Public Types | Public Member Functions | Protected Attributes
Tridiagonalization< _MatrixType > Class Template Reference

Tridiagonal decomposition of a selfadjoint matrix. More...

#include <Tridiagonalization.h>

List of all members.

Public Types

enum  {
  Size = MatrixType::RowsAtCompileTime, SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1), Options = MatrixType::Options, MaxSize = MatrixType::MaxRowsAtCompileTime,
  MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
}
typedef Matrix< Scalar,
SizeMinusOne, 1, Options
&~RowMajor, MaxSizeMinusOne, 1 > 		CoeffVectorType
typedef internal::conditional
< NumTraits< Scalar >
::IsComplex, const typename
Diagonal< const MatrixType >
::RealReturnType, const
Diagonal< const MatrixType >
>::type 		DiagonalReturnType
typedef
internal::plain_col_type
< MatrixType, RealScalar >
::type 		DiagonalType
typedef HouseholderSequence
< MatrixType, CoeffVectorType >
::ConjugateReturnType 		HouseholderSequenceType
 Return type of matrixQ()
typedef MatrixType::Index Index
typedef
internal::TridiagonalizationMatrixTReturnType
< MatrixTypeRealView > 		MatrixTReturnType
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef internal::remove_all
< typename
MatrixType::RealReturnType >
::type 		MatrixTypeRealView
typedef NumTraits< Scalar >::Real RealScalar
typedef MatrixType::Scalar Scalar
typedef internal::conditional
< NumTraits< Scalar >
::IsComplex, const typename
Diagonal< Block< const
MatrixType, SizeMinusOne,
SizeMinusOne >
>::RealReturnType, const
Diagonal< Block< const
MatrixType, SizeMinusOne,
SizeMinusOne > > >::type 		SubDiagonalReturnType
typedef Matrix< RealScalar,
SizeMinusOne, 1, Options
&~RowMajor, MaxSizeMinusOne, 1 > 		SubDiagonalType

Public Member Functions

Tridiagonalization & compute (const MatrixType &matrix)
 Computes tridiagonal decomposition of given matrix.
DiagonalReturnType diagonal () const
 Returns the diagonal of the tridiagonal matrix T in the decomposition.
CoeffVectorType householderCoefficients () const
 Returns the Householder coefficients.
HouseholderSequenceType matrixQ () const
 Returns the unitary matrix Q in the decomposition.
MatrixTReturnType matrixT () const
 Returns an expression of the tridiagonal matrix T in the decomposition.
const MatrixType & packedMatrix () const
 Returns the internal representation of the decomposition.
SubDiagonalReturnType subDiagonal () const
 Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
 Tridiagonalization (Index size=Size==Dynamic?2:Size)
 Default constructor.
 Tridiagonalization (const MatrixType &matrix)
 Constructor; computes tridiagonal decomposition of given matrix.

Protected Attributes

CoeffVectorType m_hCoeffs
bool m_isInitialized
MatrixType m_matrix

Detailed Description

template<typename _MatrixType>
class Tridiagonalization< _MatrixType >

Tridiagonal decomposition of a selfadjoint matrix.

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

This class performs a tridiagonal decomposition of a selfadjoint matrix $ A $ such that: $ A = Q T Q^* $ where $ Q $ is unitary and $ T $ a real symmetric tridiagonal matrix.

A tridiagonal matrix is a matrix which has nonzero elements only on the main diagonal and the first diagonal below and above it. The Hessenberg decomposition of a selfadjoint matrix is in fact a tridiagonal decomposition. This class is used in SelfAdjointEigenSolver to compute the eigenvalues and eigenvectors of a selfadjoint matrix.

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

The documentation of Tridiagonalization(const MatrixType&) contains an example of the typical use of this class.

See also:
class HessenbergDecomposition, class SelfAdjointEigenSolver

Definition at line 74 of file Tridiagonalization.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> Tridiagonalization< _MatrixType >::CoeffVectorType

Definition at line 93 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef internal::conditional<NumTraits<Scalar>::IsComplex, const typename Diagonal<const MatrixType>::RealReturnType, const Diagonal<const MatrixType> >::type Tridiagonalization< _MatrixType >::DiagonalReturnType

Definition at line 102 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef internal::plain_col_type<MatrixType, RealScalar>::type Tridiagonalization< _MatrixType >::DiagonalType

Definition at line 94 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType Tridiagonalization< _MatrixType >::HouseholderSequenceType

Return type of matrixQ()

Definition at line 112 of file Tridiagonalization.h.

