Public Types | Public Member Functions | Static Protected Member Functions | Protected Attributes | List of all members
Eigen::GeneralizedEigenSolver< _MatrixType > Class Template Reference
Dense linear problems and decompositions » Reference » Eigenvalues_Module

Computes the generalized eigenvalues and eigenvectors of a pair of general matrices. More...

#include <GeneralizedEigenSolver.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 > ComplexVectorType
 Type for vector of complex scalar values eigenvalues as returned by alphas(). More...
 
typedef CwiseBinaryOp< internal::scalar_quotient_op< ComplexScalar, Scalar >, ComplexVectorType, VectorType > EigenvalueType
 Expression type for the 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 Eigen::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...
 
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > VectorType
 Type for vector of real scalar values eigenvalues as returned by betas(). More...
 

Public Member Functions

ComplexVectorType alphas () const
 
VectorType betas () const
 
GeneralizedEigenSolver & compute (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
 Computes generalized eigendecomposition of given matrix. More...
 
EigenvalueType eigenvalues () const
 Returns an expression of the computed generalized eigenvalues. More...
 
EigenvectorsType eigenvectors () const
 
 GeneralizedEigenSolver ()
 Default constructor. More...
 
 GeneralizedEigenSolver (Index size)
 Default constructor with memory preallocation. More...
 
 GeneralizedEigenSolver (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
 Constructor; computes the generalized eigendecomposition of given matrix pair. More...
 
ComputationInfo info () const
 
GeneralizedEigenSolver & setMaxIterations (Index maxIters)
 

Static Protected Member Functions

static void check_template_parameters ()
 

Protected Attributes

ComplexVectorType m_alphas
 
VectorType m_betas
 
EigenvectorsType m_eivec
 
RealQZ< MatrixType > m_realQZ
 
ComplexVectorType m_tmp
 
bool m_valuesOkay
 
bool m_vectorsOkay
 

Detailed Description

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

Computes the generalized eigenvalues and eigenvectors of a pair of general matrices.

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

The generalized eigenvalues and eigenvectors of a matrix pair $ A $ and $ B $ are scalars $ \lambda $ and vectors $ v $ such that $ Av = \lambda Bv $. 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 = B V D $. The matrix $ V $ is almost always invertible, in which case we have $ A = B V D V^{-1} $. This is called the generalized eigen-decomposition.

The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex $ \alpha $ and real $ \beta $ such that: $ \lambda_i = \alpha_i / \beta_i $. If $ \beta_i $ is (nearly) zero, then one can consider the well defined left eigenvalue $ \mu = \beta_i / \alpha_i$ such that: $ \mu_i A v_i = B v_i $, or even $ \mu_i u_i^T A = u_i^T B $ where $ u_i $ is called the left eigenvector.

Call the function compute() to compute the generalized eigenvalues and eigenvectors of a given matrix pair. Alternatively, you can use the GeneralizedEigenSolver(const MatrixType&, 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.

Here is an usage example of this class: Example:

GeneralizedEigenSolver<MatrixXf> ges;
MatrixXf A = MatrixXf::Random(4,4);
MatrixXf B = MatrixXf::Random(4,4);
ges.compute(A, B);
cout << "The (complex) numerators of the generalzied eigenvalues are: " << ges.alphas().transpose() << endl;
cout << "The (real) denominatore of the generalzied eigenvalues are: " << ges.betas().transpose() << endl;
cout << "The (complex) generalzied eigenvalues are (alphas./beta): " << ges.eigenvalues().transpose() << endl;

Output: 

See also
MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver 



Definition at line 58 of file GeneralizedEigenSolver.h.


Member Typedef Documentation








template<typename _MatrixType> 

      

        

          typedef std::complex<RealScalar> Eigen::GeneralizedEigenSolver< _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 84 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::ComplexVectorType
        

      





Type for vector of complex scalar values eigenvalues as returned by alphas(). 


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



Definition at line 98 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> Eigen::GeneralizedEigenSolver< _MatrixType >::EigenvalueType
        

      





Expression type for the eigenvalues as returned by eigenvalues(). 



Definition at line 102 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::GeneralizedEigenSolver< _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 109 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef Eigen::Index Eigen::GeneralizedEigenSolver< _MatrixType >::Index
        

      




Deprecated:
since Eigen 3.3 



Definition at line 76 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef _MatrixType Eigen::GeneralizedEigenSolver< _MatrixType >::MatrixType
        

      





Synonym for the template parameter _MatrixType. 



Definition at line 63 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef NumTraits<Scalar>::Real Eigen::GeneralizedEigenSolver< _MatrixType >::RealScalar
        

      





Definition at line 75 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef MatrixType::Scalar Eigen::GeneralizedEigenSolver< _MatrixType >::Scalar
        

      





Scalar type for matrices of type MatrixType. 



Definition at line 74 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 

      

        

          typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::VectorType
        

      





Type for vector of real scalar values eigenvalues as returned by betas(). 


