Public Types | Public Member Functions | Protected Types | Protected Attributes | Private Member Functions | List of all members
Eigen::EigenSolver< _MatrixType > Class Template Reference

Computes eigenvalues and eigenvectors of general matrices. More...

#include <EigenSolver.h>

Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}
 
typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for MatrixType. More...
 
typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > EigenvalueType
 Type for vector of eigenvalues as returned by eigenvalues(). More...
 
typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime > EigenvectorsType
 Type for matrix of eigenvectors as returned by eigenvectors(). More...
 
typedef MatrixType::Index Index
 
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType. More...
 
typedef NumTraits< Scalar >::Real RealScalar
 
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType. More...
 

Public Member Functions

EigenSolver & compute (const MatrixType &matrix, bool computeEigenvectors=true)
 Computes eigendecomposition of given matrix. More...
 
 EigenSolver ()
 Default constructor. More...
 
 EigenSolver (Index size)
 Default constructor with memory preallocation. More...
 
 EigenSolver (const MatrixType &matrix, bool computeEigenvectors=true)
 Constructor; computes eigendecomposition of given matrix. More...
 
const EigenvalueType & eigenvalues () const
 Returns the eigenvalues of given matrix. More...
 
EigenvectorsType eigenvectors () const
 Returns the eigenvectors of given matrix. More...
 
Index getMaxIterations ()
 Returns the maximum number of iterations. More...
 
ComputationInfo info () const
 
MatrixType pseudoEigenvalueMatrix () const
 Returns the block-diagonal matrix in the pseudo-eigendecomposition. More...
 
const MatrixType & pseudoEigenvectors () const
 Returns the pseudo-eigenvectors of given matrix. More...
 
EigenSolver & setMaxIterations (Index maxIters)
 Sets the maximum number of iterations allowed. More...
 

Protected Types

typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > ColumnVectorType
 

Protected Attributes

bool m_eigenvectorsOk
 
EigenvalueType m_eivalues
 
MatrixType m_eivec
 
bool m_isInitialized
 
MatrixType m_matT
 
RealSchur< MatrixType > m_realSchur
 
ColumnVectorType m_tmp
 

Private Member Functions

void doComputeEigenvectors ()
 

Detailed Description

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

Computes eigenvalues and eigenvectors of general matrices.

Template Parameters
_MatrixTypethe type of the matrix of which we are computing the eigendecomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported.

The eigenvalues and eigenvectors of a matrix $ A $ are scalars $ \lambda $ and vectors $ v $ such that $ Av = \lambda v $. If $ D $ is a diagonal matrix with the eigenvalues on the diagonal, and $ V $ is a matrix with the eigenvectors as its columns, then $ A V = V D $. The matrix $ V $ is almost always invertible, in which case we have $ A = V D V^{-1} $. This is called the eigendecomposition.

The eigenvalues and eigenvectors of a matrix may be complex, even when the matrix is real. However, we can choose real matrices $ V $ and $ D $ satisfying $ A V = V D $, just like the eigendecomposition, if the matrix $ D $ is not required to be diagonal, but if it is allowed to have blocks of the form

\[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \]

(where $ u $ and $ v $ are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs $ u \pm iv $. We call this variant of the eigendecomposition the pseudo-eigendecomposition.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the EigenSolver(const MatrixType&, bool) constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues() and eigenvectors() functions. The pseudoEigenvalueMatrix() and pseudoEigenvectors() methods allow the construction of the pseudo-eigendecomposition.

The documentation for EigenSolver(const MatrixType&, bool) contains an example of the typical use of this class.

Note
The implementation is adapted from JAMA (public domain). Their code is based on EISPACK.
See also
MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver

Definition at line 64 of file EigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::ColumnVectorType
protected

Definition at line 308 of file EigenSolver.h.

template<typename _MatrixType>
typedef std::complex<RealScalar> Eigen::EigenSolver< _MatrixType >::ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Definition at line 90 of file EigenSolver.h.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::EigenvalueType

Type for vector of eigenvalues as returned by eigenvalues().

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

Definition at line 97 of file EigenSolver.h.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::EigenSolver< _MatrixType >::EigenvectorsType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Definition at line 104 of file EigenSolver.h.

template<typename _MatrixType>
typedef MatrixType::Index Eigen::EigenSolver< _MatrixType >::Index

Definition at line 82 of file EigenSolver.h.

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

Synonym for the template parameter _MatrixType.

Definition at line 69 of file EigenSolver.h.

template<typename _MatrixType>
typedef NumTraits<Scalar>::Real Eigen::EigenSolver< _MatrixType >::RealScalar

Definition at line 81 of file EigenSolver.h.

template<typename _MatrixType>
typedef MatrixType::Scalar Eigen::EigenSolver< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 80 of file EigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 71 of file EigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
Eigen::EigenSolver< _MatrixType >::EigenSolver ( )
inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

See also
compute() for an example.

Definition at line 113 of file EigenSolver.h.

template<typename _MatrixType>
Eigen::EigenSolver< _MatrixType >::EigenSolver ( Index  size)
inline

Default constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also
EigenSolver()

Definition at line 121 of file EigenSolver.h.

template<typename _MatrixType>
Eigen::EigenSolver< _MatrixType >::EigenSolver ( const MatrixType &  matrix,
bool  computeEigenvectors = true 
)
inline

Constructor; computes eigendecomposition of given matrix.

