0:13a5d365ba16
|
272
|
*
|
| ykuroda |
0:13a5d365ba16
|
273
|
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
|
| ykuroda |
0:13a5d365ba16
|
274
|
* the member function compute(const MatrixType&) has been called before
|
| ykuroda |
0:13a5d365ba16
|
275
|
* to compute the tridiagonal decomposition of a matrix.
|
| ykuroda |
0:13a5d365ba16
|
276
|
*
|
| ykuroda |
0:13a5d365ba16
|
277
|
* Example: \include Tridiagonalization_diagonal.cpp
|
| ykuroda |
0:13a5d365ba16
|
278
|
* Output: \verbinclude Tridiagonalization_diagonal.out
|
| ykuroda |
0:13a5d365ba16
|
279
|
*
|
| ykuroda |
0:13a5d365ba16
|
280
|
* \sa matrixT(), subDiagonal()
|
| ykuroda |
0:13a5d365ba16
|
281
|
*/
|
| ykuroda |
0:13a5d365ba16
|
282
|
DiagonalReturnType diagonal() const;
|
| ykuroda |
0:13a5d365ba16
|
283
|
|
| ykuroda |
0:13a5d365ba16
|
284
|
/** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
|
| ykuroda |
0:13a5d365ba16
|
285
|
*
|
| ykuroda |
0:13a5d365ba16
|
286
|
* \returns expression representing the subdiagonal of T
|
| ykuroda |
0:13a5d365ba16
|
287
|
*
|
| ykuroda |
0:13a5d365ba16
|
288
|
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
|
| ykuroda |
0:13a5d365ba16
|
289
|
* the member function compute(const MatrixType&) has been called before
|
| ykuroda |
0:13a5d365ba16
|
290
|
* to compute the tridiagonal decomposition of a matrix.
|
| ykuroda |
0:13a5d365ba16
|
291
|
*
|
| ykuroda |
0:13a5d365ba16
|
292
|
* \sa diagonal() for an example, matrixT()
|
| ykuroda |
0:13a5d365ba16
|
293
|
*/
|
| ykuroda |
0:13a5d365ba16
|
294
|
SubDiagonalReturnType subDiagonal() const;
|
| ykuroda |
0:13a5d365ba16
|
295
|
|
| ykuroda |
0:13a5d365ba16
|
296
|
protected:
|
| ykuroda |
0:13a5d365ba16
|
297
|
|
| ykuroda |
0:13a5d365ba16
|
298
|
MatrixType m_matrix;
|
| ykuroda |
0:13a5d365ba16
|
299
|
CoeffVectorType m_hCoeffs;
|
| ykuroda |
0:13a5d365ba16
|
300
|
bool m_isInitialized;
|
| ykuroda |
0:13a5d365ba16
|
301
|
};
|
| ykuroda |
0:13a5d365ba16
|
302
|
|
| ykuroda |
0:13a5d365ba16
|
303
|
template<typename MatrixType>
|
| ykuroda |
0:13a5d365ba16
|
304
|
typename Tridiagonalization<MatrixType>::DiagonalReturnType
|
| ykuroda |
0:13a5d365ba16
|
305
|
Tridiagonalization<MatrixType>::diagonal() const
|
| ykuroda |
0:13a5d365ba16
|
306
|
{
|
| ykuroda |
0:13a5d365ba16
|
307
|
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
|
| ykuroda |
0:13a5d365ba16
|
308
|
return m_matrix.diagonal();
|
| ykuroda |
0:13a5d365ba16
|
309
|
}
|
| ykuroda |
0:13a5d365ba16
|
310
|
|
| ykuroda |
0:13a5d365ba16
|
311
|
template<typename MatrixType>
|
| ykuroda |
0:13a5d365ba16
|
312
|
typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
|
| ykuroda |
0:13a5d365ba16
|
313
|
Tridiagonalization<MatrixType>::subDiagonal() const
|
| ykuroda |
0:13a5d365ba16
|
314
|
{
|
| ykuroda |
0:13a5d365ba16
|
315
|
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
|
| ykuroda |
0:13a5d365ba16
|
316
|
Index n = m_matrix.rows();
|
| ykuroda |
0:13a5d365ba16
|
317
|
return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
|
| ykuroda |
0:13a5d365ba16
|
318
|
}
|
| ykuroda |
0:13a5d365ba16
|
319
|
|
| ykuroda |
0:13a5d365ba16
|
320
|
namespace internal {
|
| ykuroda |
0:13a5d365ba16
|
321
|
|
| ykuroda |
0:13a5d365ba16
|
322
|
/** \internal
|
| ykuroda |
0:13a5d365ba16
|
323
|
* Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
|
| ykuroda |
0:13a5d365ba16
|
324
|
*
|
| ykuroda |
0:13a5d365ba16
|
325
|
* \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
|
| ykuroda |
0:13a5d365ba16
|
326
|
* On output, the strict upper part is left unchanged, and the lower triangular part
|
| ykuroda |
0:13a5d365ba16
|
327
|
* represents the T and Q matrices in packed format has detailed below.
