Public Types | Public Member Functions | Protected Types | Protected Attributes | Private Member Functions
EigenSolver< _MatrixType > Class Template Reference

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

#include <EigenSolver.h>

List of all members.

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

Public Member Functions

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

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 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 78 of file EigenSolver.h.

Member Typedef Documentation

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

Definition at line 309 of file EigenSolver.h.

template<typename _MatrixType>
typedef std::complex<RealScalar> 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 104 of file EigenSolver.h.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> 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 111 of file EigenSolver.h.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> 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 118 of file EigenSolver.h.

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

Definition at line 96 of file EigenSolver.h.

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

Synonym for the template parameter _MatrixType.

Definition at line 83 of file EigenSolver.h.

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

Definition at line 95 of file EigenSolver.h.

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

Scalar type for matrices of type MatrixType.

Definition at line 94 of file EigenSolver.h.

Member Enumeration Documentation

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

Definition at line 85 of file EigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
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 127 of file EigenSolver.h.

template<typename _MatrixType>
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 135 of file EigenSolver.h.

template<typename _MatrixType>
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: 

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;

EigenSolver<MatrixXd> es(A);
cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;

complex<double> lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcd v = es.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A.cast<complex<double> >() * v << endl << endl;

MatrixXcd D = es.eigenvalues().asDiagonal();
MatrixXcd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output: 

See also:
compute() 



Definition at line 160 of file EigenSolver.h.







Member Function Documentation








template<typename MatrixType > 

      

        

          EigenSolver< MatrixType > & 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: 


EigenSolver<MatrixXf> es;
MatrixXf A = MatrixXf::Random(4,4);
es.compute(A, /* computeEigenvectors = */ false);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4), false); // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

 Output: 


 

Definition at line 365 of file EigenSolver.h.













template<typename MatrixType > 

      

        

          void EigenSolver< MatrixType >::doComputeEigenvectors 
          (
          )
           [private]
        

      







Definition at line 429 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          const EigenvalueType& 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: 


MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones, false);
cout << "The eigenvalues of the 3x3 matrix of ones are:" 
     << endl << es.eigenvalues() << endl;

 Output: 


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



Definition at line 257 of file EigenSolver.h.













template<typename MatrixType > 

      

        

          EigenSolver< MatrixType >::EigenvectorsType 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: 


MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:" 
     << endl << es.eigenvectors().col(1) << endl;

 Output: 


See also:
eigenvalues(), pseudoEigenvectors() 



Definition at line 334 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          ComputationInfo EigenSolver< _MatrixType >::info 
          (
          )
           const [inline]
        

      







Definition at line 292 of file EigenSolver.h.













template<typename MatrixType > 

      

        

          MatrixType 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 314 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          const MatrixType& 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: 


MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;

EigenSolver<MatrixXd> es(A);
MatrixXd D = es.pseudoEigenvalueMatrix();
MatrixXd V = es.pseudoEigenvectors();
cout << "The pseudo-eigenvalue matrix D is:" << endl << D << endl;
cout << "The pseudo-eigenvector matrix V is:" << endl << V << endl;
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

 Output: 


See also:
pseudoEigenvalueMatrix(), eigenvectors() 



Definition at line 212 of file EigenSolver.h.







Member Data Documentation








template<typename _MatrixType> 

      

        

          bool EigenSolver< _MatrixType >::m_eigenvectorsOk [protected]
        

      







Definition at line 305 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          EigenvalueType EigenSolver< _MatrixType >::m_eivalues [protected]
        

      







Definition at line 303 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          MatrixType EigenSolver< _MatrixType >::m_eivec [protected]
        

      







Definition at line 302 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          bool EigenSolver< _MatrixType >::m_isInitialized [protected]
        

      







Definition at line 304 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          MatrixType EigenSolver< _MatrixType >::m_matT [protected]
        

      







Definition at line 307 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          RealSchur<MatrixType> EigenSolver< _MatrixType >::m_realSchur [protected]
        

      







Definition at line 306 of file EigenSolver.h.













template<typename _MatrixType> 

      

        

          ColumnVectorType EigenSolver< _MatrixType >::m_tmp [protected]
        

      







Definition at line 310 of file EigenSolver.h.







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







libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:15