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Extend documentation for HessenbergDecomposition.

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Jitse Niesen 2010-03-28 17:33:56 +01:00
parent 0a5c2d8a54
commit e6300efb5c
4 changed files with 158 additions and 30 deletions
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165
Eigen/src/Eigenvalues/HessenbergDecomposition.h
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@ -30,14 +30,30 @@
*
* \class HessenbergDecomposition
*
* \brief Reduces a squared matrix to an Hessemberg form
* \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation
*
* \param MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
* \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$ such that:
* \f$ A = Q H Q^* \f$ where \f$ Q \f$ is unitary and \f$ H \f$ a Hessenberg matrix.
* 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$).
*
* \sa class Tridiagonalization, class Qr
* 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
{
@ -51,13 +67,28 @@ template<typename _MatrixType> class HessenbergDecomposition
MaxSize = MatrixType::MaxRowsAtCompileTime,
MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1
};
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef Matrix<Scalar, SizeMinusOne, 1, Options, MaxSizeMinusOne, 1> CoeffVectorType;
typedef Matrix<Scalar, 1, Size, Options, 1, MaxSize> VectorType;
/** This constructor initializes a HessenbergDecomposition object for
* further use with HessenbergDecomposition::compute()
/** \brief Scalar type for matrices of type \p _MatrixType. */
typedef typename MatrixType::Scalar Scalar;
/** \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 \p _MatrixType, if it is a
* fixed-side type.
*/
typedef Matrix<Scalar, SizeMinusOne, 1, Options, MaxSizeMinusOne, 1> CoeffVectorType;
/** \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(int size = Size==Dynamic ? 2 : Size)
: m_matrix(size,size)
@ -66,6 +97,15 @@ template<typename _MatrixType> class HessenbergDecomposition
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)
{
@ -75,9 +115,21 @@ template<typename _MatrixType> class HessenbergDecomposition
_compute(m_matrix, m_hCoeffs);
}
/** Computes or re-compute the Hessenberg decomposition for the matrix \a matrix.
/** \brief Computes Hessenberg decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed.
*
* This method allows to re-use the allocated data.
* 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
*/
void compute(const MatrixType& matrix)
{
@ -88,36 +140,95 @@ template<typename _MatrixType> class HessenbergDecomposition
_compute(m_matrix, m_hCoeffs);
}
/** \returns a const reference to the householder coefficients allowing to
* reconstruct the matrix Q from the packed data.
/** \brief Returns the Householder coefficients.
*
* \sa packedMatrix()
* \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 { return m_hCoeffs; }
/** \returns a const reference to the internal representation of the decomposition.
/** \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 follow:
* Q = H_{N-1} ... H_1 H_0
* where the matrices H are the Householder transformation:
* H_i = (I - h_i * v_i * v_i')
* where h_i == householderCoefficients()[i] and v_i is a Householder vector:
* v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]
* 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(void) const { return m_matrix; }
/** \brief Reconstructs the orthogonal matrix Q in the decomposition
*
* \returns 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 reconstructs the matrix Q from the Householder
* coefficients and the packed matrix stored internally. This
* reconstruction requires \f$ 4n^3 / 3 \f$ flops.
*
* \sa matrixH() for an example
*/
MatrixType matrixQ() const;
/** \brief Constructs the Hessenberg matrix H in the decomposition
*
* \returns 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.
*
* This function copies the matrix H from internal data. The upper part
* (including the subdiagonal) of the packed matrix as returned by
* packedMatrix() contains the matrix H. This function copies those
* entries in a newly created matrix and sets the remaining entries to
* zero. It may sometimes be sufficient to directly use the packed matrix
* instead of creating a new one.
*
* Example: \include HessenbergDecomposition_matrixH.cpp
* Output: \verbinclude HessenbergDecomposition_matrixH.out
*
* \sa matrixQ(), packedMatrix()
*/
MatrixType matrixH() const;
private:
static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
typedef Matrix<Scalar, 1, Size, Options, 1, MaxSize> VectorType;
typedef typename NumTraits<Scalar>::Real RealScalar;
protected:
MatrixType m_matrix;
@ -134,7 +245,7 @@ template<typename _MatrixType> class HessenbergDecomposition
*
* The result is written in the lower triangular part of \a matA.
*
* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
* Implemented from Golub's "%Matrix Computations", algorithm 8.3.1.
*
* \sa packedMatrix()
*/
@ -167,7 +278,6 @@ void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVector
}
}
/** reconstructs and returns the matrix Q */
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
HessenbergDecomposition<MatrixType>::matrixQ() const
@ -185,11 +295,6 @@ HessenbergDecomposition<MatrixType>::matrixQ() const
#endif // EIGEN_HIDE_HEAVY_CODE
/** constructs and returns the matrix H.
* Note that the matrix H is equivalent to the upper part of the packed matrix
* (including the lower sub-diagonal). Therefore, it might be often sufficient
* to directly use the packed matrix instead of creating a new one.
*/
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
HessenbergDecomposition<MatrixType>::matrixH() const

6
doc/snippets/HessenbergDecomposition_compute.cpp Normal file
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@ -0,0 +1,6 @@
MatrixXcf A = MatrixXcf::Random(4,4);
HessenbergDecomposition<MatrixXcf> hd(4);
hd.compute(A);
cout << "The matrix H in the decomposition of A is:" << endl << hd.matrixH() << endl;
hd.compute(2*A); // re-use hd to compute and store decomposition of 2A
cout << "The matrix H in the decomposition of 2A is:" << endl << hd.matrixH() << endl;

8
doc/snippets/HessenbergDecomposition_matrixH.cpp Normal file
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@ -0,0 +1,8 @@
Matrix4f A = MatrixXf::Random(4,4);
cout << "Here is a random 4x4 matrix:" << endl << A << endl;
HessenbergDecomposition<MatrixXf> hessOfA(A);
MatrixXf H = hessOfA.matrixH();
cout << "The Hessenberg matrix H is:" << endl << H << endl;
MatrixXf Q = hessOfA.matrixQ();
cout << "The orthogonal matrix Q is:" << endl << Q << endl;
cout << "Q H Q^T is:" << endl << Q * H * Q.transpose() << endl;

9
doc/snippets/HessenbergDecomposition_packedMatrix.cpp Normal file
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@ -0,0 +1,9 @@
Matrix4d A = Matrix4d::Random(4,4);
cout << "Here is a random 4x4 matrix:" << endl << A << endl;
HessenbergDecomposition<Matrix4d> hessOfA(A);
Matrix4d pm = hessOfA.packedMatrix();
cout << "The packed matrix M is:" << endl << pm << endl;
cout << "The upper Hessenberg part corresponds to the matrix H, which is:"
<< endl << hessOfA.matrixH() << endl;
Vector3d hc = hessOfA.householderCoefficients();
cout << "The vector of Householder coefficients is:" << endl << hc << endl;