|
| ykuroda |
0:13a5d365ba16
|
328
|
* \param[out] hCoeffs returned Householder coefficients (see below)
|
| ykuroda |
0:13a5d365ba16
|
329
|
*
|
| ykuroda |
0:13a5d365ba16
|
330
|
* On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
|
| ykuroda |
0:13a5d365ba16
|
331
|
* and lower sub-diagonal of the matrix \a matA.
|
| ykuroda |
0:13a5d365ba16
|
332
|
* The unitary matrix Q is represented in a compact way as a product of
|
| ykuroda |
0:13a5d365ba16
|
333
|
* Householder reflectors \f$ H_i \f$ such that:
|
| ykuroda |
0:13a5d365ba16
|
334
|
* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
|
| ykuroda |
0:13a5d365ba16
|
335
|
* The Householder reflectors are defined as
|
| ykuroda |
0:13a5d365ba16
|
336
|
* \f$ H_i = (I - h_i v_i v_i^T) \f$
|
| ykuroda |
0:13a5d365ba16
|
337
|
* where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
|
| ykuroda |
0:13a5d365ba16
|
338
|
* \f$ v_i \f$ is the Householder vector defined by
|
| ykuroda |
0:13a5d365ba16
|
339
|
* \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
|
| ykuroda |
0:13a5d365ba16
|
340
|
*
|
| ykuroda |
0:13a5d365ba16
|
341
|
* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
|
| ykuroda |
0:13a5d365ba16
|
342
|
*
|
| ykuroda |
0:13a5d365ba16
|
343
|
* \sa Tridiagonalization::packedMatrix()
|
| ykuroda |
0:13a5d365ba16
|
344
|
*/
|
| ykuroda |
0:13a5d365ba16
|
345
|
template<typename MatrixType, typename CoeffVectorType>
|
| ykuroda |
0:13a5d365ba16
|
346
|
void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
|
| ykuroda |
0:13a5d365ba16
|
347
|
{
|
| ykuroda |
0:13a5d365ba16
|
348
|
using numext::conj;
|
| ykuroda |
0:13a5d365ba16
|
349
|
typedef typename MatrixType::Index Index;
|
| ykuroda |
0:13a5d365ba16
|
350
|
typedef typename MatrixType::Scalar Scalar;
|
| ykuroda |
0:13a5d365ba16
|
351
|
typedef typename MatrixType::RealScalar RealScalar;
|
| ykuroda |
0:13a5d365ba16
|
352
|
Index n = matA.rows();
|
| ykuroda |
0:13a5d365ba16
|
353
|
eigen_assert(n==matA.cols());
|
| ykuroda |
0:13a5d365ba16
|
354
|
eigen_assert(n==hCoeffs.size()+1 || n==1);
|
| ykuroda |
0:13a5d365ba16
|
355
|
|
| ykuroda |
0:13a5d365ba16
|
356
|
for (Index i = 0; i<n-1; ++i)
|
| ykuroda |
0:13a5d365ba16
|
357
|
{
|
| ykuroda |
0:13a5d365ba16
|
358
|
Index remainingSize = n-i-1;
|
| ykuroda |
0:13a5d365ba16
|
359
|
RealScalar beta;
|
| ykuroda |
0:13a5d365ba16
|
360
|
Scalar h;
|
| ykuroda |
0:13a5d365ba16
|
361
|
matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
|
| ykuroda |
0:13a5d365ba16
|
362
|
|
| ykuroda |
0:13a5d365ba16
|
363
|
// Apply similarity transformation to remaining columns,
|
| ykuroda |
0:13a5d365ba16
|
364
|
// i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
|
| ykuroda |
0:13a5d365ba16
|
365
|
matA.col(i).coeffRef(i+1) = 1;
|
| ykuroda |
0:13a5d365ba16
|
366
|
|
| ykuroda |
0:13a5d365ba16
|
367
|
hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
|
| ykuroda |
0:13a5d365ba16
|
368
|
* (conj(h) * matA.col(i).tail(remainingSize)));
|
| ykuroda |
0:13a5d365ba16
|
369
|
|
| ykuroda |
0:13a5d365ba16
|
370
|
hCoeffs.tail(n-i-1) += (conj(h)*Scalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
|
| ykuroda |
0:13a5d365ba16
|
371
|
|
| ykuroda |
0:13a5d365ba16
