Jaesung Oh / Eigen

Important changes to repositories hosted on mbed.com

Mbed hosted mercurial repositories are deprecated and are due to be permanently deleted in July 2026.

To keep a copy of this software download the repository Zip archive or clone locally using Mercurial.

It is also possible to export all your personal repositories from the account settings page.

Dependents:   MPC_current_control HydraulicControlBoard_SW AHRS Test_ekf ... more

Embed: (wiki syntax)

« Back to documentation index

Tridiagonalization< _MatrixType > Class Template Reference
Public Types | Public Member Functions

Tridiagonalization< _MatrixType > Class Template Reference
[Eigenvalues module]

More...

#include <Tridiagonalization.h>

Public Types

typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef HouseholderSequence
< MatrixType, typename
internal::remove_all< typename
CoeffVectorType::ConjugateReturnType >
::type > 		HouseholderSequenceType
 Return type of matrixQ()

Public Member Functions

 Tridiagonalization (Index size=Size==Dynamic?2:Size)
 Default constructor.
 Tridiagonalization (const MatrixType &matrix)
 Constructor; computes tridiagonal decomposition of given matrix.
Tridiagonalization & compute (const MatrixType &matrix)
 Computes tridiagonal decomposition of given matrix.
CoeffVectorType householderCoefficients () const
 Returns the Householder coefficients.
const MatrixType & packedMatrix () const
 Returns the internal representation of the decomposition.
HouseholderSequenceType matrixQ () const
 Returns the unitary matrix Q in the decomposition.
MatrixTReturnType matrixT () const
 Returns an expression of the tridiagonal matrix T in the decomposition.
DiagonalReturnType diagonal () const
 Returns the diagonal of the tridiagonal matrix T in the decomposition.
SubDiagonalReturnType subDiagonal () const
 Returns the subdiagonal of the tridiagonal matrix T in the decomposition.

Detailed Description

template<typename _MatrixType>
class Eigen::Tridiagonalization< _MatrixType >

Tridiagonal decomposition of a selfadjoint matrix

Template Parameters:
_MatrixTypethe 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 $ A $ such that: $ A = Q T Q^* $ where $ Q $ is unitary and $ T $ 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.

See also:
class HessenbergDecomposition, class SelfAdjointEigenSolver

Definition at line 61 of file Tridiagonalization.h.

Member Typedef Documentation

typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType

Return type of matrixQ()

Definition at line 99 of file Tridiagonalization.h.

typedef _MatrixType MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 66 of file Tridiagonalization.h.

Constructor & Destructor Documentation

Tridiagonalization ( Index  size = Size==Dynamic ? 2 : Size )

Default constructor.

Parameters:
[in]sizePositive 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 size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also:
compute() for an example.

Definition at line 113 of file Tridiagonalization.h.

Tridiagonalization ( const MatrixType &  matrix )

Constructor; computes tridiagonal decomposition of given matrix.

Parameters:
[in]matrixSelfadjoint matrix whose tridiagonal decomposition is to be computed.

This constructor calls compute() to compute the tridiagonal decomposition.

Example: 

 Output: 


 

Definition at line 129 of file Tridiagonalization.h.







Member Function Documentation






      

        

          Tridiagonalization& compute 
          (
          const MatrixType & 
           matrix )
          
        

      







Computes tridiagonal decomposition of given matrix. 


Parameters:

  

    
[in]matrixSelfadjoint matrix whose tridiagonal decomposition is to be computed. 

  

  




Returns:
Reference to *this 


The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is $ 4n^3/3 $ flops, where $ n $ 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: 


 Output: 


 

Definition at line 155 of file Tridiagonalization.h.











      

        

          Tridiagonalization< MatrixType >::DiagonalReturnType diagonal 
          (
           )
           const
        

      







Returns the diagonal of the tridiagonal matrix T in the decomposition. 


Returns:
expression representing the diagonal of T


Precondition:
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: 


 Output: 


See also:
matrixT(), subDiagonal() 



Definition at line 305 of file Tridiagonalization.h.











      

        

          CoeffVectorType householderCoefficients 
          (
           )
           const
        

      







Returns the Householder coefficients. 


Returns:
a const reference to the vector of Householder coefficients


Precondition:
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 $ Q $ in the tridiagonal decomposition from the packed data.


Example: 


 Output: 


See also:
packedMatrix(), Householder module 



Definition at line 180 of file Tridiagonalization.h.











      

        

          HouseholderSequenceType matrixQ 
          (
           )
           const
        

      







Returns the unitary matrix Q in the decomposition. 


Returns:
object representing the matrix Q


Precondition:
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.


See also:
Tridiagonalization(const MatrixType&) for an example, matrixT(), class HouseholderSequence 



Definition at line 238 of file Tridiagonalization.h.











      

        

          MatrixTReturnType matrixT 
          (
           )
           const
        

      







Returns an expression of the tridiagonal matrix T in the decomposition. 


Returns:
expression object representing the matrix T


Precondition:
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.


See also:
Tridiagonalization(const MatrixType&) for an example, matrixQ(), packedMatrix(), diagonal(), subDiagonal() 



Definition at line 263 of file Tridiagonalization.h.











      

        

          const MatrixType& packedMatrix 
          (
           )
           const
        

      







Returns the internal representation of the decomposition. 


Returns:
a const reference to a matrix with the internal representation of the decomposition.


Precondition:
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 $ Q = H_{N-1} \ldots H_1 H_0 $. Here, the matrices $ H_i $ are the Householder transformations $ H_i = (I - h_i v_i v_i^T) $ where $ h_i $ is the $ i $th Householder coefficient and $ v_i $ is the Householder vector defined by $ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T $ with M the matrix returned by this function.
    • 



See LAPACK for further details on this packed storage.


Example: 


 Output: 


See also:
householderCoefficients() 



Definition at line 217 of file Tridiagonalization.h.











      

        

          Tridiagonalization< MatrixType >::SubDiagonalReturnType subDiagonal 
          (
           )
           const
        

      







Returns the subdiagonal of the tridiagonal matrix T in the decomposition. 


Returns:
expression representing the subdiagonal of T


Precondition:
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.


See also:
diagonal() for an example, matrixT() 



Definition at line 313 of file Tridiagonalization.h.









Generated on Tue Jul 12 2022 17:34:20 by 

doxygen 1.7.2