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Eigen::GeneralizedEigenSolver< _MatrixType > Class Template Reference

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

#include <GeneralizedEigenSolver.h>

+ Collaboration diagram for Eigen::GeneralizedEigenSolver< _MatrixType >:

Public Types

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

Public Member Functions

 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...
 
EigenvalueType eigenvalues () const
 Returns an expression of the computed generalized eigenvalues. More...
 
ComplexVectorType alphas () const
 
VectorType betas () const
 
GeneralizedEigenSolver & compute (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
 Computes generalized eigendecomposition of given matrix. More...
 
ComputationInfo info () const
 
GeneralizedEigenSolver & setMaxIterations (Index maxIters)
 

Protected Types

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

Protected Attributes

MatrixType m_eivec
 
ComplexVectorType m_alphas
 
VectorType m_betas
 
bool m_isInitialized
 
bool m_eigenvectorsOk
 
RealQZ< MatrixType > m_realQZ
 
MatrixType m_matS
 
ColumnVectorType m_tmp
 

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:

Output: 

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



Definition at line 57 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 83 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 betas(). 


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 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 108 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 90 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 117 of file GeneralizedEigenSolver.h.


  117 : m_eivec(), m_alphas(), m_betas(), m_isInitialized(false), m_realQZ(), m_matS(), m_tmp() {}












template<typename _MatrixType > 



  

  
      

        

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



Definition at line 125 of file GeneralizedEigenSolver.h.


  126       : m_eivec(size, size),
  127         m_alphas(size),
  128         m_betas(size),
  129         m_isInitialized(false),
  130         m_eigenvectorsOk(false),
  131         m_realQZ(size),
  132         m_matS(size, size),
  133         m_tmp(size)
  134     {}












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 148 of file GeneralizedEigenSolver.h.


  149       : m_eivec(A.rows(), A.cols()),
  150         m_alphas(A.cols()),
  151         m_betas(A.cols()),
  152         m_isInitialized(false),
  153         m_eigenvectorsOk(false),
  154         m_realQZ(A.cols()),
  155         m_matS(A.rows(), A.cols()),
  156         m_tmp(A.cols())
  157     {
  158       compute(A, B, computeEigenvectors);
  159     }
Eigen::GeneralizedEigenSolver::compute
GeneralizedEigenSolver & compute(const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
Computes generalized eigendecomposition of given matrix. 
Definition: GeneralizedEigenSolver.h:291








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 209 of file GeneralizedEigenSolver.h.


  210     {
  211       eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
  212       return m_alphas;
  213     }












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 220 of file GeneralizedEigenSolver.h.


  221     {
  222       eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
  223       return m_betas;
  224     }












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 291 of file GeneralizedEigenSolver.h.


  292 {
  293   using std::sqrt;
  294   using std::abs;
  295   eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
  296 
  297   // Reduce to generalized real Schur form:
  298   // A = Q S Z and B = Q T Z
  299   m_realQZ.compute(A, B, computeEigenvectors);
  300 
  301   if (m_realQZ.info() == Success)
  302   {
  303     m_matS = m_realQZ.matrixS();
  304     if (computeEigenvectors)
  305       m_eivec = m_realQZ.matrixZ().transpose();
  306   
  307     // Compute eigenvalues from matS
  308     m_alphas.resize(A.cols());
  309     m_betas.resize(A.cols());
  310     Index i = 0;
  311     while (i < A.cols())
  312     {
  313       if (i == A.cols() - 1 || m_matS.coeff(i+1, i) == Scalar(0))
  314       {
  315         m_alphas.coeffRef(i) = m_matS.coeff(i, i);
  316         m_betas.coeffRef(i)  = m_realQZ.matrixT().coeff(i,i);
  317         ++i;
  318       }
  319       else
  320       {
  321         Scalar p = Scalar(0.5) * (m_matS.coeff(i, i) - m_matS.coeff(i+1, i+1));
  322         Scalar z = sqrt(abs(p * p + m_matS.coeff(i+1, i) * m_matS.coeff(i, i+1)));
  323         m_alphas.coeffRef(i)   = ComplexScalar(m_matS.coeff(i+1, i+1) + p, z);
  324         m_alphas.coeffRef(i+1) = ComplexScalar(m_matS.coeff(i+1, i+1) + p, -z);
  325 
  326         m_betas.coeffRef(i)   = m_realQZ.matrixT().coeff(i,i);
  327         m_betas.coeffRef(i+1) = m_realQZ.matrixT().coeff(i,i);
  328         i += 2;
  329       }
  330     }
  331   }
  332 
  333   m_isInitialized = true;
  334   m_eigenvectorsOk = false;//computeEigenvectors;
  335 
  336   return *this;
  337 }
Eigen::RealQZ::info
ComputationInfo info() const 
Reports whether previous computation was successful. 
Definition: RealQZ.h:166


Eigen::GeneralizedEigenSolver::ComplexScalar
std::complex< RealScalar > ComplexScalar
Complex scalar type for MatrixType. 
Definition: GeneralizedEigenSolver.h:83


Eigen::GeneralizedEigenSolver::Scalar
MatrixType::Scalar Scalar
Scalar type for matrices of type MatrixType. 
Definition: GeneralizedEigenSolver.h:73


Eigen::RealQZ::matrixT
const MatrixType & matrixT() const 
Returns matrix S in the QZ decomposition. 
Definition: RealQZ.h:148


Eigen::RealQZ::matrixS
const MatrixType & matrixS() const 
Returns matrix S in the QZ decomposition. 
Definition: RealQZ.h:139


Eigen::RealQZ::matrixZ
const MatrixType & matrixZ() const 
Returns matrix Z in the QZ decomposition. 
Definition: RealQZ.h:129


Eigen::PlainObjectBase::resize
EIGEN_STRONG_INLINE void resize(Index nbRows, Index nbCols)
Definition: PlainObjectBase.h:232


Eigen::Success
Definition: Constants.h:376


Eigen::RealQZ::compute
RealQZ & compute(const MatrixType &A, const MatrixType &B, bool computeQZ=true)
Computes QZ decomposition of given matrix. 
Definition: RealQZ.h:557














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 198 of file GeneralizedEigenSolver.h.


  199     {
  200       eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
  201       return EigenvalueType(m_alphas,m_betas);
  202     }
Eigen::GeneralizedEigenSolver::EigenvalueType
CwiseBinaryOp< internal::scalar_quotient_op< ComplexScalar, Scalar >, ComplexVectorType, VectorType > EigenvalueType
Expression type for the eigenvalues as returned by eigenvalues(). 
Definition: GeneralizedEigenSolver.h:101














template<typename _MatrixType > 



  

  
      

        

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

      

  		
  
inline  
  






Sets the maximal number of iterations allowed. 



Definition at line 259 of file GeneralizedEigenSolver.h.


  260     {
  261       m_realQZ.setMaxIterations(maxIters);
  262       return *this;
  263     }
Eigen::RealQZ::setMaxIterations
RealQZ & setMaxIterations(Index maxIters)
Definition: RealQZ.h:183








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





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