|
372
|
matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
|
| ykuroda |
0:13a5d365ba16
|
373
|
.rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
|
| ykuroda |
0:13a5d365ba16
|
374
|
|
| ykuroda |
0:13a5d365ba16
|
375
|
matA.col(i).coeffRef(i+1) = beta;
|
| ykuroda |
0:13a5d365ba16
|
376
|
hCoeffs.coeffRef(i) = h;
|
| ykuroda |
0:13a5d365ba16
|
377
|
}
|
| ykuroda |
0:13a5d365ba16
|
378
|
}
|
| ykuroda |
0:13a5d365ba16
|
379
|
|
| ykuroda |
0:13a5d365ba16
|
380
|
// forward declaration, implementation at the end of this file
|
| ykuroda |
0:13a5d365ba16
|
381
|
template<typename MatrixType,
|
| ykuroda |
0:13a5d365ba16
|
382
|
int Size=MatrixType::ColsAtCompileTime,
|
| ykuroda |
0:13a5d365ba16
|
383
|
bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
|
| ykuroda |
0:13a5d365ba16
|
384
|
struct tridiagonalization_inplace_selector;
|
| ykuroda |
0:13a5d365ba16
|
385
|
|
| ykuroda |
0:13a5d365ba16
|
386
|
/** \brief Performs a full tridiagonalization in place
|
| ykuroda |
0:13a5d365ba16
|
387
|
*
|
| ykuroda |
0:13a5d365ba16
|
388
|
* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
|
| ykuroda |
0:13a5d365ba16
|
389
|
* decomposition is to be computed. Only the lower triangular part referenced.
|
| ykuroda |
0:13a5d365ba16
|
390
|
* The rest is left unchanged. On output, the orthogonal matrix Q
|
| ykuroda |
0:13a5d365ba16
|
391
|
* in the decomposition if \p extractQ is true.
|
| ykuroda |
0:13a5d365ba16
|
392
|
* \param[out] diag The diagonal of the tridiagonal matrix T in the
|
| ykuroda |
0:13a5d365ba16
|
393
|
* decomposition.
|
| ykuroda |
0:13a5d365ba16
|
394
|
* \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
|
| ykuroda |
0:13a5d365ba16
|
395
|
* the decomposition.
|
| ykuroda |
0:13a5d365ba16
|
396
|
* \param[in] extractQ If true, the orthogonal matrix Q in the
|
| ykuroda |
0:13a5d365ba16
|
397
|
* decomposition is computed and stored in \p mat.
|
| ykuroda |
0:13a5d365ba16
|
398
|
*
|
| ykuroda |
0:13a5d365ba16
|
399
|
* Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
|
| ykuroda |
0:13a5d365ba16
|
400
|
* such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
|
| ykuroda |
0:13a5d365ba16
|
401
|
* symmetric tridiagonal matrix.
|
| ykuroda |
0:13a5d365ba16
|
402
|
*
|
| ykuroda |
0:13a5d365ba16
|
403
|
* The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
|
| ykuroda |
0:13a5d365ba16
|
404
|
* \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
|
| ykuroda |
0:13a5d365ba16
|
405
|
* part of the matrix \p mat is destroyed.
|
| ykuroda |
0:13a5d365ba16
|
406
|
*
|
| ykuroda |
0:13a5d365ba16
|
407
|
* The vectors \p diag and \p subdiag are not resized. The function
|
| ykuroda |
0:13a5d365ba16
|
408
|
* assumes that they are already of the correct size. The length of the
|
| ykuroda |
0:13a5d365ba16
|
409
|
* vector \p diag should equal the number of rows in \p mat, and the
|
| ykuroda |
0:13a5d365ba16
|
410
|
* length of the vector \p subdiag should be one left.
|
| ykuroda |
0:13a5d365ba16
|
411
|
*
|
| ykuroda |
0:13a5d365ba16
|
412
|
* This implementation contains an optimized path for 3-by-3 matrices
|
| ykuroda |
0:13a5d365ba16
|
413
|
* which is especially useful for plane fitting.