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

Definition at line 83 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> Tridiagonalization< _MatrixType >::MatrixTReturnType

Definition at line 97 of file Tridiagonalization.h.

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

Synonym for the template parameter _MatrixType.

Definition at line 79 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef internal::remove_all<typename MatrixType::RealReturnType>::type Tridiagonalization< _MatrixType >::MatrixTypeRealView

Definition at line 96 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef NumTraits<Scalar>::Real Tridiagonalization< _MatrixType >::RealScalar

Definition at line 82 of file Tridiagonalization.h.

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

Definition at line 81 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef internal::conditional<NumTraits<Scalar>::IsComplex, const typename Diagonal< Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType, const Diagonal< Block<const MatrixType,SizeMinusOne,SizeMinusOne> > >::type Tridiagonalization< _MatrixType >::SubDiagonalReturnType

Definition at line 109 of file Tridiagonalization.h.

template<typename _MatrixType>
typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> Tridiagonalization< _MatrixType >::SubDiagonalType

Definition at line 95 of file Tridiagonalization.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
Size 
SizeMinusOne 
Options 
MaxSize 
MaxSizeMinusOne 

Definition at line 85 of file Tridiagonalization.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
Tridiagonalization< _MatrixType >::Tridiagonalization ( Index  size = Size==Dynamic ? 2 : Size) [inline]

Default constructor.

Parameters:
[in]sizePositive integer, size of the matrix whose tridiagonal 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 126 of file Tridiagonalization.h.

template<typename _MatrixType>
Tridiagonalization< _MatrixType >::Tridiagonalization ( const MatrixType &  matrix) [inline]

Constructor; computes tridiagonal decomposition of given matrix.

Parameters:
[in]matrixSelfadjoint matrix whose tridiagonal decomposition is to be computed.

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

Example: 

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix:" << endl << A << endl << endl;
Tridiagonalization<MatrixXd> triOfA(A);
MatrixXd Q = triOfA.matrixQ();
cout << "The orthogonal matrix Q is:" << endl << Q << endl;
MatrixXd T = triOfA.matrixT();
cout << "The tridiagonal matrix T is:" << endl << T << endl << endl;
cout << "Q * T * Q^T = " << endl << Q * T * Q.transpose() << endl;

Output: 

 

Definition at line 142 of file Tridiagonalization.h.







Member Function Documentation








template<typename _MatrixType> 

      

        

          Tridiagonalization& Tridiagonalization< _MatrixType >::compute 
          (
          const MatrixType & 
          matrix)
           [inline]
        

      







Computes tridiagonal decomposition of given matrix. 


Parameters:

  

    
[in]matrixSelfadjoint matrix whose tridiagonal decomposition is to be computed. 

  

  




Returns:
Reference to *this 


The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is $ 4n^3/3 $ flops, where $ n $ denotes the size of the given matrix.


This method reuses of the allocated data in the Tridiagonalization object, if the size of the matrix does not change.


Example: 


Tridiagonalization<MatrixXf> tri;
MatrixXf X = MatrixXf::Random(4,4);
MatrixXf A = X + X.transpose();
tri.compute(A);
cout << "The matrix T in the tridiagonal decomposition of A is: " << endl;
cout << tri.matrixT() << endl;
tri.compute(2*A); // re-use tri to compute eigenvalues of 2A
cout << "The matrix T in the tridiagonal decomposition of 2A is: " << endl;
cout << tri.matrixT() << endl;

 Output: 


 

Definition at line 168 of file Tridiagonalization.h.













template<typename MatrixType > 

      

        

          Tridiagonalization< MatrixType >::DiagonalReturnType Tridiagonalization< MatrixType >::diagonal 
          (
          )
           const
        

      







Returns the diagonal of the tridiagonal matrix T in the decomposition. 


Returns:
expression representing the diagonal of T


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


Example: 


MatrixXcd X = MatrixXcd::Random(4,4);
MatrixXcd A = X + X.adjoint();
cout << "Here is a random self-adjoint 4x4 matrix:" << endl << A << endl << endl;

Tridiagonalization<MatrixXcd> triOfA(A);
MatrixXd T = triOfA.matrixT();
cout << "The tridiagonal matrix T is:" << endl << T << endl << endl;

cout << "We can also extract the diagonals of T directly ..." << endl;
VectorXd diag = triOfA.diagonal();
cout << "The diagonal is:" << endl << diag << endl; 
VectorXd subdiag = triOfA.subDiagonal();
cout << "The subdiagonal is:" << endl << subdiag << endl;

 Output: 


See also:
matrixT(), subDiagonal() 



Definition at line 318 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          CoeffVectorType Tridiagonalization< _MatrixType >::householderCoefficients 
          (
          )
           const [inline]
        

      







Returns the Householder coefficients. 