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



Definition at line 91 of file GeneralizedEigenSolver.h.







Member Enumeration Documentation








template<typename _MatrixType> 

      

        

          anonymous enum
        

      






Enumerator
RowsAtCompileTime 



ColsAtCompileTime 



Options 



MaxRowsAtCompileTime 



MaxColsAtCompileTime 






Definition at line 65 of file GeneralizedEigenSolver.h.







Constructor & Destructor Documentation








template<typename _MatrixType> 



  

  
      

        

          Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver 
          (
          )
          
        

      

  		
  
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 118 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver 
          (
          Index 
          size)
          
        

      

  		
  
inlineexplicit  
  







Default constructor with memory preallocation. 


Like the default constructor but with preallocation of the internal data according to the specified problem size. 
See also
GeneralizedEigenSolver() 



Definition at line 133 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver 
          (
          const MatrixType & 
          A, 
        

        

          
          
          const MatrixType & 
          B, 
        

        

          
          
          bool 
          computeEigenvectors = true 
        

        

          
          )
          
        

      

  		
  
inline  
  







Constructor; computes the generalized eigendecomposition of given matrix pair. 


Parameters

  

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

    
[in]BSquare 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 generalized eigenvalues and eigenvectors.


See also
compute() 



Definition at line 155 of file GeneralizedEigenSolver.h.







Member Function Documentation








template<typename _MatrixType> 



  

  
      

        

          ComplexVectorType Eigen::GeneralizedEigenSolver< _MatrixType >::alphas 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
A const reference to the vectors containing the alpha values


This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).


See also
betas(), eigenvalues() 



Definition at line 213 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          VectorType Eigen::GeneralizedEigenSolver< _MatrixType >::betas 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
A const reference to the vectors containing the beta values


This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).


See also
alphas(), eigenvalues() 



Definition at line 224 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          static void Eigen::GeneralizedEigenSolver< _MatrixType >::check_template_parameters 
          (
          )
          
        

      

  		
  
inlinestaticprotected  
  







Definition at line 271 of file GeneralizedEigenSolver.h.













template<typename MatrixType > 

      

        

          GeneralizedEigenSolver< MatrixType > & Eigen::GeneralizedEigenSolver< MatrixType >::compute 
          (
          const MatrixType & 
          A, 
        

        

          
          
          const MatrixType & 
          B, 
        

        

          
          
          bool 
          computeEigenvectors = true 
        

        

          
          )
          
        

      





Computes generalized eigendecomposition of given matrix. 


Parameters

  

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

    
[in]BSquare 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 generalized Schur form using the RealQZ class. The generalized Schur decomposition is then used to compute the eigenvalues and eigenvectors.


The cost of the computation is dominated by the cost of the generalized Schur decomposition.


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



Definition at line 287 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          EigenvalueType Eigen::GeneralizedEigenSolver< _MatrixType >::eigenvalues 
          (
          )
           const
        

      

  		
  
inline  
  







Returns an expression of the computed generalized eigenvalues. 


Returns
An expression of the column vector containing the eigenvalues.


It is a shortcut for
this->alphas().cwiseQuotient(this->betas()); 
 Not that betas might contain zeros. It is therefore not recommended to use this function, but rather directly deal with the alphas and betas vectors.


Precondition
Either the constructor GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function compute(const MatrixType&,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.


See also
alphas(), betas(), eigenvectors() 



Definition at line 202 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          EigenvectorsType Eigen::GeneralizedEigenSolver< _MatrixType >::eigenvectors 
          (
          )
           const
        

      

  		
  
inline  
  







Definition at line 179 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  







Definition at line 255 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  






Sets the maximal number of iterations allowed. 



Definition at line 263 of file GeneralizedEigenSolver.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          ComplexVectorType Eigen::GeneralizedEigenSolver< _MatrixType >::m_alphas
        

      

  		
  
protected  
  







Definition at line 278 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          VectorType Eigen::GeneralizedEigenSolver< _MatrixType >::m_betas
        

      

  		
  
protected  
  







Definition at line 279 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          EigenvectorsType Eigen::GeneralizedEigenSolver< _MatrixType >::m_eivec
        

      

  		
  
protected  
  







Definition at line 277 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          RealQZ<MatrixType> Eigen::GeneralizedEigenSolver< _MatrixType >::m_realQZ
        

      

  		
  
protected  
  







Definition at line 281 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          ComplexVectorType Eigen::GeneralizedEigenSolver< _MatrixType >::m_tmp
        

      

  		
  
protected  
  







Definition at line 282 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::GeneralizedEigenSolver< _MatrixType >::m_valuesOkay
        

      

  		
  
protected  
  







Definition at line 280 of file GeneralizedEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::GeneralizedEigenSolver< _MatrixType >::m_vectorsOkay
        

      

  		
  
protected  
  







Definition at line 280 of file GeneralizedEigenSolver.h.







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







gtsam

Author(s): 

autogenerated on Sat May 8 2021 02:52:37