Public Types | Public Member Functions | Private Types | Private Member Functions | Private Attributes
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

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

Public Member Functions

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.
const MatrixType & matrixQ () const
 Returns matrix Q 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.
const MatrixType & matrixZ () const
 Returns matrix Z in the QZ decomposition.
 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.
RealQZ & setMaxIterations (Index maxIters)

Private Types

typedef JacobiRotation< Scalar > JRs
typedef Matrix< Scalar, 2, 2 > Matrix2s
typedef Matrix< Scalar, 2, 1 > Vector2s
typedef Matrix< Scalar, 3, 1 > Vector3s

Private Member Functions

void computeNorms ()
Index findSmallDiagEntry (Index f, Index l)
Index findSmallSubdiagEntry (Index iu)
void hessenbergTriangular ()
void pushDownZero (Index z, Index f, Index l)
void splitOffTwoRows (Index i)
void step (Index f, Index l, Index iter)

Private Attributes

bool m_computeQZ
Index m_global_iter
ComputationInfo m_info
bool m_isInitialized
Index m_maxIters
Scalar m_normOfS
Scalar m_normOfT
MatrixType m_Q
MatrixType m_S
MatrixType m_T
Matrix< Scalar, Dynamic, 1 > m_workspace
MatrixType m_Z

Detailed Description

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

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

Template Parameters:
_MatrixTypethe 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 JacobiRotation<Scalar> Eigen::RealQZ< _MatrixType >::JRs [private]
        

      







Definition at line 203 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          typedef Matrix<Scalar,2,2> Eigen::RealQZ< _MatrixType >::Matrix2s [private]
        

      







Definition at line 202 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.













template<typename _MatrixType> 

      

        

          typedef Matrix<Scalar,2,1> Eigen::RealQZ< _MatrixType >::Vector2s [private]
        

      







Definition at line 201 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          typedef Matrix<Scalar,3,1> Eigen::RealQZ< _MatrixType >::Vector3s [private]
        

      







Definition at line 200 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]sizePositive 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]AMatrix A. 

    
[in]BMatrix B. 

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

  

  




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



Definition at line 104 of file RealQZ.h.







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]AMatrix A. 

    
[in]BMatrix B. 

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

  

  




Returns:
Reference to *this 



Definition at line 557 of file RealQZ.h.













template<typename MatrixType > 

      

        

          void Eigen::RealQZ< MatrixType >::computeNorms 
          (
          )
           [inline, private]
        

      







Definition at line 264 of file RealQZ.h.













template<typename MatrixType > 

      

        

          MatrixType::Index Eigen::RealQZ< MatrixType >::findSmallDiagEntry 
          (
          Index 
          f, 
        

        

          
          
          Index 
          l 
        

        

          
          )
           [inline, private]
        

      







Definition at line 297 of file RealQZ.h.













template<typename MatrixType > 

      

        

          MatrixType::Index Eigen::RealQZ< MatrixType >::findSmallSubdiagEntry 
          (
          Index 
          iu)
           [inline, private]
        

      







Definition at line 279 of file RealQZ.h.













template<typename MatrixType > 

      

        

          void Eigen::RealQZ< MatrixType >::hessenbergTriangular 
          (
          )
           [private]
        

      







Definition at line 217 of file RealQZ.h.













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.













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.













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.













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.













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.













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.













template<typename MatrixType > 

      

        

          void Eigen::RealQZ< MatrixType >::pushDownZero 
          (
          Index 
          z, 
        

        

          
          
          Index 
          f, 
        

        

          
          
          Index 
          l 
        

        

          
          )
           [inline, private]
        

      







Definition at line 361 of file RealQZ.h.













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.













template<typename MatrixType > 

      

        

          void Eigen::RealQZ< MatrixType >::splitOffTwoRows 
          (
          Index 
          i)
           [inline, private]
        

      







Definition at line 311 of file RealQZ.h.













template<typename MatrixType > 

      

        

          void Eigen::RealQZ< MatrixType >::step 
          (
          Index 
          f, 
        

        

          
          
          Index 
          l, 
        

        

          
          
          Index 
          iter 
        

        

          
          )
           [inline, private]
        

      







Definition at line 400 of file RealQZ.h.







Member Data Documentation








template<typename _MatrixType> 

      

        

          bool Eigen::RealQZ< _MatrixType >::m_computeQZ [private]
        

      







Definition at line 196 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          Index Eigen::RealQZ< _MatrixType >::m_global_iter [private]
        

      







Definition at line 198 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          ComputationInfo Eigen::RealQZ< _MatrixType >::m_info [private]
        

      







Definition at line 193 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          bool Eigen::RealQZ< _MatrixType >::m_isInitialized [private]
        

      







Definition at line 195 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          Index Eigen::RealQZ< _MatrixType >::m_maxIters [private]
        

      







Definition at line 194 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          Scalar Eigen::RealQZ< _MatrixType >::m_normOfS [private]
        

      







Definition at line 197 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          Scalar Eigen::RealQZ< _MatrixType >::m_normOfT [private]
        

      







Definition at line 197 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::RealQZ< _MatrixType >::m_Q [private]
        

      







Definition at line 191 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::RealQZ< _MatrixType >::m_S [private]
        

      







Definition at line 191 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::RealQZ< _MatrixType >::m_T [private]
        

      







Definition at line 191 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          Matrix<Scalar,Dynamic,1> Eigen::RealQZ< _MatrixType >::m_workspace [private]
        

      







Definition at line 192 of file RealQZ.h.













template<typename _MatrixType> 

      

        

          MatrixType Eigen::RealQZ< _MatrixType >::m_Z [private]
        

      







Definition at line 191 of file RealQZ.h.







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







acado

Author(s): Milan Vukov, Rien Quirynen

autogenerated on Sat Jun 8 2019 19:40:52