Public Types | Public Member Functions | Static Public Attributes | Protected Attributes | Friends | List of all members
Eigen::SelfAdjointEigenSolver< _MatrixType > Class Template Reference

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

#include <SelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::SelfAdjointEigenSolver< _MatrixType >:
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Public Types

enum  { Size = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
 
typedef MatrixType::Index Index
 
typedef _MatrixType MatrixType
 
typedef NumTraits< Scalar >::Real RealScalar
 Real scalar type for _MatrixType. More...
 
typedef internal::plain_col_type< MatrixType, RealScalar >::type RealVectorType
 Type for vector of eigenvalues as returned by eigenvalues(). More...
 
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type _MatrixType. More...
 
typedef Tridiagonalization< MatrixType > TridiagonalizationType
 

Public Member Functions

SelfAdjointEigenSolver & compute (const MatrixType &matrix, int options=ComputeEigenvectors)
 Computes eigendecomposition of given matrix. More...
 
SelfAdjointEigenSolver & computeDirect (const MatrixType &matrix, int options=ComputeEigenvectors)
 Computes eigendecomposition of given matrix using a direct algorithm. More...
 
const RealVectorType & eigenvalues () const
 Returns the eigenvalues of given matrix. More...
 
const MatrixType & eigenvectors () const
 Returns the eigenvectors of given matrix. More...
 
ComputationInfo info () const
 Reports whether previous computation was successful. More...
 
MatrixType operatorInverseSqrt () const
 Computes the inverse square root of the matrix. More...
 
MatrixType operatorSqrt () const
 Computes the positive-definite square root of the matrix. More...
 
 SelfAdjointEigenSolver ()
 Default constructor for fixed-size matrices. More...
 
 SelfAdjointEigenSolver (Index size)
 Constructor, pre-allocates memory for dynamic-size matrices. More...
 
 SelfAdjointEigenSolver (const MatrixType &matrix, int options=ComputeEigenvectors)
 Constructor; computes eigendecomposition of given matrix. More...
 

Static Public Attributes

static const int m_maxIterations = 30
 Maximum number of iterations. More...
 

Protected Attributes

bool m_eigenvectorsOk
 
RealVectorType m_eivalues
 
MatrixType m_eivec
 
ComputationInfo m_info
 
bool m_isInitialized
 
TridiagonalizationType::SubDiagonalType m_subdiag
 

Friends

struct internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >
 

Detailed Description

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

Computes eigenvalues and eigenvectors of selfadjoint 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.

A matrix $ A $ is selfadjoint if it equals its adjoint. For real matrices, this means that the matrix is symmetric: it equals its transpose. This class computes the eigenvalues and eigenvectors of a selfadjoint matrix. These are the scalars $ \lambda $ and vectors $ v $ such that $ Av = \lambda v $. The eigenvalues of a selfadjoint matrix are always real. 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 D V^{-1} $ (for selfadjoint matrices, the matrix $ V $ is always invertible). This is called the eigendecomposition.

The algorithm exploits the fact that the matrix is selfadjoint, making it faster and more accurate than the general purpose eigenvalue algorithms implemented in EigenSolver and ComplexEigenSolver.

Only the lower triangular part of the input matrix is referenced.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the SelfAdjointEigenSolver(const MatrixType&, int) 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 documentation for SelfAdjointEigenSolver(const MatrixType&, int) contains an example of the typical use of this class.

To solve the generalized eigenvalue problem $ Av = \lambda Bv $ and the likes, see the class GeneralizedSelfAdjointEigenSolver.

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

Definition at line 68 of file SelfAdjointEigenSolver.h.

Member Typedef Documentation

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

Definition at line 82 of file SelfAdjointEigenSolver.h.

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

Definition at line 72 of file SelfAdjointEigenSolver.h.

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

Real scalar type for _MatrixType.

This is just Scalar if Scalar is real (e.g., float or double), and the type of the real part of Scalar if Scalar is complex.

