Eigen::RealQZ

Eigen::RealQZ< _MatrixType > Class Template Reference

Performs a real QZ decomposition of a pair of square matrices. More...

#include <RealQZ.h>

List of all members.

Public Types

typedef _MatrixType MatrixType
typedef MatrixType::Scalar Scalar
typedef std::complex< typename
NumTraits< Scalar >::Real > 		ComplexScalar
typedef MatrixType::Index Index
typedef Matrix< ComplexScalar,
ColsAtCompileTime, 1, Options
&~RowMajor, MaxColsAtCompileTime, 1 > 		EigenvalueType
typedef Matrix< Scalar, ColsAtCompileTime,
1, Options &~RowMajor, MaxColsAtCompileTime, 1 > 		ColumnVectorType
 RowsAtCompileTime = MatrixType::RowsAtCompileTime
 ColsAtCompileTime = MatrixType::ColsAtCompileTime
 Options = MatrixType::Options
 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}

Public Member Functions

 RealQZ (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
 Default constructor.
 RealQZ (const MatrixType &A, const MatrixType &B, bool computeQZ=true)
 Constructor; computes real QZ decomposition of given matrices.
const MatrixType & matrixQ () const
 Returns matrix Q in the QZ decomposition.
const MatrixType & matrixZ () const
 Returns matrix Z in the QZ decomposition.
const MatrixType & matrixS () const
 Returns matrix S in the QZ decomposition.
const MatrixType & matrixT () const
 Returns matrix S in the QZ decomposition.
RealQZ & compute (const MatrixType &A, const MatrixType &B, bool computeQZ=true)
 Computes QZ decomposition of given matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.
Index iterations () const
 Returns number of performed QR-like iterations.
RealQZ & setMaxIterations (Index maxIters)

Detailed Description

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

Performs a real QZ decomposition of a pair of square matrices.

_MatrixType the type of the matrix of which we are computing the real QZ decomposition; this is expected to be an instantiation of the Matrix class template.

Given a real square matrices A and B, this class computes the real QZ decomposition: $ A = Q S Z $, $ B = Q T Z $ where Q and Z are real orthogonal matrixes, T is upper-triangular matrix, and S is upper quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose, $ U^{-1} = U^T $. A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks where further reduction is impossible due to complex eigenvalues.

The eigenvalues of the pencil $ A - z B $ can be obtained from 1x1 and 2x2 blocks on the diagonals of S and T.

Call the function compute() to compute the real QZ decomposition of a given pair of matrices. Alternatively, you can use the RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ) constructor which computes the real QZ decomposition at construction time. Once the decomposition is computed, you can use the matrixS(), matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices S, T, Q and Z in the decomposition. If computeQZ==false, some time is saved by not computing matrices Q and Z.

Example:

 Output:


Note:
The implementation is based on the algorithm in "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.


See also:
class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver 







Definition at line 57 of file RealQZ.h.
Member Typedef Documentation








template<typename _MatrixType> 

      

        

          typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::RealQZ< _MatrixType >::ColumnVectorType          
        

      











Definition at line 73 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          typedef std::complex<typename NumTraits<Scalar>::Real> Eigen::RealQZ< _MatrixType >::ComplexScalar          
        

      











Definition at line 69 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::RealQZ< _MatrixType >::EigenvalueType          
        

      











Definition at line 72 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          typedef MatrixType::Index Eigen::RealQZ< _MatrixType >::Index          
        

      











Definition at line 70 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          typedef _MatrixType Eigen::RealQZ< _MatrixType >::MatrixType          
        

      











Definition at line 60 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          typedef MatrixType::Scalar Eigen::RealQZ< _MatrixType >::Scalar          
        

      











Definition at line 68 of file RealQZ.h.





