Eigen
3.2.9

Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation.
This is defined in the Eigenvalues module.
_MatrixType  the type of the matrix of which we are computing the Hessenberg decomposition 
This class performs an Hessenberg decomposition of a matrix . In the real case, the Hessenberg decomposition consists of an orthogonal matrix and a Hessenberg matrix such that . An orthogonal matrix is a matrix whose inverse equals its transpose ( ). A Hessenberg matrix has zeros below the subdiagonal, so it is almost upper triangular. The Hessenberg decomposition of a complex matrix is with unitary (that is, ).
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.
Public Types  
typedef Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >  CoeffVectorType 
Type for vector of Householder coefficients. More...  
typedef HouseholderSequence < MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType > ::type >  HouseholderSequenceType 
Return type of matrixQ()  
typedef _MatrixType  MatrixType 
Synonym for the template parameter _MatrixType .  
typedef MatrixType::Scalar  Scalar 
Scalar type for matrices of type MatrixType.  
Public Member Functions  
HessenbergDecomposition &  compute (const MatrixType &matrix) 
Computes Hessenberg decomposition of given matrix. More...  
HessenbergDecomposition (Index size=Size==Dynamic?2:Size)  
Default constructor; the decomposition will be computed later. More...  
HessenbergDecomposition (const MatrixType &matrix)  
Constructor; computes Hessenberg decomposition of given matrix. More...  
const CoeffVectorType &  householderCoefficients () const 
Returns the Householder coefficients. More...  
MatrixHReturnType  matrixH () const 
Constructs the Hessenberg matrix H in the decomposition. More...  
HouseholderSequenceType  matrixQ () const 
Reconstructs the orthogonal matrix Q in the decomposition. More...  
const MatrixType &  packedMatrix () const 
Returns the internal representation of the decomposition. More...  
typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType 
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 fixedside type.

inline 
Default constructor; the decomposition will be computed later.
[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 size
parameter is only used as a hint. It is not an error to give a wrong size
, but it may impair performance.

inline 

inline 
Computes Hessenberg decomposition of given matrix.
[in]  matrix  Square matrix whose Hessenberg decomposition is to be computed. 
*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, Matrix Computations). The cost is flops, where denotes the size of the given matrix.
This method reuses of the allocated data in the HessenbergDecomposition object.
Example:
Output:
The matrix H in the decomposition of A is: (0.211,0.68) (0.346,0.216) (0.688,0.00979) (0.0451,0.584) (1.45,0) (0.0574,0.0123) (0.196,0.385) (0.395,0.389) (0,0) (1.68,0) (0.397,0.552) (0.156,0.241) (0,0) (0,0) (1.56,0) (0.876,0.423) The matrix H in the decomposition of 2A is: (0.422,1.36) (0.691,0.431) (1.38,0.0196) (0.0902,1.17) (2.91,0) (0.115,0.0246) (0.392,0.77) (0.791,0.777) (0,0) (3.36,0) (0.795,1.1) (0.311,0.482) (0,0) (0,0) (3.12,0) (1.75,0.846)

inline 
Returns the Householder coefficients.
The Householder coefficients allow the reconstruction of the matrix in the Hessenberg decomposition from the packed data.

inline 
Constructs the Hessenberg matrix H in the decomposition.
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:
Output:
Here is a random 4x4 matrix: 0.68 0.823 0.444 0.27 0.211 0.605 0.108 0.0268 0.566 0.33 0.0452 0.904 0.597 0.536 0.258 0.832 The Hessenberg matrix H is: 0.68 0.691 0.645 0.235 0.849 0.836 0.419 0.794 0 0.469 0.547 0.0731 0 0 0.559 0.107 The orthogonal matrix Q is: 1 0 0 0 0 0.249 0.958 0.144 0 0.667 0.277 0.692 0 0.703 0.0761 0.707 Q H Q^T is: 0.68 0.823 0.444 0.27 0.211 0.605 0.108 0.0268 0.566 0.33 0.0452 0.904 0.597 0.536 0.258 0.832

inline 
Reconstructs the orthogonal matrix Q in the decomposition.
This function returns a lightweight 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.

inline 
Returns the internal representation of the decomposition.
The returned matrix contains the following information:
See LAPACK for further details on this packed storage.
Example:
Output:
Here is a random 4x4 matrix: 0.68 0.823 0.444 0.27 0.211 0.605 0.108 0.0268 0.566 0.33 0.0452 0.904 0.597 0.536 0.258 0.832 The packed matrix M is: 0.68 0.691 0.645 0.235 0.849 0.836 0.419 0.794 0.534 0.469 0.547 0.0731 0.563 0.344 0.559 0.107 The upper Hessenberg part corresponds to the matrix H, which is: 0.68 0.691 0.645 0.235 0.849 0.836 0.419 0.794 0 0.469 0.547 0.0731 0 0 0.559 0.107 The vector of Householder coefficients is: 1.25 1.79 0