// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 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_HESSENBERGDECOMPOSITION_H
#define EIGEN_HESSENBERGDECOMPOSITION_H
namespace Eigen {
namespace internal {
template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType;
template<typename MatrixType>
struct traits<HessenbergDecompositionMatrixHReturnType<MatrixType> >
{
typedef MatrixType ReturnType;
};
}
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class HessenbergDecomposition
*
* \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation
*
* \tparam _MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
*
* This class performs an Hessenberg decomposition of a matrix \f$ A \f$. In
* the real case, the Hessenberg decomposition consists of an orthogonal
* matrix \f$ Q \f$ and a Hessenberg matrix \f$ H \f$ such that \f$ A = Q H
* Q^T \f$. An orthogonal matrix is a matrix whose inverse equals its
* transpose (\f$ Q^{-1} = Q^T \f$). A Hessenberg matrix has zeros below the
* subdiagonal, so it is almost upper triangular. The Hessenberg decomposition
* of a complex matrix is \f$ A = Q H Q^* \f$ with \f$ Q \f$ unitary (that is,
* \f$ Q^{-1} = Q^* \f$).
*
* Call the function compute() to compute the Hessenberg decomposition of a
* given matrix. Alternatively, you can use the
* HessenbergDecomposition(const MatrixType&) constructor which computes the
* Hessenberg decomposition at construction time. Once the decomposition is
* computed, you can use the matrixH() and matrixQ() functions to construct
* the matrices H and Q in the decomposition.
*
* The documentation for matrixH() contains an example of the typical use of
* this class.
*
* \sa class ComplexSchur, class Tridiagonalization, \ref QR_Module "QR Module"
*/
template<typename _MatrixType> class HessenbergDecomposition
{
public:
/** \brief Synonym for the template parameter \p _MatrixType. */
typedef _MatrixType MatrixType;
enum {
Size = MatrixType::RowsAtCompileTime,
SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1,
Options = MatrixType::Options,
MaxSize = MatrixType::MaxRowsAtCompileTime,
MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1
};
/** \brief Scalar type for matrices of type #MatrixType. */
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
/** \brief Type for vector of Householder coefficients.
*
* This is column vector with entries of type #Scalar. The length of the
* vector is one less than the size of #MatrixType, if it is a fixed-side
* type.
*/
typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
/** \brief Return type of matrixQ() */
typedef typename HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType HouseholderSequenceType;
typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> MatrixHReturnType;
/** \brief Default constructor; the decomposition will be computed later.
*
* \param [in] size The size of the matrix whose Hessenberg 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.
*/
HessenbergDecomposition(Index size = Size==Dynamic ? 2 : Size)
: m_matrix(size,size),
m_temp(size),
m_isInitialized(false)
{
if(size>1)
m_hCoeffs.resize(size-1);
}
/** \brief Constructor; computes Hessenberg decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed.
*
* This constructor calls compute() to compute the Hessenberg
* decomposition.
*
* \sa matrixH() for an example.
*/
HessenbergDecomposition(const MatrixType& matrix)
: m_matrix(matrix),
m_temp(matrix.rows()),
m_isInitialized(false)
{
if(matrix.rows()<2)
{
m_isInitialized = true;
return;
}
m_hCoeffs.resize(matrix.rows()-1,1);
_compute(m_matrix, m_hCoeffs, m_temp);
m_isInitialized = true;
}
/** \brief Computes Hessenberg decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed.
* \returns Reference to \c *this
*
* The Hessenberg decomposition is computed by bringing the columns of the
* matrix successively in the required form using Householder reflections
* (see, e.g., Algorithm 7.4.2 in Golub \& Van Loan, <i>%Matrix
* Computations</i>). The cost is \f$ 10n^3/3 \f$ flops, where \f$ n \f$
* denotes the size of the given matrix.
*
* This method reuses of the allocated data in the HessenbergDecomposition
* object.
*
* Example: \include HessenbergDecomposition_compute.cpp
* Output: \verbinclude HessenbergDecomposition_compute.out
*/
HessenbergDecomposition& compute(const MatrixType& matrix)
{
m_matrix = matrix;
if(matrix.rows()<2)
{
m_isInitialized = true;
return *this;
}
m_hCoeffs.resize(matrix.rows()-1,1);
_compute(m_matrix, m_hCoeffs, m_temp);
m_isInitialized = true;
return *this;
}
/** \brief Returns the Householder coefficients.
*
* \returns a const reference to the vector of Householder coefficients
*
* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
* or the member function compute(const MatrixType&) has been called
* before to compute the Hessenberg decomposition of a matrix.
*
* The Householder coefficients allow the reconstruction of the matrix
* \f$ Q \f$ in the Hessenberg decomposition from the packed data.
