Public Types | Public Member Functions | Static Public Attributes | Private Types | Private Member Functions | Private Attributes
RealSchur< _MatrixType > Class Template Reference

Performs a real Schur decomposition of a square matrix. More...

#include <RealSchur.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

RealSchur & compute (const MatrixType &matrix, bool computeU=true)
 Computes Schur decomposition of given matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.
const MatrixType & matrixT () const
 Returns the quasi-triangular matrix in the Schur decomposition.
const MatrixType & matrixU () const
 Returns the orthogonal matrix in the Schur decomposition.
 RealSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
 Default constructor.
 RealSchur (const MatrixType &matrix, bool computeU=true)
 Constructor; computes real Schur decomposition of given matrix.

Static Public Attributes

static const int m_maxIterations = 40
 Maximum number of iterations.

Private Types

typedef Matrix< Scalar, 3, 1 > Vector3s

Private Member Functions

Scalar computeNormOfT ()
void computeShift (Index iu, Index iter, Scalar &exshift, Vector3s &shiftInfo)
Index findSmallSubdiagEntry (Index iu, Scalar norm)
void initFrancisQRStep (Index il, Index iu, const Vector3s &shiftInfo, Index &im, Vector3s &firstHouseholderVector)
void performFrancisQRStep (Index il, Index im, Index iu, bool computeU, const Vector3s &firstHouseholderVector, Scalar *workspace)
void splitOffTwoRows (Index iu, bool computeU, Scalar exshift)

Private Attributes

HessenbergDecomposition
< MatrixType > 		m_hess
ComputationInfo m_info
bool m_isInitialized
MatrixType m_matT
MatrixType m_matU
bool m_matUisUptodate
ColumnVectorType m_workspaceVector

Detailed Description

template<typename _MatrixType>
class RealSchur< _MatrixType >

Performs a real Schur decomposition of a square matrix.

Template Parameters:
_MatrixTypethe type of the matrix of which we are computing the real Schur decomposition; this is expected to be an instantiation of the Matrix class template.

Given a real square matrix A, this class computes the real Schur decomposition: $ A = U T U^T $ where U is a real orthogonal matrix and T is a real 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 with complex eigenvalues. The eigenvalues of the blocks on the diagonal of T are the same as the eigenvalues of the matrix A, and thus the real Schur decomposition is used in EigenSolver to compute the eigendecomposition of a matrix.

Call the function compute() to compute the real Schur decomposition of a given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool) constructor which computes the real Schur decomposition at construction time. Once the decomposition is computed, you can use the matrixU() and matrixT() functions to retrieve the matrices U and T in the decomposition.

The documentation of RealSchur(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:
class ComplexSchur, class EigenSolver, class ComplexEigenSolver

Definition at line 68 of file RealSchur.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> RealSchur< _MatrixType >::ColumnVectorType

Definition at line 84 of file RealSchur.h.

template<typename _MatrixType>
typedef std::complex<typename NumTraits<Scalar>::Real> RealSchur< _MatrixType >::ComplexScalar

Definition at line 80 of file RealSchur.h.

template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> RealSchur< _MatrixType >::EigenvalueType

Definition at line 83 of file RealSchur.h.

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

Definition at line 81 of file RealSchur.h.

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

Definition at line 71 of file RealSchur.h.

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

Definition at line 79 of file RealSchur.h.

template<typename _MatrixType>
typedef Matrix<Scalar,3,1> RealSchur< _MatrixType >::Vector3s [private]

Definition at line 206 of file RealSchur.h.

Member Enumeration Documentation

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

Definition at line 72 of file RealSchur.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
RealSchur< _MatrixType >::RealSchur ( Index  size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) [inline]

Default constructor.

Parameters:
[in]sizePositive integer, size of the matrix whose Schur 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 97 of file RealSchur.h.

template<typename _MatrixType>
RealSchur< _MatrixType >::RealSchur ( const MatrixType &  matrix,
bool  computeU = true 
) [inline]

Constructor; computes real Schur decomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose Schur decomposition is to be computed.
[in]computeUIf true, both T and U are computed; if false, only T is computed.

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

Example: 

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

RealSchur<MatrixXd> schur(A);
cout << "The orthogonal matrix U is:" << endl << schur.matrixU() << endl;
cout << "The quasi-triangular matrix T is:" << endl << schur.matrixT() << endl << endl;

MatrixXd U = schur.matrixU();
MatrixXd T = schur.matrixT();
cout << "U * T * U^T = " << endl << U * T * U.transpose() << endl;

Output: 

 

Definition at line 116 of file RealSchur.h.







Member Function Documentation








template<typename _MatrixType> 

      

        

          RealSchur& RealSchur< _MatrixType >::compute 
          (
          const MatrixType & 
          matrix, 
        

        

          
          
          bool 
          computeU = true 
        

        

          
          )
          
        

      







Computes Schur decomposition of given matrix. 