|
| ykuroda |
0:13a5d365ba16
|
414
|
*
|
| ykuroda |
0:13a5d365ba16
|
415
|
* \note Currently, it requires two temporary vectors to hold the intermediate
|
| ykuroda |
0:13a5d365ba16
|
416
|
* Householder coefficients, and to reconstruct the matrix Q from the Householder
|
| ykuroda |
0:13a5d365ba16
|
417
|
* reflectors.
|
| ykuroda |
0:13a5d365ba16
|
418
|
*
|
| ykuroda |
0:13a5d365ba16
|
419
|
* Example (this uses the same matrix as the example in
|
| ykuroda |
0:13a5d365ba16
|
420
|
* Tridiagonalization::Tridiagonalization(const MatrixType&)):
|
| ykuroda |
0:13a5d365ba16
|
421
|
* \include Tridiagonalization_decomposeInPlace.cpp
|
| ykuroda |
0:13a5d365ba16
|
422
|
* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
|
| ykuroda |
0:13a5d365ba16
|
423
|
*
|
| ykuroda |
0:13a5d365ba16
|
424
|
* \sa class Tridiagonalization
|
| ykuroda |
0:13a5d365ba16
|
425
|
*/
|
| ykuroda |
0:13a5d365ba16
|
426
|
template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
|
| ykuroda |
0:13a5d365ba16
|
427
|
void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
|
| ykuroda |
0:13a5d365ba16
|
428
|
{
|
| ykuroda |
0:13a5d365ba16
|
429
|
eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
|
| ykuroda |
0:13a5d365ba16
|
430
|
tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
|
| ykuroda |
0:13a5d365ba16
|
431
|
}
|
| ykuroda |
0:13a5d365ba16
|
432
|
|
| ykuroda |
0:13a5d365ba16
|
433
|
/** \internal
|
| ykuroda |
0:13a5d365ba16
|
434
|
* General full tridiagonalization
|
| ykuroda |
0:13a5d365ba16
|
435
|
*/
|
| ykuroda |
0:13a5d365ba16
|
436
|
template<typename MatrixType, int Size, bool IsComplex>
|
| ykuroda |
0:13a5d365ba16
|
437
|
struct tridiagonalization_inplace_selector
|
| ykuroda |
0:13a5d365ba16
|
438
|
{
|
| ykuroda |
0:13a5d365ba16
|
439
|
typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
|
| ykuroda |
0:13a5d365ba16
|
440
|
typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
|
| ykuroda |
0:13a5d365ba16
|
441
|
typedef typename MatrixType::Index Index;
|
| ykuroda |
0:13a5d365ba16
|
442
|
template<typename DiagonalType, typename SubDiagonalType>
|
| ykuroda |
0:13a5d365ba16
|
443
|
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
|
| ykuroda |
0:13a5d365ba16
|
444
|
{
|
| ykuroda |
0:13a5d365ba16
|
445
|
CoeffVectorType hCoeffs(mat.cols()-1);
|
| ykuroda |
0:13a5d365ba16
|
446
|
tridiagonalization_inplace(mat,hCoeffs);
|
| ykuroda |
0:13a5d365ba16
|
447
|
diag = mat.diagonal().real();
|
| ykuroda |
0:13a5d365ba16
|
448
|
subdiag = mat.template diagonal<-1>().real();
|
| ykuroda |
0:13a5d365ba16
|
449
|
if(extractQ)
|
| ykuroda |
0:13a5d365ba16
|
450
|
mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
|
| ykuroda |
0:13a5d365ba16
|
451
|
.setLength(mat.rows() - 1)
|
| ykuroda |
0:13a5d365ba16
|
452
|
.setShift(1);
|
| ykuroda |
0:13a5d365ba16
|
453
|
}
|
| ykuroda |
0:13a5d365ba16
|
454
|
};
|
| ykuroda |
0:13a5d365ba16
|
455
|
|
| ykuroda |
0:13a5d365ba16
|
456
|
/** \internal
|
| ykuroda |
0:13a5d365ba16
|
457
|
* Specialization for 3x3 real matrices.
|
| ykuroda |
0:13a5d365ba16
|
458
|
* Especially useful for plane fitting.
