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This is to avoid dynamic memory allocations in the compute() methods of ComplexEigenSolver, EigenSolver, and SelfAdjointEigenSolver where possible. As a result, Tridiagonalization::decomposeInPlace() is no longer used. Biggest remaining issue is the allocation in HouseholderSequence::evalTo().
457 lines
18 KiB
C++
457 lines
18 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 <g.gael@free.fr>
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// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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//
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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//
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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//
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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//
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_TRIDIAGONALIZATION_H
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#define EIGEN_TRIDIAGONALIZATION_H
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/** \eigenvalues_module \ingroup Eigenvalues_Module
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* \nonstableyet
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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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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 ei_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 ei_meta_if<NumTraits<Scalar>::IsComplex,
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typename Diagonal<MatrixType,0>::RealReturnType,
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Diagonal<MatrixType,0>
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>::ret DiagonalReturnType;
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typedef typename ei_meta_if<NumTraits<Scalar>::IsComplex,
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typename Diagonal<
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Block<MatrixType,SizeMinusOne,SizeMinusOne>,0 >::RealReturnType,
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Diagonal<
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Block<MatrixType,SizeMinusOne,SizeMinusOne>,0 >
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>::ret SubDiagonalReturnType;
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/** \brief Return type of matrixQ() */
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typedef typename HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType 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(int size = Size==Dynamic ? 2 : Size)
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: m_matrix(size,size), m_hCoeffs(size > 1 ? size-1 : 1)
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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), m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1)
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{
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_compute(m_matrix, m_hCoeffs);
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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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*
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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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void 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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_compute(m_matrix, m_hCoeffs);
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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 { return m_hCoeffs; }
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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 { return m_matrix; }
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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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return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate(), false, m_matrix.rows() - 1, 1);
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}
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/** \brief Constructs the tridiagonal matrix T in the decomposition
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*
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* \returns 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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* This function copies the matrix T from internal data. The diagonal and
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* subdiagonal of the packed matrix as returned by packedMatrix()
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* represents the matrix T. It may sometimes be sufficient to directly use
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* the packed matrix or the vector expressions returned by diagonal()
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* and subDiagonal() instead of creating a new 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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MatrixType matrixT() const;
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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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const 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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const SubDiagonalReturnType subDiagonal() const;
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/** \brief Performs a full decomposition 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. 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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* Compute the tridiagonal matrix of \p mat in place. The tridiagonal
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* 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.
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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 real 3-by-3 matrices
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* which is especially useful for plane fitting.
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*
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* \note Notwithstanding the name, the current implementation copies
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* \p mat to a temporary matrix and uses that matrix to compute the
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* decomposition.
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*
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* Example (this uses the same matrix as the example in
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* 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 Tridiagonalization(const MatrixType&), compute()
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*/
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static void decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);
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protected:
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static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
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static void _decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);
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MatrixType m_matrix;
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CoeffVectorType m_hCoeffs;
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};
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template<typename MatrixType>
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const typename Tridiagonalization<MatrixType>::DiagonalReturnType
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Tridiagonalization<MatrixType>::diagonal() const
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{
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return m_matrix.diagonal();
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}
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template<typename MatrixType>
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const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
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Tridiagonalization<MatrixType>::subDiagonal() const
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{
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int n = m_matrix.rows();
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return Block<MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
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}
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template<typename MatrixType>
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typename Tridiagonalization<MatrixType>::MatrixType
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Tridiagonalization<MatrixType>::matrixT() const
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{
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// FIXME should this function (and other similar ones) rather take a matrix as argument
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// and fill it ? (to avoid temporaries)
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int n = m_matrix.rows();
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MatrixType matT = m_matrix;
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matT.topRightCorner(n-1, n-1).diagonal() = subDiagonal().template cast<Scalar>().conjugate();
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if (n>2)
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{
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matT.topRightCorner(n-2, n-2).template triangularView<Upper>().setZero();
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matT.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero();
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}
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return matT;
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}
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#ifndef EIGEN_HIDE_HEAVY_CODE
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/** \internal
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* Performs a tridiagonal decomposition of \a matA in place.
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*
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* \param matA the input selfadjoint matrix
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* \param hCoeffs returned Householder coefficients
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*
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* The result is written in the lower triangular part of \a matA.
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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 packedMatrix()
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*/
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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
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{
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assert(matA.rows()==matA.cols());
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int n = matA.rows();
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for (int i = 0; i<n-1; ++i)
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{
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int 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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* (ei_conj(h) * matA.col(i).tail(remainingSize)));
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hCoeffs.tail(n-i-1) += (ei_conj(h)*Scalar(-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), -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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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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int n = mat.rows();
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ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
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if (n==3 && (!NumTraits<Scalar>::IsComplex) )
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{
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_decomposeInPlace3x3(mat, diag, subdiag, extractQ);
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}
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else
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{
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Tridiagonalization tridiag(mat);
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diag = tridiag.diagonal();
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subdiag = tridiag.subDiagonal();
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if (extractQ)
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mat = tridiag.matrixQ();
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}
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}
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/** \internal
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* Optimized path for 3x3 matrices.
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* Especially useful for plane fitting.
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*/
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template<typename MatrixType>
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void Tridiagonalization<MatrixType>::_decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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{
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diag[0] = ei_real(mat(0,0));
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RealScalar v1norm2 = ei_abs2(mat(0,2));
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if (ei_isMuchSmallerThan(v1norm2, RealScalar(1)))
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{
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diag[1] = ei_real(mat(1,1));
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diag[2] = ei_real(mat(2,2));
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subdiag[0] = ei_real(mat(0,1));
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subdiag[1] = ei_real(mat(1,2));
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if (extractQ)
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mat.setIdentity();
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}
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else
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{
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RealScalar beta = ei_sqrt(ei_abs2(mat(0,1))+v1norm2);
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RealScalar invBeta = RealScalar(1)/beta;
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Scalar m01 = mat(0,1) * invBeta;
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Scalar m02 = mat(0,2) * invBeta;
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Scalar q = RealScalar(2)*m01*mat(1,2) + m02*(mat(2,2) - mat(1,1));
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diag[1] = ei_real(mat(1,1) + m02*q);
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diag[2] = ei_real(mat(2,2) - m02*q);
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subdiag[0] = beta;
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subdiag[1] = ei_real(mat(1,2) - m01 * q);
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if (extractQ)
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{
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mat(0,0) = 1;
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mat(0,1) = 0;
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mat(0,2) = 0;
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mat(1,0) = 0;
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mat(1,1) = m01;
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mat(1,2) = m02;
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mat(2,0) = 0;
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mat(2,1) = m02;
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mat(2,2) = -m01;
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}
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}
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}
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#endif // EIGEN_HIDE_HEAVY_CODE
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#endif // EIGEN_TRIDIAGONALIZATION_H
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