Definition at line 90 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>
typedef internal::plain_col_type<MatrixType, RealScalar>::type Eigen::SelfAdjointEigenSolver< _MatrixType >::RealVectorType

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

This is a column vector with entries of type RealScalar. The length of the vector is the size of _MatrixType.

Definition at line 99 of file SelfAdjointEigenSolver.h.

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

Scalar type for matrices of type _MatrixType.

Definition at line 81 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>
typedef Tridiagonalization<MatrixType> Eigen::SelfAdjointEigenSolver< _MatrixType >::TridiagonalizationType

Definition at line 100 of file SelfAdjointEigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator
Size 
ColsAtCompileTime 
Options 
MaxColsAtCompileTime 

Definition at line 73 of file SelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

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

Default constructor for fixed-size matrices.

The default constructor is useful in cases in which the user intends to perform decompositions via compute(). This constructor can only be used if _MatrixType is a fixed-size matrix; use SelfAdjointEigenSolver(Index) for dynamic-size matrices.

Example:

Output: 

 

Definition at line 112 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  







Constructor, pre-allocates memory for dynamic-size matrices. 


Parameters

  

    
[in]sizePositive integer, size of the matrix whose eigenvalues and eigenvectors will be computed.

  

  




This constructor is useful for dynamic-size matrices, when the user intends to perform decompositions via compute(). The size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.


See also
compute() for an example 



Definition at line 131 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          Eigen::SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver 
          (
          const MatrixType & 
          matrix, 
        

        

          
          
          int 
          options = ComputeEigenvectors 
        

        

          
          )
          
        

      

  		
  
inline  
  







Constructor; computes eigendecomposition of given matrix. 


Parameters

  

    
[in]matrixSelfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced. 

    
[in]optionsCan be ComputeEigenvectors (default) or EigenvaluesOnly.

  

  




This constructor calls compute(const MatrixType&, int) to compute the eigenvalues of the matrix matrix. The eigenvectors are computed if options equals ComputeEigenvectors.


Example: 
 Output: 
See also
compute(const MatrixType&, int) 



Definition at line 153 of file SelfAdjointEigenSolver.h.







Member Function Documentation








template<typename MatrixType > 

      

        

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

        

          
          
          int 
          options = ComputeEigenvectors 
        

        

          
          )
          
        

      





Computes eigendecomposition of given matrix. 


Parameters

  

    
[in]matrixSelfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced. 

    
[in]optionsCan be ComputeEigenvectors (default) or EigenvaluesOnly. 

  

  




Returns
Reference to *this 


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


This implementation uses a symmetric QR algorithm. The matrix is first reduced to tridiagonal form using the Tridiagonalization class. The tridiagonal matrix is then brought to diagonal form with implicit symmetric QR steps with Wilkinson shift. Details can be found in Section 8.3 of Golub & Van Loan, Matrix Computations.


The cost of the computation is about $ 9n^3 $ if the eigenvectors are required and $ 4n^3/3 $ if they are not required.


This method reuses the memory in the SelfAdjointEigenSolver object that was allocated when the object was constructed, if the size of the matrix does not change.


Example: 
 Output: 
See also
SelfAdjointEigenSolver(const MatrixType&, int) 



Definition at line 385 of file SelfAdjointEigenSolver.h.













template<typename MatrixType > 

      

        

          SelfAdjointEigenSolver< MatrixType > & Eigen::SelfAdjointEigenSolver< MatrixType >::computeDirect 
          (
          const MatrixType & 
          matrix, 
        

        

          
          
          int 
          options = ComputeEigenvectors 
        

        

          
          )
          
        

      





Computes eigendecomposition of given matrix using a direct algorithm. 


This is a variant of compute(const MatrixType&, int options) which directly solves the underlying polynomial equation.


Currently only 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).


This method is usually significantly faster than the QR algorithm but it might also be less accurate. It is also worth noting that for 3x3 matrices it involves trigonometric operations which are not necessarily available for all scalar types.


