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Tridiagonalization.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_TRIDIAGONALIZATION_H
#define EIGEN_TRIDIAGONALIZATION_H
namespace Eigen {
namespace internal {
template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
template<typename MatrixType>
struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
{
typedef typename MatrixType::PlainObject ReturnType;
};
template<typename MatrixType, typename CoeffVectorType>
void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
}
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class Tridiagonalization
*
* \brief Tridiagonal decomposition of a selfadjoint matrix
*
* \tparam _MatrixType the type of the matrix of which we are computing the
* tridiagonal decomposition; this is expected to be an instantiation of the
* Matrix class template.
*
* This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
* \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
*
* A tridiagonal matrix is a matrix which has nonzero elements only on the
* main diagonal and the first diagonal below and above it. The Hessenberg
* decomposition of a selfadjoint matrix is in fact a tridiagonal
* decomposition. This class is used in SelfAdjointEigenSolver to compute the
* eigenvalues and eigenvectors of a selfadjoint matrix.
*
* Call the function compute() to compute the tridiagonal decomposition of a
* given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
* constructor which computes the tridiagonal Schur decomposition at
* construction time. Once the decomposition is computed, you can use the
* matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
* decomposition.
*
* The documentation of Tridiagonalization(const MatrixType&) contains an
* example of the typical use of this class.
*
* \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
*/
template<typename _MatrixType> class Tridiagonalization
{
public:
/** \brief Synonym for the template parameter \p _MatrixType. */
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef typename MatrixType::Index Index;
enum {
Size = MatrixType::RowsAtCompileTime,
SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
Options = MatrixType::Options,
MaxSize = MatrixType::MaxRowsAtCompileTime,
MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
};
typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
const Diagonal<const MatrixType>
>::type DiagonalReturnType;
typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
typename internal::add_const_on_value_type<typename Diagonal<
Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
const Diagonal<
Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
>::type SubDiagonalReturnType;
/** \brief Return type of matrixQ() */
typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
/** \brief Default constructor.
*
* \param [in] size Positive integer, size of the matrix whose tridiagonal
* decomposition will be computed.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via compute(). The \p size parameter is only
* used as a hint. It is not an error to give a wrong \p size, but it may
* impair performance.
*
* \sa compute() for an example.
*/
Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
: m_matrix(size,size),
m_hCoeffs(size > 1 ? size-1 : 1),
m_isInitialized(false)
{}
/** \brief Constructor; computes tridiagonal decomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
* is to be computed.
*
* This constructor calls compute() to compute the tridiagonal decomposition.
*
* Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
* Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
*/
Tridiagonalization(const MatrixType& matrix)
: m_matrix(matrix),
m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
m_isInitialized(false)
{
internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
m_isInitialized = true;
}
/** \brief Computes tridiagonal decomposition of given matrix.
*
* \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
* is to be computed.
* \returns Reference to \c *this
*
* The tridiagonal decomposition is computed by bringing the columns of
* the matrix successively in the required form using Householder
* reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
* the size of the given matrix.
*
* This method reuses of the allocated data in the Tridiagonalization
* object, if the size of the matrix does not change.
*
* Example: \include Tridiagonalization_compute.cpp
* Output: \verbinclude Tridiagonalization_compute.out
*/
Tridiagonalization& compute(const MatrixType& matrix)
{
m_matrix = matrix;
m_hCoeffs.resize(matrix.rows()-1, 1);
internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
m_isInitialized = true;
return *this;
}
/** \brief Returns the Householder coefficients.
*
* \returns a const reference to the vector of Householder coefficients
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* The Householder coefficients allow the reconstruction of the matrix
* \f$ Q \f$ in the tridiagonal decomposition from the packed data.
*
* Example: \include Tridiagonalization_householderCoefficients.cpp
* Output: \verbinclude Tridiagonalization_householderCoefficients.out
*
* \sa packedMatrix(), \ref Householder_Module "Householder module"
*/
inline CoeffVectorType householderCoefficients() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
return m_hCoeffs;
}
/** \brief Returns the internal representation of the decomposition
*
* \returns a const reference to a matrix with the internal representation
* of the decomposition.
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* The returned matrix contains the following information:
* - the strict upper triangular part is equal to the input matrix A.
* - the diagonal and lower sub-diagonal represent the real tridiagonal
* symmetric matrix T.
* - the rest of the lower part contains the Householder vectors that,
* combined with Householder coefficients returned by
* householderCoefficients(), allows to reconstruct the matrix Q as
* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
* Here, the matrices \f$ H_i \f$ are the Householder transformations
* \f$ H_i = (I - h_i v_i v_i^T) \f$
* where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
* \f$ v_i \f$ is the Householder vector defined by
* \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
* with M the matrix returned by this function.
*
* See LAPACK for further details on this packed storage.
*
* Example: \include Tridiagonalization_packedMatrix.cpp
* Output: \verbinclude Tridiagonalization_packedMatrix.out
*
* \sa householderCoefficients()
*/
inline const MatrixType& packedMatrix() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
return m_matrix;
}
/** \brief Returns the unitary matrix Q in the decomposition
*
* \returns object representing the matrix Q
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* This function returns a light-weight object of template class
* HouseholderSequence. You can either apply it directly to a matrix or
* you can convert it to a matrix of type #MatrixType.