*
* \sa packedMatrix(), \ref Householder_Module "Householder module"
*/
const CoeffVectorType& householderCoefficients() const
{
eigen_assert(m_isInitialized && "HessenbergDecomposition 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 HessenbergDecomposition(const MatrixType&)
* or the member function compute(const MatrixType&) has been called
* before to compute the Hessenberg decomposition of a matrix.
*
* The returned matrix contains the following information:
* - the upper part and lower sub-diagonal represent the Hessenberg matrix H
* - 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 HessenbergDecomposition_packedMatrix.cpp
* Output: \verbinclude HessenbergDecomposition_packedMatrix.out
*
* \sa householderCoefficients()
*/
const MatrixType& packedMatrix() const
{
eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
return m_matrix;
}
/** \brief Reconstructs the orthogonal matrix Q in the decomposition
*
* \returns object representing the matrix Q
*
* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
* or the member function compute(const MatrixType&) has been called
* before to compute the Hessenberg 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 matrixH() for an example, class HouseholderSequence
*/
HouseholderSequenceType matrixQ() const
{
eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
.setLength(m_matrix.rows() - 1)
.setShift(1);
}
/** \brief Constructs the Hessenberg matrix H in the decomposition
*
* \returns expression object representing the matrix H
*
* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
* or the member function compute(const MatrixType&) has been called
* before to compute the Hessenberg decomposition of a matrix.
*
* The object returned by this function constructs the Hessenberg matrix H
* when it is assigned to a matrix or otherwise evaluated. The matrix H is
* constructed from the packed matrix as returned by packedMatrix(): The
* upper part (including the subdiagonal) of the packed matrix contains
* the matrix H. It may sometimes be better to directly use the packed
* matrix instead of constructing the matrix H.
*
* Example: \include HessenbergDecomposition_matrixH.cpp
* Output: \verbinclude HessenbergDecomposition_matrixH.out
*
* \sa matrixQ(), packedMatrix()
*/
MatrixHReturnType matrixH() const
{
eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
return MatrixHReturnType(*this);
}
private:
typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> VectorType;
typedef typename NumTraits<Scalar>::Real RealScalar;
static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp);
protected:
MatrixType m_matrix;
CoeffVectorType m_hCoeffs;
VectorType m_temp;
bool m_isInitialized;
};
/** \internal
* Performs a tridiagonal decomposition of \a matA in place.
*
* \param matA the input selfadjoint matrix
* \param hCoeffs returned Householder coefficients
*
* The result is written in the lower triangular part of \a matA.
*
* Implemented from Golub's "%Matrix Computations", algorithm 8.3.1.
*
* \sa packedMatrix()
*/
template<typename MatrixType>
void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp)
{
assert(matA.rows()==matA.cols());
Index n = matA.rows();
temp.resize(n);
for (Index i = 0; i<n-1; ++i)
{
// let's consider the vector v = i-th column starting at position i+1
Index remainingSize = n-i-1;
RealScalar beta;
Scalar h;
matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
matA.col(i).coeffRef(i+1) = beta;
hCoeffs.coeffRef(i) = h;
// Apply similarity transformation to remaining columns,
// i.e., compute A = H A H'
// A = H A
matA.bottomRightCorner(remainingSize, remainingSize)
.applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0));
// A = A H'
matA.rightCols(remainingSize)
.applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1).conjugate(), internal::conj(h), &temp.coeffRef(0));
}
}
namespace internal {
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \brief Expression type for return value of HessenbergDecomposition::matrixH()
*
* \tparam MatrixType type of matrix in the Hessenberg decomposition
*
* Objects of this type represent the Hessenberg matrix in the Hessenberg
* decomposition of some matrix. The object holds a reference to the
* HessenbergDecomposition class until the it is assigned or evaluated for
* some other reason (the reference should remain valid during the life time
* of this object). This class is the return type of
* HessenbergDecomposition::matrixH(); there is probably no other use for this
* class.
*/
template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType
: public ReturnByValue<HessenbergDecompositionMatrixHReturnType<MatrixType> >
{
typedef typename MatrixType::Index Index;
public:
/** \brief Constructor.
*
* \param[in] hess Hessenberg decomposition
*/
HessenbergDecompositionMatrixHReturnType(const HessenbergDecomposition<MatrixType>& hess) : m_hess(hess) { }
/** \brief Hessenberg matrix in decomposition.
*
* \param[out] result Hessenberg matrix in decomposition \p hess which
* was passed to the constructor
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
result = m_hess.packedMatrix();
Index n = result.rows();
if (n>2)
result.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero();
}
Index rows() const { return m_hess.packedMatrix().rows(); }
Index cols() const { return m_hess.packedMatrix().cols(); }
protected:
const HessenbergDecomposition<MatrixType>& m_hess;
};
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_HESSENBERGDECOMPOSITION_H