|
| ykuroda |
0:13a5d365ba16
|
459
|
*/
|
| ykuroda |
0:13a5d365ba16
|
460
|
template<typename MatrixType>
|
| ykuroda |
0:13a5d365ba16
|
461
|
struct tridiagonalization_inplace_selector<MatrixType,3,false>
|
| ykuroda |
0:13a5d365ba16
|
462
|
{
|
| ykuroda |
0:13a5d365ba16
|
463
|
typedef typename MatrixType::Scalar Scalar;
|
| ykuroda |
0:13a5d365ba16
|
464
|
typedef typename MatrixType::RealScalar RealScalar;
|
| ykuroda |
0:13a5d365ba16
|
465
|
|
| ykuroda |
0:13a5d365ba16
|
466
|
template<typename DiagonalType, typename SubDiagonalType>
|
| ykuroda |
0:13a5d365ba16
|
467
|
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
|
| ykuroda |
0:13a5d365ba16
|
468
|
{
|
| ykuroda |
0:13a5d365ba16
|
469
|
using std::sqrt;
|
| ykuroda |
0:13a5d365ba16
|
470
|
diag[0] = mat(0,0);
|
| ykuroda |
0:13a5d365ba16
|
471
|
RealScalar v1norm2 = numext::abs2(mat(2,0));
|
| ykuroda |
0:13a5d365ba16
|
472
|
if(v1norm2 == RealScalar(0))
|
| ykuroda |
0:13a5d365ba16
|
473
|
{
|
| ykuroda |
0:13a5d365ba16
|
474
|
diag[1] = mat(1,1);
|
| ykuroda |
0:13a5d365ba16
|
475
|
diag[2] = mat(2,2);
|
| ykuroda |
0:13a5d365ba16
|
476
|
subdiag[0] = mat(1,0);
|
| ykuroda |
0:13a5d365ba16
|
477
|
subdiag[1] = mat(2,1);
|
| ykuroda |
0:13a5d365ba16
|
478
|
if (extractQ)
|
| ykuroda |
0:13a5d365ba16
|
479
|
mat.setIdentity();
|
| ykuroda |
0:13a5d365ba16
|
480
|
}
|
| ykuroda |
0:13a5d365ba16
|
481
|
else
|
| ykuroda |
0:13a5d365ba16
|
482
|
{
|
| ykuroda |
0:13a5d365ba16
|
483
|
RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
|
| ykuroda |
0:13a5d365ba16
|
484
|
RealScalar invBeta = RealScalar(1)/beta;
|
| ykuroda |
0:13a5d365ba16
|
485
|
Scalar m01 = mat(1,0) * invBeta;
|
| ykuroda |
0:13a5d365ba16
|
486
|
Scalar m02 = mat(2,0) * invBeta;
|
| ykuroda |
0:13a5d365ba16
|
487
|
Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
|
| ykuroda |
0:13a5d365ba16
|
488
|
diag[1] = mat(1,1) + m02*q;
|
| ykuroda |
0:13a5d365ba16
|
489
|
diag[2] = mat(2,2) - m02*q;
|
| ykuroda |
0:13a5d365ba16
|
490
|
subdiag[0] = beta;
|
| ykuroda |
0:13a5d365ba16
|
491
|
subdiag[1] = mat(2,1) - m01 * q;
|
| ykuroda |
0:13a5d365ba16
|
492
|
if (extractQ)
|
| ykuroda |
0:13a5d365ba16
|
493
|
{
|
| ykuroda |
0:13a5d365ba16
|
494
|
mat << 1, 0, 0,
|
| ykuroda |
0:13a5d365ba16
|
495
|
0, m01, m02,
|
| ykuroda |
0:13a5d365ba16
|
496
|
0, m02, -m01;
|
| ykuroda |
0:13a5d365ba16
|
497
|
}
|
| ykuroda |
0:13a5d365ba16
|
498
|
}
|
| ykuroda |
0:13a5d365ba16
|
499
|
}
|
| ykuroda |
0:13a5d365ba16
|
500
|
};
|
| ykuroda |
0:13a5d365ba16
|
501
|
|
| ykuroda |
0:13a5d365ba16
|
502
|
/** \internal
|
| ykuroda |
0:13a5d365ba16
|
503
|
* Trivial specialization for 1x1 matrices
|
| ykuroda |
0:13a5d365ba16
|
504
|
*/
|
| ykuroda |
0:13a5d365ba16
|
505
|
template<typename MatrixType, bool IsComplex>
|
| ykuroda |
0:13a5d365ba16
|
506
|
struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
|
| ykuroda |
0:13a5d365ba16
|
507
|
{
|
| ykuroda |
0:13a5d365ba16
|
508
|
typedef typename MatrixType::Scalar Scalar;
|