Parameters:

  

    
[in]matrixSquare matrix whose Schur decomposition is to be computed. 

    
[in]computeUIf true, both T and U are computed; if false, only T is computed. 

  

  




Returns:
Reference to *this 


The Schur decomposition is computed by first reducing the matrix to Hessenberg form using the class HessenbergDecomposition. The Hessenberg matrix is then reduced to triangular form by performing Francis QR iterations with implicit double shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken to be $25n^3$ flops if computeU is true and $10n^3$ flops if computeU is false.


Example: 


MatrixXf A = MatrixXf::Random(4,4);
RealSchur<MatrixXf> schur(4);
schur.compute(A, /* computeU = */ false);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse(), /* computeU = */ false);
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;

 Output: 


 











template<typename _MatrixType> 

      

        

          Scalar RealSchur< _MatrixType >::computeNormOfT 
          (
          )
           [private]
        

      

















template<typename _MatrixType> 

      

        

          void RealSchur< _MatrixType >::computeShift 
          (
          Index 
          iu, 
        

        

          
          
          Index 
          iter, 
        

        

          
          
          Scalar & 
          exshift, 
        

        

          
          
          Vector3s & 
          shiftInfo 
        

        

          
          )
           [private]
        

      

















template<typename _MatrixType> 

      

        

          Index RealSchur< _MatrixType >::findSmallSubdiagEntry 
          (
          Index 
          iu, 
        

        

          
          
          Scalar 
          norm 
        

        

          
          )
           [private]
        

      

















template<typename _MatrixType> 

      

        

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

      







Reports whether previous computation was successful. 


Returns:
Success if computation was succesful, NoConvergence otherwise. 



Definition at line 184 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          void RealSchur< _MatrixType >::initFrancisQRStep 
          (
          Index 
          il, 
        

        

          
          
          Index 
          iu, 
        

        

          
          
          const Vector3s & 
          shiftInfo, 
        

        

          
          
          Index & 
          im, 
        

        

          
          
          Vector3s & 
          firstHouseholderVector 
        

        

          
          )
           [private]
        

      

















template<typename _MatrixType> 

      

        

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

      







Returns the quasi-triangular matrix in the Schur decomposition. 


Returns:
A const reference to the matrix T.


Precondition:
Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix.


See also:
RealSchur(const MatrixType&, bool) for an example 



Definition at line 155 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          const MatrixType& RealSchur< _MatrixType >::matrixU 
          (
          )
           const [inline]
        

      







Returns the orthogonal matrix in the Schur decomposition. 


Returns:
A const reference to the matrix U.


Precondition:
Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix, and computeU was set to true (the default value).


See also:
RealSchur(const MatrixType&, bool) for an example 



Definition at line 138 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          void RealSchur< _MatrixType >::performFrancisQRStep 
          (
          Index 
          il, 
        

        

          
          
          Index 
          im, 
        

        

          
          
          Index 
          iu, 
        

        

          
          
          bool 
          computeU, 
        

        

          
          
          const Vector3s & 
          firstHouseholderVector, 
        

        

          
          
          Scalar * 
          workspace 
        

        

          
          )
           [private]
        

      

















template<typename _MatrixType> 

      

        

          void RealSchur< _MatrixType >::splitOffTwoRows 
          (
          Index 
          iu, 
        

        

          
          
          bool 
          computeU, 
        

        

          
          
          Scalar 
          exshift 
        

        

          
          )
           [private]
        

      











Member Data Documentation








template<typename _MatrixType> 

      

        

          HessenbergDecomposition<MatrixType> RealSchur< _MatrixType >::m_hess [private]
        

      







Definition at line 201 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          ComputationInfo RealSchur< _MatrixType >::m_info [private]
        

      







Definition at line 202 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          bool RealSchur< _MatrixType >::m_isInitialized [private]
        

      







Definition at line 203 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          MatrixType RealSchur< _MatrixType >::m_matT [private]
        

      







Definition at line 198 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          MatrixType RealSchur< _MatrixType >::m_matU [private]
        

      







Definition at line 199 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          bool RealSchur< _MatrixType >::m_matUisUptodate [private]
        

      







Definition at line 204 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          const int RealSchur< _MatrixType >::m_maxIterations = 40 [static]
        

      







Maximum number of iterations. 


Maximum number of iterations allowed for an eigenvalue to converge. 



Definition at line 194 of file RealSchur.h.













template<typename _MatrixType> 

      

        

          ColumnVectorType RealSchur< _MatrixType >::m_workspaceVector [private]
        

      







Definition at line 200 of file RealSchur.h.







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







libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:24