See also
compute(const MatrixType&, int options) 



Definition at line 732 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          const RealVectorType& Eigen::SelfAdjointEigenSolver< _MatrixType >::eigenvalues 
          (
          )
           const
        

      

  		
  
inline  
  







Returns the eigenvalues of given matrix. 


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


Precondition
The eigenvalues have been computed before.


The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are sorted in increasing order.


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



Definition at line 250 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::SelfAdjointEigenSolver< _MatrixType >::eigenvectors 
          (
          )
           const
        

      

  		
  
inline  
  







Returns the eigenvectors of given matrix. 


Returns
A const reference to the matrix whose columns are the eigenvectors.


Precondition
The eigenvectors have been computed before.


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. If this object was used to solve the eigenproblem for the selfadjoint matrix $ A $, then the matrix returned by this function is the matrix $ V $ in the eigendecomposition $ A = V D V^{-1} $.


Example: 
 Output: 
See also
eigenvalues() 



Definition at line 228 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  







Reports whether previous computation was successful. 


Returns
Success if computation was succesful, NoConvergence otherwise. 



Definition at line 310 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorInverseSqrt 
          (
          )
           const
        

      

  		
  
inline  
  







Computes the inverse square root of the matrix. 


Returns
the inverse positive-definite square root of the matrix


Precondition
The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.


This function uses the eigendecomposition $ A = V D V^{-1} $ to compute the inverse square root as $ V D^{-1/2} V^{-1} $. This is cheaper than first computing the square root with operatorSqrt() and then its inverse with MatrixBase::inverse().


Example: 
 Output: 
See also
operatorSqrt(), MatrixBase::inverse(), MatrixFunctions Module 



Definition at line 299 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::SelfAdjointEigenSolver< _MatrixType >::operatorSqrt 
          (
          )
           const
        

      

  		
  
inline  
  







Computes the positive-definite square root of the matrix. 


Returns
the positive-definite square root of the matrix


Precondition
The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.


The square root of a positive-definite matrix $ A $ is the positive-definite matrix whose square equals $ A $. This function uses the eigendecomposition $ A = V D V^{-1} $ to compute the square root as $ A^{1/2} = V D^{1/2} V^{-1} $.


Example: 
 Output: 
See also
operatorInverseSqrt(), MatrixFunctions Module 



Definition at line 274 of file SelfAdjointEigenSolver.h.







Friends And Related Function Documentation








template<typename _MatrixType> 



  

  
      

        

          friend struct internal::direct_selfadjoint_eigenvalues< SelfAdjointEigenSolver, Size, NumTraits< Scalar >::IsComplex >
        

      

  		
  
friend  
  







Definition at line 92 of file SelfAdjointEigenSolver.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          bool Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eigenvectorsOk
        

      

  		
  
protected  
  







Definition at line 359 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          RealVectorType Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eivalues
        

      

  		
  
protected  
  







Definition at line 355 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::SelfAdjointEigenSolver< _MatrixType >::m_eivec
        

      

  		
  
protected  
  







Definition at line 354 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          ComputationInfo Eigen::SelfAdjointEigenSolver< _MatrixType >::m_info
        

      

  		
  
protected  
  







Definition at line 357 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::SelfAdjointEigenSolver< _MatrixType >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 358 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          const int Eigen::SelfAdjointEigenSolver< _MatrixType >::m_maxIterations = 30
        

      

  		
  
static  
  







Maximum number of iterations. 


The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK). 



Definition at line 321 of file SelfAdjointEigenSolver.h.













template<typename _MatrixType> 



  

  
      

        

          TridiagonalizationType::SubDiagonalType Eigen::SelfAdjointEigenSolver< _MatrixType >::m_subdiag
        

      

  		
  
protected  
  







Definition at line 356 of file SelfAdjointEigenSolver.h.







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







acado

Author(s): Milan Vukov, Rien Quirynen

autogenerated on Mon Jun 10 2019 12:35:39