*
* \sa Tridiagonalization(const MatrixType&) for an example,
* matrixT(), class HouseholderSequence
*/
HouseholderSequenceType matrixQ() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
.setLength(m_matrix.rows() - 1)
.setShift(1);
}
/** \brief Returns an expression of the tridiagonal matrix T in the decomposition
*
* \returns expression object representing the matrix T
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* Currently, this function can be used to extract the matrix T from internal
* data and copy it to a dense matrix object. In most cases, it may be
* sufficient to directly use the packed matrix or the vector expressions
* returned by diagonal() and subDiagonal() instead of creating a new
* dense copy matrix with this function.
*
* \sa Tridiagonalization(const MatrixType&) for an example,
* matrixQ(), packedMatrix(), diagonal(), subDiagonal()
*/
MatrixTReturnType matrixT() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
return MatrixTReturnType(m_matrix.real());
}
/** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
*
* \returns expression representing the diagonal of T
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* Example: \include Tridiagonalization_diagonal.cpp
* Output: \verbinclude Tridiagonalization_diagonal.out
*
* \sa matrixT(), subDiagonal()
*/
DiagonalReturnType diagonal() const;
/** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
*
* \returns expression representing the subdiagonal of T
*
* \pre Either the constructor Tridiagonalization(const MatrixType&) or
* the member function compute(const MatrixType&) has been called before
* to compute the tridiagonal decomposition of a matrix.
*
* \sa diagonal() for an example, matrixT()
*/
SubDiagonalReturnType subDiagonal() const;
protected:
MatrixType m_matrix;
CoeffVectorType m_hCoeffs;
bool m_isInitialized;
};
template<typename MatrixType>
typename Tridiagonalization<MatrixType>::DiagonalReturnType
Tridiagonalization<MatrixType>::diagonal() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
return m_matrix.diagonal();
}
template<typename MatrixType>
typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
Tridiagonalization<MatrixType>::subDiagonal() const
{
eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
Index n = m_matrix.rows();
return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
}
namespace internal {
/** \internal
* Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
*
* \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
* On output, the strict upper part is left unchanged, and the lower triangular part
* represents the T and Q matrices in packed format has detailed below.
* \param[out] hCoeffs returned Householder coefficients (see below)
*
* On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
* and lower sub-diagonal of the matrix \a matA.
* The unitary matrix Q is represented in a compact way as a product of
* Householder reflectors \f$ H_i \f$ such that:
* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
* The Householder reflectors are defined as
* \f$ H_i = (I - h_i v_i v_i^T) \f$
* where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
* \f$ v_i \f$ is the Householder vector defined by
* \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
*
* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
*
* \sa Tridiagonalization::packedMatrix()
*/
template<typename MatrixType, typename CoeffVectorType>
void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
{
using numext::conj;
typedef typename MatrixType::Index Index;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
Index n = matA.rows();
eigen_assert(n==matA.cols());
eigen_assert(n==hCoeffs.size()+1 || n==1);
for (Index i = 0; i<n-1; ++i)
{
Index remainingSize = n-i-1;
RealScalar beta;
Scalar h;
matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
// Apply similarity transformation to remaining columns,
// i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
matA.col(i).coeffRef(i+1) = 1;
hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
* (conj(h) * matA.col(i).tail(remainingSize)));
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);
matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
.rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
matA.col(i).coeffRef(i+1) = beta;
hCoeffs.coeffRef(i) = h;
}
}
// forward declaration, implementation at the end of this file
template<typename MatrixType,
int Size=MatrixType::ColsAtCompileTime,
bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
struct tridiagonalization_inplace_selector;
/** \brief Performs a full tridiagonalization in place
*
* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
* decomposition is to be computed. Only the lower triangular part referenced.
* The rest is left unchanged. On output, the orthogonal matrix Q
* in the decomposition if \p extractQ is true.
* \param[out] diag The diagonal of the tridiagonal matrix T in the
* decomposition.
* \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
* the decomposition.
* \param[in] extractQ If true, the orthogonal matrix Q in the
* decomposition is computed and stored in \p mat.
*
* Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
* such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
* symmetric tridiagonal matrix.
*
* The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
* \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
* part of the matrix \p mat is destroyed.
*
* The vectors \p diag and \p subdiag are not resized. The function
* assumes that they are already of the correct size. The length of the
* vector \p diag should equal the number of rows in \p mat, and the
* length of the vector \p subdiag should be one left.
*
* This implementation contains an optimized path for 3-by-3 matrices
* which is especially useful for plane fitting.
*
* \note Currently, it requires two temporary vectors to hold the intermediate
* Householder coefficients, and to reconstruct the matrix Q from the Householder
* reflectors.
*
* Example (this uses the same matrix as the example in
* Tridiagonalization::Tridiagonalization(const MatrixType&)):
* \include Tridiagonalization_decomposeInPlace.cpp
* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
*
* \sa class Tridiagonalization
*/
template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
{
eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
}
/** \internal
* General full tridiagonalization
*/
template<typename MatrixType, int Size, bool IsComplex>
struct tridiagonalization_inplace_selector
{
typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
typedef typename MatrixType::Index Index;
template<typename DiagonalType, typename SubDiagonalType>
static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
{
CoeffVectorType hCoeffs(mat.cols()-1);
tridiagonalization_inplace(mat,hCoeffs);
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
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