Returns:
a const reference to the vector of Householder coefficients


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


The Householder coefficients allow the reconstruction of the matrix $ Q $ in the tridiagonal decomposition from the packed data.


Example: 


Matrix4d X = Matrix4d::Random(4,4);
Matrix4d A = X + X.transpose();
cout << "Here is a random symmetric 4x4 matrix:" << endl << A << endl;
Tridiagonalization<Matrix4d> triOfA(A);
Vector3d hc = triOfA.householderCoefficients();
cout << "The vector of Householder coefficients is:" << endl << hc << endl;

 Output: 


See also:
packedMatrix(), Householder module 



Definition at line 193 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          HouseholderSequenceType Tridiagonalization< _MatrixType >::matrixQ 
          (
          void 
          )
           const [inline]
        

      







Returns the unitary matrix Q in the decomposition. 


Returns:
object representing the matrix Q


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


This function returns a light-weight object of template class HouseholderSequence. You can either apply it directly to a matrix or you can convert it to a matrix of type MatrixType.


See also:
Tridiagonalization(const MatrixType&) for an example, matrixT(), class HouseholderSequence 



Definition at line 251 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          MatrixTReturnType Tridiagonalization< _MatrixType >::matrixT 
          (
          )
           const [inline]
        

      







Returns an expression of the tridiagonal matrix T in the decomposition. 


Returns:
expression object representing the matrix T


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


Currently, this function can be used to extract the matrix T from internal data and copy it to a dense matrix object. In most cases, it may be sufficient to directly use the packed matrix or the vector expressions returned by diagonal() and subDiagonal() instead of creating a new dense copy matrix with this function.


See also:
Tridiagonalization(const MatrixType&) for an example, matrixQ(), packedMatrix(), diagonal(), subDiagonal() 



Definition at line 276 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          const MatrixType& Tridiagonalization< _MatrixType >::packedMatrix 
          (
          )
           const [inline]
        

      







Returns the internal representation of the decomposition. 


Returns:
a const reference to a matrix with the internal representation of the decomposition.


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


The returned matrix contains the following information:


    

  • the strict upper triangular part is equal to the input matrix A.
    • 

  • the diagonal and lower sub-diagonal represent the real tridiagonal symmetric matrix T.
    • 

  • the rest of the lower part contains the Householder vectors that, combined with Householder coefficients returned by householderCoefficients(), allows to reconstruct the matrix Q as $ Q = H_{N-1} \ldots H_1 H_0 $. Here, the matrices $ H_i $ are the Householder transformations $ H_i = (I - h_i v_i v_i^T) $ where $ h_i $ is the $ i $th Householder coefficient and $ v_i $ is the Householder vector defined by $ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T $ with M the matrix returned by this function.
    • 



See LAPACK for further details on this packed storage.


Example: 


Matrix4d X = Matrix4d::Random(4,4);
Matrix4d A = X + X.transpose();
cout << "Here is a random symmetric 4x4 matrix:" << endl << A << endl;
Tridiagonalization<Matrix4d> triOfA(A);
Matrix4d pm = triOfA.packedMatrix();
cout << "The packed matrix M is:" << endl << pm << endl;
cout << "The diagonal and subdiagonal corresponds to the matrix T, which is:" 
     << endl << triOfA.matrixT() << endl;

 Output: 


See also:
householderCoefficients() 



Definition at line 230 of file Tridiagonalization.h.













template<typename MatrixType > 

      

        

          Tridiagonalization< MatrixType >::SubDiagonalReturnType Tridiagonalization< MatrixType >::subDiagonal 
          (
          )
           const
        

      







Returns the subdiagonal of the tridiagonal matrix T in the decomposition. 


Returns:
expression representing the subdiagonal of T


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


See also:
diagonal() for an example, matrixT() 



Definition at line 326 of file Tridiagonalization.h.







Member Data Documentation








template<typename _MatrixType> 

      

        

          CoeffVectorType Tridiagonalization< _MatrixType >::m_hCoeffs [protected]
        

      







Definition at line 312 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          bool Tridiagonalization< _MatrixType >::m_isInitialized [protected]
        

      







Definition at line 313 of file Tridiagonalization.h.













template<typename _MatrixType> 

      

        

          MatrixType Tridiagonalization< _MatrixType >::m_matrix [protected]
        

      







Definition at line 311 of file Tridiagonalization.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 11:34:29