558 lines
22 KiB
C++
558 lines
22 KiB
C++
// 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
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* to compute the tridiagonal decomposition of a matrix.
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*
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* The returned matrix contains the following information:
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* - the strict upper triangular part is equal to the input matrix A.
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* - the diagonal and lower sub-diagonal represent the real tridiagonal
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* 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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* householderCoefficients(), allows to reconstruct the matrix Q as
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* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
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* Here, the matrices \f$ H_i \f$ are the Householder transformations
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* \f$ H_i = (I - h_i v_i v_i^T) \f$
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* 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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* \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
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* with M the matrix returned by this function.
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*
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* See LAPACK for further details on this packed storage.
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*
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* Example: \include Tridiagonalization_packedMatrix.cpp
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* Output: \verbinclude Tridiagonalization_packedMatrix.out
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*
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* \sa householderCoefficients()
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*/
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inline const MatrixType& packedMatrix() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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return m_matrix;
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}
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/** \brief Returns the unitary matrix Q in the decomposition
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*
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* \returns object representing the matrix Q
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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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* This function returns a light-weight object of template class
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* HouseholderSequence. You can either apply it directly to a matrix or
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* you can convert it to a matrix of type #MatrixType.
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*
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* \sa Tridiagonalization(const MatrixType&) for an example,
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* matrixT(), class HouseholderSequence
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*/
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HouseholderSequenceType matrixQ() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
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.setLength(m_matrix.rows() - 1)
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.setShift(1);
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}
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/** \brief Returns an expression of the tridiagonal matrix T in the decomposition
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*
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* \returns expression object representing the matrix T
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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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* Currently, this function can be used to extract the matrix T from internal
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* data and copy it to a dense matrix object. In most cases, it may be
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* sufficient to directly use the packed matrix or the vector expressions
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* returned by diagonal() and subDiagonal() instead of creating a new
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* dense copy matrix with this function.
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*
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* \sa Tridiagonalization(const MatrixType&) for an example,
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* matrixQ(), packedMatrix(), diagonal(), subDiagonal()
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*/
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MatrixTReturnType matrixT() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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return MatrixTReturnType(m_matrix.real());
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}
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/** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
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*
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* \returns expression representing the diagonal of T
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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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* Example: \include Tridiagonalization_diagonal.cpp
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* Output: \verbinclude Tridiagonalization_diagonal.out
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*
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* \sa matrixT(), subDiagonal()
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*/
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DiagonalReturnType diagonal() const;
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/** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
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*
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* \returns expression representing the subdiagonal of T
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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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* \sa diagonal() for an example, matrixT()
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*/
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SubDiagonalReturnType subDiagonal() const;
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protected:
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MatrixType m_matrix;
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CoeffVectorType m_hCoeffs;
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bool m_isInitialized;
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};
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template<typename MatrixType>
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typename Tridiagonalization<MatrixType>::DiagonalReturnType
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Tridiagonalization<MatrixType>::diagonal() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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return m_matrix.diagonal();
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}
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template<typename MatrixType>
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typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
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Tridiagonalization<MatrixType>::subDiagonal() const
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{
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eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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Index n = m_matrix.rows();
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return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
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}
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namespace internal {
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/** \internal
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* Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
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*
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* \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
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* On output, the strict upper part is left unchanged, and the lower triangular part
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* represents the T and Q matrices in packed format has detailed below.
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* \param[out] hCoeffs returned Householder coefficients (see below)
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*
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* On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
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* and lower sub-diagonal of the matrix \a matA.
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* The unitary matrix Q is represented in a compact way as a product of
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* Householder reflectors \f$ H_i \f$ such that:
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* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
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* The Householder reflectors are defined as
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* \f$ H_i = (I - h_i v_i v_i^T) \f$
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* where \f$ h_i = hCoeffs[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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* \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
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*
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* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
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*
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* \sa Tridiagonalization::packedMatrix()
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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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using numext::conj;
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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Index n = matA.rows();
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eigen_assert(n==matA.cols());
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eigen_assert(n==hCoeffs.size()+1 || n==1);
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for (Index i = 0; i<n-1; ++i)
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{
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Index remainingSize = n-i-1;
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RealScalar beta;
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Scalar h;
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matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
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// Apply similarity transformation to remaining columns,
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// i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
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matA.col(i).coeffRef(i+1) = 1;
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hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
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* (conj(h) * matA.col(i).tail(remainingSize)));
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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);
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matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
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.rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
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matA.col(i).coeffRef(i+1) = beta;
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hCoeffs.coeffRef(i) = h;
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}
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}
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// forward declaration, implementation at the end of this file
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template<typename MatrixType,
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int Size=MatrixType::ColsAtCompileTime,
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bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
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struct tridiagonalization_inplace_selector;
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/** \brief Performs a full tridiagonalization in place
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*
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* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
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* decomposition is to be computed. Only the lower triangular part referenced.
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* The rest is left unchanged. On output, the orthogonal matrix Q
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* in the decomposition if \p extractQ is true.
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* \param[out] diag The diagonal of the tridiagonal matrix T in the
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* decomposition.
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* \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
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* the decomposition.