| ykuroda |
0:13a5d365ba16
|
509
|
|
| ykuroda |
0:13a5d365ba16
|
510
|
template<typename DiagonalType, typename SubDiagonalType>
|
| ykuroda |
0:13a5d365ba16
|
511
|
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
|
| ykuroda |
0:13a5d365ba16
|
512
|
{
|
| ykuroda |
0:13a5d365ba16
|
513
|
diag(0,0) = numext::real(mat(0,0));
|
| ykuroda |
0:13a5d365ba16
|
514
|
if(extractQ)
|
| ykuroda |
0:13a5d365ba16
|
515
|
mat(0,0) = Scalar(1);
|
| ykuroda |
0:13a5d365ba16
|
516
|
}
|
| ykuroda |
0:13a5d365ba16
|
517
|
};
|
| ykuroda |
0:13a5d365ba16
|
518
|
|
| ykuroda |
0:13a5d365ba16
|
519
|
/** \internal
|
| ykuroda |
0:13a5d365ba16
|
520
|
* \eigenvalues_module \ingroup Eigenvalues_Module
|
| ykuroda |
0:13a5d365ba16
|
521
|
*
|
| ykuroda |
0:13a5d365ba16
|
522
|
* \brief Expression type for return value of Tridiagonalization::matrixT()
|
| ykuroda |
0:13a5d365ba16
|
523
|
*
|
| ykuroda |
0:13a5d365ba16
|
524
|
* \tparam MatrixType type of underlying dense matrix
|
| ykuroda |
0:13a5d365ba16
|
525
|
*/
|
| ykuroda |
0:13a5d365ba16
|
526
|
template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
|
| ykuroda |
0:13a5d365ba16
|
527
|
: public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
|
| ykuroda |
0:13a5d365ba16
|
528
|
{
|
| ykuroda |
0:13a5d365ba16
|
529
|
typedef typename MatrixType::Index Index;
|
| ykuroda |
0:13a5d365ba16
|
530
|
public:
|
| ykuroda |
0:13a5d365ba16
|
531
|
/** \brief Constructor.
|
| ykuroda |
0:13a5d365ba16
|
532
|
*
|
| ykuroda |
0:13a5d365ba16
|
533
|
* \param[in] mat The underlying dense matrix
|
| ykuroda |
0:13a5d365ba16
|
534
|
*/
|
| ykuroda |
0:13a5d365ba16
|
535
|
TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
|
| ykuroda |
0:13a5d365ba16
|
536
|
|
| ykuroda |
0:13a5d365ba16
|
537
|
template <typename ResultType>
|
| ykuroda |
0:13a5d365ba16
|
538
|
inline void evalTo(ResultType& result) const
|
| ykuroda |
0:13a5d365ba16
|
539
|
{
|
| ykuroda |
0:13a5d365ba16
|
540
|
result.setZero();
|
| ykuroda |
0:13a5d365ba16
|
541
|
result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
|
| ykuroda |
0:13a5d365ba16
|
542
|
result.diagonal() = m_matrix.diagonal();
|
| ykuroda |
0:13a5d365ba16
|
543
|
result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
|
| ykuroda |
0:13a5d365ba16
|
544
|
}
|
| ykuroda |
0:13a5d365ba16
|
545
|
|
| ykuroda |
0:13a5d365ba16
|
546
|
Index rows() const { return m_matrix.rows(); }
|
| ykuroda |
0:13a5d365ba16
|
547
|
Index cols() const { return m_matrix.cols(); }
|
| ykuroda |
0:13a5d365ba16
|
548
|
|
| ykuroda |
0:13a5d365ba16
|
549
|
protected:
|
| ykuroda |
0:13a5d365ba16
|
550
|
typename MatrixType::Nested m_matrix;
|
| ykuroda |
0:13a5d365ba16
|
551
|
};
|
| ykuroda |
0:13a5d365ba16
|
552
|
|
| ykuroda |
0:13a5d365ba16
|
553
|
} // end namespace internal
|
| ykuroda |
0:13a5d365ba16
|
554
|
|
| ykuroda |
0:13a5d365ba16
|
555
|
} // end namespace Eigen
|
| ykuroda |
0:13a5d365ba16
|
556
|
|
| ykuroda |
0:13a5d365ba16
|
557
|
#endif // EIGEN_TRIDIAGONALIZATION_H |