Public Types | Public Member Functions | Private Types
GeneralizedSelfAdjointEigenSolver< _MatrixType > Class Template Reference

Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem. More...

#include <GeneralizedSelfAdjointEigenSolver.h>

Inheritance diagram for GeneralizedSelfAdjointEigenSolver< _MatrixType >:
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List of all members.

Public Types

typedef Base::Index Index
typedef _MatrixType MatrixType

Public Member Functions

GeneralizedSelfAdjointEigenSolver & compute (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors|Ax_lBx)
 Computes generalized eigendecomposition of given matrix pencil.
 GeneralizedSelfAdjointEigenSolver ()
 Default constructor for fixed-size matrices.
 GeneralizedSelfAdjointEigenSolver (Index size)
 Constructor, pre-allocates memory for dynamic-size matrices.
 GeneralizedSelfAdjointEigenSolver (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors|Ax_lBx)
 Constructor; computes generalized eigendecomposition of given matrix pencil.

Private Types

typedef SelfAdjointEigenSolver
< _MatrixType > 		Base

Detailed Description

template<typename _MatrixType>
class GeneralizedSelfAdjointEigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem.

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.

This class solves the generalized eigenvalue problem $ Av = \lambda Bv $. In this case, the matrix $ A $ should be selfadjoint and the matrix $ B $ should be positive definite.

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 GeneralizedSelfAdjointEigenSolver(const MatrixType&, 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 GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int) contains an example of the typical use of this class.

See also:
class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver

Definition at line 62 of file GeneralizedSelfAdjointEigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef SelfAdjointEigenSolver<_MatrixType> GeneralizedSelfAdjointEigenSolver< _MatrixType >::Base [private]

Definition at line 64 of file GeneralizedSelfAdjointEigenSolver.h.

template<typename _MatrixType>
typedef Base::Index GeneralizedSelfAdjointEigenSolver< _MatrixType >::Index

Reimplemented from SelfAdjointEigenSolver< _MatrixType >.

Definition at line 67 of file GeneralizedSelfAdjointEigenSolver.h.

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

Reimplemented from SelfAdjointEigenSolver< _MatrixType >.

Definition at line 68 of file GeneralizedSelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( ) [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 GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.

Definition at line 77 of file GeneralizedSelfAdjointEigenSolver.h.

template<typename _MatrixType>
GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( 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 91 of file GeneralizedSelfAdjointEigenSolver.h.

template<typename _MatrixType>
GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( const MatrixType &  matA,
const MatrixType &  matB,
int  options = ComputeEigenvectors|Ax_lBx 
) [inline]

Constructor; computes generalized eigendecomposition of given matrix pencil.

Parameters:
[in]matASelfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]matBPositive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]optionsA or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} | {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors|Ax_lBx.

This constructor calls compute(const MatrixType&, const MatrixType&, int) to compute the eigenvalues and (if requested) the eigenvectors of the generalized eigenproblem $ Ax = \lambda B x $ with matA the selfadjoint matrix $ A $ and matB the positive definite matrix $ B $. Each eigenvector $ x $ satisfies the property $ x^* B x = 1 $. The eigenvectors are computed if options contains ComputeEigenvectors.

In addition, the two following variants can be solved via options:

  • ABx_lx: $ ABx = \lambda x $
  • BAx_lx: $ BAx = \lambda x $

Example: 

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric matrix, A:" << endl << A << endl;
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();
cout << "and a random postive-definite matrix, B:" << endl << B << endl << endl;

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

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

Output: 

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



Definition at line 121 of file GeneralizedSelfAdjointEigenSolver.h.







Member Function Documentation








template<typename MatrixType > 

      

        

          GeneralizedSelfAdjointEigenSolver< MatrixType > & GeneralizedSelfAdjointEigenSolver< MatrixType >::compute 
          (
          const MatrixType & 
          matA, 
        

        

          
          
          const MatrixType & 
          matB, 
        

        

          
          
          int 
          options = ComputeEigenvectors|Ax_lBx 
        

        

          
          )
          
        

      







Computes generalized eigendecomposition of given matrix pencil. 


Parameters:

  

    
[in]matASelfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced. 

    
[in]matBPositive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced. 

    
[in]optionsA or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} | {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors|Ax_lBx.

  

  




Returns:
Reference to *this 


Accoring to options, this function computes eigenvalues and (if requested) the eigenvectors of one of the following three generalized eigenproblems:


    

  • Ax_lBx: $ Ax = \lambda B x $
    • 

  • ABx_lx: $ ABx = \lambda x $
    • 

  • BAx_lx: $ BAx = \lambda x $ with matA the selfadjoint matrix $ A $ and matB the positive definite matrix $ B $. In addition, each eigenvector $ x $ satisfies the property $ x^* B x = 1 $.
    • 



The eigenvalues() function can be used to retrieve the eigenvalues. If options contains ComputeEigenvectors, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().


The implementation uses LLT to compute the Cholesky decomposition $ B = LL^* $ and computes the classical eigendecomposition of the selfadjoint matrix $ L^{-1} A (L^*)^{-1} $ if options contains Ax_lBx and of $ L^{*} A L $ otherwise. This solves the generalized eigenproblem, because any solution of the generalized eigenproblem $ Ax = \lambda B x $ corresponds to a solution $ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) $ of the eigenproblem for $ L^{-1} A (L^*)^{-1} $. Similar statements can be made for the two other variants.


Example: 


MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X * X.transpose();
X = MatrixXd::Random(5,5);
MatrixXd B = X * X.transpose();

GeneralizedSelfAdjointEigenSolver<MatrixXd> es(A,B,EigenvaluesOnly);
cout << "The eigenvalues of the pencil (A,B) are:" << endl << es.eigenvalues() << endl;
es.compute(B,A,false);
cout << "The eigenvalues of the pencil (B,A) are:" << endl << es.eigenvalues() << endl;

 Output: 


See also:
GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int) 



Definition at line 178 of file GeneralizedSelfAdjointEigenSolver.h.







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







re_vision

Author(s): Dorian Galvez-Lopez

autogenerated on Sun Jan 5 2014 11:34:07