Member Enumeration Documentation








template<typename _MatrixType> 

      

        

          anonymous enum          
        

      









Enumerator: 




RowsAtCompileTime 



ColsAtCompileTime 



Options 



MaxRowsAtCompileTime 



MaxColsAtCompileTime 









Definition at line 61 of file RealQZ.h.





Constructor & Destructor Documentation








template<typename _MatrixType> 

      

        

          Eigen::RealQZ< _MatrixType >::RealQZ           
          (
          Index 
           size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime          
           ) 
           [inline]
        

      








Default constructor. 



Parameters:

  

    
[in] size Positive integer, size of the matrix whose QZ decomposition will be computed.

  



The default constructor is useful in cases in which 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 86 of file RealQZ.h.











template<typename _MatrixType> 

      

        

          Eigen::RealQZ< _MatrixType >::RealQZ           
          (
          const MatrixType & 
           A, 
        

        

          
          
          const MatrixType & 
           B, 
        

        

          
          
          bool 
           computeQZ = true 
        

        

          
          )
           [inline]
        

      








Constructor; computes real QZ decomposition of given matrices. 



Parameters:

  

    
[in] A Matrix A. 

    
[in] B Matrix B. 

    
[in] computeQZ If false, A and Z are not computed.

  



This constructor calls compute() to compute the QZ decomposition. 


Definition at line 104 of file RealQZ.h.


References Eigen::RealQZ< _MatrixType >::compute().





Member Function Documentation








template<typename MatrixType> 

      

        

          RealQZ< MatrixType > & Eigen::RealQZ< MatrixType >::compute           
          (
          const MatrixType & 
           A, 
        

        

          
          
          const MatrixType & 
           B, 
        

        

          
          
          bool 
           computeQZ = true 
        

        

          
          )
          
        

      








Computes QZ decomposition of given matrix. 



Parameters:

  

    
[in] A Matrix A. 

    
[in] B Matrix B. 

    
[in] computeQZ If false, A and Z are not computed. 

  




Returns:
Reference to *this 




Definition at line 557 of file RealQZ.h.


References eigen_assert, Eigen::NoConvergence, and Eigen::Success.


Referenced by Eigen::RealQZ< _MatrixType >::RealQZ().











template<typename _MatrixType> 

      

        

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

      








Reports whether previous computation was successful. 



Returns:
Success if computation was succesful, NoConvergence otherwise. 




Definition at line 166 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          Index Eigen::RealQZ< _MatrixType >::iterations           
          (
                    
           ) 
           const [inline]
        

      








Returns number of performed QR-like iterations. 





Definition at line 174 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          const MatrixType& Eigen::RealQZ< _MatrixType >::matrixQ           
          (
          void 
                    
           ) 
           const [inline]
        

      








Returns matrix Q in the QZ decomposition. 



Returns:
A const reference to the matrix Q. 




Definition at line 119 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          const MatrixType& Eigen::RealQZ< _MatrixType >::matrixS           
          (
                    
           ) 
           const [inline]
        

      








Returns matrix S in the QZ decomposition. 



Returns:
A const reference to the matrix S. 




Definition at line 139 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          const MatrixType& Eigen::RealQZ< _MatrixType >::matrixT           
          (
                    
           ) 
           const [inline]
        

      








Returns matrix S in the QZ decomposition. 



Returns:
A const reference to the matrix S. 




Definition at line 148 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          const MatrixType& Eigen::RealQZ< _MatrixType >::matrixZ           
          (
                    
           ) 
           const [inline]
        

      








Returns matrix Z in the QZ decomposition. 



Returns:
A const reference to the matrix Z. 




Definition at line 129 of file RealQZ.h.


References eigen_assert.











template<typename _MatrixType> 

      

        

          RealQZ& Eigen::RealQZ< _MatrixType >::setMaxIterations           
          (
          Index 
           maxIters          
           ) 
           [inline]
        

      








Sets the maximal number of iterations allowed to converge to one eigenvalue or decouple the problem. 


Definition at line 183 of file RealQZ.h.





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