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Eigen::Tridiagonalization< _MatrixType > Class Template Reference
Dense linear problems and decompositions » Reference » Eigenvalues_Module

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#include <Tridiagonalization.h>

Public Types

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 _MatrixType MatrixType
 Synonym for the template parameter _MatrixType. More...
 
typedef MatrixType::Scalar Scalar
 
typedef NumTraits< Scalar >::Real RealScalar
 
typedef MatrixType::Index Index
 
typedef Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 > CoeffVectorType
 
typedef internal::plain_col_type< MatrixType, RealScalar >::type DiagonalType
 
typedef Matrix< RealScalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 > SubDiagonalType
 
typedef internal::remove_all< typename MatrixType::RealReturnType >::type MatrixTypeRealView
 
typedef internal::TridiagonalizationMatrixTReturnType< MatrixTypeRealView > MatrixTReturnType
 
typedef 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 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
 
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. More...
 
 Tridiagonalization (const MatrixType &matrix)
 Constructor; computes tridiagonal decomposition of given matrix. More...
 
Tridiagonalization & compute (const MatrixType &matrix)
 Computes tridiagonal decomposition of given matrix. More...
 
CoeffVectorType householderCoefficients () const
 Returns the Householder coefficients. More...
 
const MatrixType & packedMatrix () const
 Returns the internal representation of the decomposition. More...
 
HouseholderSequenceType matrixQ () const
 Returns the unitary matrix Q in the decomposition. More...
 
MatrixTReturnType matrixT () const
 Returns an expression of the tridiagonal matrix T in the decomposition. More...
 
DiagonalReturnType diagonal () const
 Returns the diagonal of the tridiagonal matrix T in the decomposition. More...
 
SubDiagonalReturnType subDiagonal () const
 Returns the subdiagonal of the tridiagonal matrix T in the decomposition. More...
 

Protected Attributes

MatrixType m_matrix
 
CoeffVectorType m_hCoeffs
 
bool m_isInitialized
 

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

Member Typedef Documentation

§ MatrixType

template<typename _MatrixType>
typedef _MatrixType Eigen::Tridiagonalization< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

Constructor & Destructor Documentation

§ Tridiagonalization() [1/2]

template<typename _MatrixType>
Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( Index  size = Size==Dynamic ? 2 : Size)
inline

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.

§ Tridiagonalization() [2/2]

template<typename _MatrixType>
Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( const MatrixType &  matrix)
inline

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: 

 





Member Function Documentation



§ compute()








template<typename _MatrixType> 



  

  
      

        

          Tridiagonalization& Eigen::Tridiagonalization< _MatrixType >::compute 
          (
          const MatrixType & 
          matrix)
          
        

      

  		
  
inline  
  







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: 
 






§ diagonal()








template<typename MatrixType > 

      

        

          Tridiagonalization< MatrixType >::DiagonalReturnType Eigen::Tridiagonalization< MatrixType >::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() 








§ householderCoefficients()








template<typename _MatrixType> 



  

  
      

        

          CoeffVectorType Eigen::Tridiagonalization< _MatrixType >::householderCoefficients 
          (
          )
           const
        

      

  		
  
inline  
  







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 








§ matrixQ()








template<typename _MatrixType> 



  

  
      

        

          HouseholderSequenceType Eigen::Tridiagonalization< _MatrixType >::matrixQ 
          (
          void 
          )
           const
        

      

  		
  
inline  
  







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 








§ matrixT()








template<typename _MatrixType> 



  

  
      

        

          MatrixTReturnType Eigen::Tridiagonalization< _MatrixType >::matrixT 
          (
          )
           const
        

      

  		
  
inline  
  







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() 








§ packedMatrix()








template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::Tridiagonalization< _MatrixType >::packedMatrix 
          (
          )
           const
        

      

  		
  
inline  
  







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() 








§ subDiagonal()








template<typename MatrixType > 

      

        

          Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::Tridiagonalization< MatrixType >::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() 







The documentation for this class was generated from the following file:





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