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

GeneralizedEigenSolver< _MatrixType > Class Template Reference
[Eigenvalues module]

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#include <GeneralizedEigenSolver.h>

Public Types

typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.
typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for MatrixType.
typedef Matrix< Scalar,
ColsAtCompileTime, 1, Options
&~RowMajor,
MaxColsAtCompileTime, 1 > 		VectorType
 Type for vector of real scalar values eigenvalues as returned by betas().
typedef Matrix< ComplexScalar,
ColsAtCompileTime, 1, Options
&~RowMajor,
MaxColsAtCompileTime, 1 > 		ComplexVectorType
 Type for vector of complex scalar values eigenvalues as returned by betas().
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().

Public Member Functions

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

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






      

        

          typedef std::complex<RealScalar> 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.











      

        

          typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> 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.











      

        

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

      







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



Definition at line 101 of file GeneralizedEigenSolver.h.











      

        

          typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> 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.











      

        

          typedef _MatrixType MatrixType
        

      







Synonym for the template parameter _MatrixType. 



Definition at line 62 of file GeneralizedEigenSolver.h.











      

        

          typedef MatrixType::Scalar Scalar
        

      







Scalar type for matrices of type MatrixType. 



Definition at line 73 of file GeneralizedEigenSolver.h.











      

        

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






      

        

          GeneralizedEigenSolver 
          (
           )
          
        

      







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.











      

        

          GeneralizedEigenSolver 
          (
          Index 
           size )
          
        

      







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.











      

        

          GeneralizedEigenSolver 
          (
          const MatrixType & 
           A, 
        

        

          
          
          const MatrixType & 
           B, 
        

        

          
          
          bool 
           computeEigenvectors = true 
        

        

          
          )
          
        

      







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.







Member Function Documentation






      

        

          ComplexVectorType alphas 
          (
           )
           const
        

      






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.











      

        

          VectorType betas 
          (
           )
           const
        

      






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.











      

        

          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 298 of file GeneralizedEigenSolver.h.











      

        

          EigenvalueType eigenvalues 
          (
           )
           const
        

      







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.











      

        

          GeneralizedEigenSolver& setMaxIterations 
          (
          Index 
           maxIters )
          
        

      







Sets the maximal number of iterations allowed. 



Definition at line 259 of file GeneralizedEigenSolver.h.









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