Parameters
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigenvalues and eigenvectors.

Example:

Output: 

See also
compute() 



Definition at line 146 of file EigenSolver.h.







Member Function Documentation








template<typename MatrixType > 

      

        

          EigenSolver< MatrixType > & Eigen::EigenSolver< MatrixType >::compute 
          (
          const MatrixType & 
          matrix, 
        

        

          
          
          bool 
          computeEigenvectors = true 
        

        

          
          )
          
        

      





Computes eigendecomposition of given matrix. 


Parameters

  

    
[in]matrixSquare matrix whose eigendecomposition is to be computed. 

    
[in]computeEigenvectorsIf true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed. 

  

  




Returns
Reference to *this 


This function computes the eigenvalues of the real matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().


The matrix is first reduced to real Schur form using the RealSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.


The cost of the computation is dominated by the cost of the Schur decomposition, which is very approximately $ 25n^3 $ (where $ n $ is the size of the matrix) if computeEigenvectors is true, and $ 10n^3 $ if computeEigenvectors is false.


This method reuses of the allocated data in the EigenSolver object.


Example: 
 Output: 
 

Definition at line 365 of file EigenSolver.h.













template<typename MatrixType > 



  

  
      

        

          void Eigen::EigenSolver< MatrixType >::doComputeEigenvectors 
          (
          )
          
        

      

  		
  
private  
  







Definition at line 433 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          const EigenvalueType& Eigen::EigenSolver< _MatrixType >::eigenvalues 
          (
          )
           const
        

      

  		
  
inline  
  







Returns the eigenvalues of given matrix. 


Returns
A const reference to the column vector containing the eigenvalues.


Precondition
Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.


The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.


Example: 
 Output: 
See also
eigenvectors(), pseudoEigenvalueMatrix(), MatrixBase::eigenvalues() 



Definition at line 243 of file EigenSolver.h.













template<typename MatrixType > 

      

        

          EigenSolver< MatrixType >::EigenvectorsType Eigen::EigenSolver< MatrixType >::eigenvectors 
          (
          )
           const
        

      





Returns the eigenvectors of given matrix. 


Returns
Matrix whose columns are the (possibly complex) eigenvectors.


Precondition
Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).


Column $ k $ of the returned matrix is an eigenvector corresponding to eigenvalue number $ k $ as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. The matrix returned by this function is the matrix $ V $ in the eigendecomposition $ A = V D V^{-1} $, if it exists.


Example: 
 Output: 
See also
eigenvalues(), pseudoEigenvectors() 



Definition at line 333 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::EigenSolver< _MatrixType >::getMaxIterations 
          (
          )
          
        

      

  		
  
inline  
  







Returns the maximum number of iterations. 



Definition at line 292 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          ComputationInfo Eigen::EigenSolver< _MatrixType >::info 
          (
          )
           const
        

      

  		
  
inline  
  







Definition at line 278 of file EigenSolver.h.













template<typename MatrixType > 

      

        

          MatrixType Eigen::EigenSolver< MatrixType >::pseudoEigenvalueMatrix 
          (
          )
           const
        

      





Returns the block-diagonal matrix in the pseudo-eigendecomposition. 


Returns
A block-diagonal matrix.


Precondition
Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.


The matrix $ D $ returned by this function is real and block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 blocks of the form $ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} $. These blocks are not sorted in any particular order. The matrix $ D $ and the matrix $ V $ returned by pseudoEigenvectors() satisfy $ AV = VD $.


See also
pseudoEigenvectors() for an example, eigenvalues() 



Definition at line 313 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::EigenSolver< _MatrixType >::pseudoEigenvectors 
          (
          )
           const
        

      

  		
  
inline  
  







Returns the pseudo-eigenvectors of given matrix. 


Returns
Const reference to matrix whose columns are the pseudo-eigenvectors.


Precondition
Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).


The real matrix $ V $ returned by this function and the block-diagonal matrix $ D $ returned by pseudoEigenvalueMatrix() satisfy $ AV = VD $.


Example: 
 Output: 
See also
pseudoEigenvalueMatrix(), eigenvectors() 



Definition at line 198 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          EigenSolver& Eigen::EigenSolver< _MatrixType >::setMaxIterations 
          (
          Index 
          maxIters)
          
        

      

  		
  
inline  
  







Sets the maximum number of iterations allowed. 



Definition at line 285 of file EigenSolver.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          bool Eigen::EigenSolver< _MatrixType >::m_eigenvectorsOk
        

      

  		
  
protected  
  







Definition at line 304 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          EigenvalueType Eigen::EigenSolver< _MatrixType >::m_eivalues
        

      

  		
  
protected  
  







Definition at line 302 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::EigenSolver< _MatrixType >::m_eivec
        

      

  		
  
protected  
  







Definition at line 301 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::EigenSolver< _MatrixType >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 303 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::EigenSolver< _MatrixType >::m_matT
        

      

  		
  
protected  
  







Definition at line 306 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          RealSchur<MatrixType> Eigen::EigenSolver< _MatrixType >::m_realSchur
        

      

  		
  
protected  
  







Definition at line 305 of file EigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          ColumnVectorType Eigen::EigenSolver< _MatrixType >::m_tmp
        

      

  		
  
protected  
  







Definition at line 309 of file EigenSolver.h.







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







tuw_aruco

Author(s): Lukas Pfeifhofer

autogenerated on Mon Jun 10 2019 15:41:06