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* \param[in] extractQ If true, the orthogonal matrix Q in the
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* decomposition is computed and stored in \p mat.
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*
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* Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
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* such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
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* symmetric tridiagonal matrix.
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*
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* The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
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* \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
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* part of the matrix \p mat is destroyed.
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*
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* The vectors \p diag and \p subdiag are not resized. The function
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* assumes that they are already of the correct size. The length of the
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* vector \p diag should equal the number of rows in \p mat, and the
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* length of the vector \p subdiag should be one left.
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*
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* This implementation contains an optimized path for 3-by-3 matrices
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* which is especially useful for plane fitting.
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*
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* \note Currently, it requires two temporary vectors to hold the intermediate
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* Householder coefficients, and to reconstruct the matrix Q from the Householder
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* reflectors.
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*
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* Example (this uses the same matrix as the example in
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* Tridiagonalization::Tridiagonalization(const MatrixType&)):
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* \include Tridiagonalization_decomposeInPlace.cpp
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* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
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*
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* \sa class Tridiagonalization
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*/
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template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
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void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
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tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
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}
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/** \internal
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* General full tridiagonalization
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*/
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template<typename MatrixType, int Size, bool IsComplex>
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struct tridiagonalization_inplace_selector
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{
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typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
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typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
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typedef typename MatrixType::Index Index;
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template<typename DiagonalType, typename SubDiagonalType>
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static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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CoeffVectorType hCoeffs(mat.cols()-1);
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tridiagonalization_inplace(mat,hCoeffs);
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diag = mat.diagonal().real();
|
|
subdiag = mat.template diagonal<-1>().real();
|
|
if(extractQ)
|
|
mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
|
|
.setLength(mat.rows() - 1)
|
|
.setShift(1);
|
|
}
|
|
};
|
|
|
|
/** \internal
|
|
* Specialization for 3x3 real matrices.
|
|
* Especially useful for plane fitting.
|
|
*/
|
|
template<typename MatrixType>
|
|
struct tridiagonalization_inplace_selector<MatrixType,3,false>
|
|
{
|
|
typedef typename MatrixType::Scalar Scalar;
|
|
typedef typename MatrixType::RealScalar RealScalar;
|
|
|
|
template<typename DiagonalType, typename SubDiagonalType>
|
|
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
|
|
{
|
|
using std::sqrt;
|
|
diag[0] = mat(0,0);
|
|
RealScalar v1norm2 = numext::abs2(mat(2,0));
|
|
if(v1norm2 == RealScalar(0))
|
|
{
|
|
diag[1] = mat(1,1);
|
|
diag[2] = mat(2,2);
|
|
subdiag[0] = mat(1,0);
|
|
subdiag[1] = mat(2,1);
|
|
if (extractQ)
|
|
mat.setIdentity();
|
|
}
|
|
else
|
|
{
|
|
RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
|
|
RealScalar invBeta = RealScalar(1)/beta;
|
|
Scalar m01 = mat(1,0) * invBeta;
|
|
Scalar m02 = mat(2,0) * invBeta;
|
|
Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
|
|
diag[1] = mat(1,1) + m02*q;
|
|
diag[2] = mat(2,2) - m02*q;
|
|
subdiag[0] = beta;
|
|
subdiag[1] = mat(2,1) - m01 * q;
|
|
if (extractQ)
|
|
{
|
|
mat << 1, 0, 0,
|
|
0, m01, m02,
|
|
0, m02, -m01;
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
/** \internal
|
|
* Trivial specialization for 1x1 matrices
|
|
*/
|
|
template<typename MatrixType, bool IsComplex>
|
|
struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
|
|
{
|
|
typedef typename MatrixType::Scalar Scalar;
|
|
|
|
template<typename DiagonalType, typename SubDiagonalType>
|
|
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
|
|
{
|
|
diag(0,0) = numext::real(mat(0,0));
|
|
if(extractQ)
|
|
mat(0,0) = Scalar(1);
|
|
}
|
|
};
|
|
|
|
/** \internal
|
|
* \eigenvalues_module \ingroup Eigenvalues_Module
|
|
*
|
|
* \brief Expression type for return value of Tridiagonalization::matrixT()
|
|
*
|
|
* \tparam MatrixType type of underlying dense matrix
|
|
*/
|
|
template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
|
|
: public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
|
|
{
|
|
typedef typename MatrixType::Index Index;
|
|
public:
|
|
/** \brief Constructor.
|
|
*
|
|
* \param[in] mat The underlying dense matrix
|
|
*/
|
|
TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
|
|
|
|
template <typename ResultType>
|
|
inline void evalTo(ResultType& result) const
|
|
{
|
|
result.setZero();
|
|
result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
|
|
result.diagonal() = m_matrix.diagonal();
|
|
result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
|
|
}
|
|
|
|
Index rows() const { return m_matrix.rows(); }
|
|
Index cols() const { return m_matrix.cols(); }
|
|
|
|
protected:
|
|
typename MatrixType::Nested m_matrix;
|
|
};
|
|
|
|
} // end namespace internal
|
|
|
|
} // end namespace Eigen
|
|
|
|
#endif // EIGEN_TRIDIAGONALIZATION_H
|