Public Types | Public Member Functions | Protected Attributes | List of all members
Eigen::Tridiagonalization< _MatrixType > Class Template Reference

Tridiagonal decomposition of a selfadjoint matrix. More...

#include <Tridiagonalization.h>

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, typename internal::add_const_on_value_type< typename Diagonal< const MatrixType >::RealReturnType >::type, const Diagonal< const MatrixType > >::type DiagonalReturnType
 
typedef internal::plain_col_type< MatrixType, RealScalar >::type DiagonalType
 
typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType >::typeHouseholderSequenceType
 Return type of matrixQ() More...
 
typedef Eigen::Index Index
 
typedef internal::TridiagonalizationMatrixTReturnType< MatrixTypeRealViewMatrixTReturnType
 
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType. More...
 
typedef internal::remove_all< typename MatrixType::RealReturnType >::type MatrixTypeRealView
 
typedef NumTraits< Scalar >::Real RealScalar
 
typedef MatrixType::Scalar Scalar
 
typedef internal::conditional< NumTraits< Scalar >::IsComplex, typename internal::add_const_on_value_type< typename Diagonal< const MatrixType, -1 >::RealReturnType >::type, const Diagonal< const MatrixType, -1 > >::type SubDiagonalReturnType
 
typedef Matrix< RealScalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 > SubDiagonalType
 

Public Member Functions

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

Protected Attributes

CoeffVectorType m_hCoeffs
 
bool m_isInitialized
 
MatrixType m_matrix
 

Detailed Description

template<typename _MatrixType>
class Eigen::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 64 of file Tridiagonalization.h.

Member Typedef Documentation

◆ CoeffVectorType

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

Definition at line 83 of file Tridiagonalization.h.

◆ DiagonalReturnType

Definition at line 92 of file Tridiagonalization.h.

◆ DiagonalType

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

Definition at line 84 of file Tridiagonalization.h.

◆ HouseholderSequenceType

Return type of matrixQ()

Definition at line 100 of file Tridiagonalization.h.

◆ Index

template<typename _MatrixType>
typedef Eigen::Index Eigen::Tridiagonalization< _MatrixType >::Index
Deprecated:
since Eigen 3.3

Definition at line 73 of file Tridiagonalization.h.

◆ MatrixTReturnType

Definition at line 87 of file Tridiagonalization.h.

◆ MatrixType

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

Synonym for the template parameter _MatrixType.

Definition at line 69 of file Tridiagonalization.h.

◆ MatrixTypeRealView

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

Definition at line 86 of file Tridiagonalization.h.

◆ RealScalar

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

Definition at line 72 of file Tridiagonalization.h.

◆ Scalar

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

Definition at line 71 of file Tridiagonalization.h.

◆ SubDiagonalReturnType

template<typename _MatrixType>
typedef internal::conditional<NumTraits<Scalar>::IsComplex, typename internal::add_const_on_value_type<typename Diagonal<const MatrixType, -1>::RealReturnType>::type, const Diagonal<const MatrixType, -1> >::type Eigen::Tridiagonalization< _MatrixType >::SubDiagonalReturnType

Definition at line 97 of file Tridiagonalization.h.

◆ SubDiagonalType

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

Definition at line 85 of file Tridiagonalization.h.

Member Enumeration Documentation

◆ anonymous enum

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

Definition at line 75 of file Tridiagonalization.h.

Constructor & Destructor Documentation

◆ Tridiagonalization() [1/2]

template<typename _MatrixType>
Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( Index  size = Size==Dynamic ? 2 : Size)
inlineexplicit

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 114 of file Tridiagonalization.h.

◆ Tridiagonalization() [2/2]

template<typename _MatrixType>
template<typename InputType >
Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( const EigenBase< InputType > &  matrix)
inlineexplicit

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 131 of file Tridiagonalization.h.

Member Function Documentation

◆ compute()

template<typename _MatrixType>
template<typename InputType >
Tridiagonalization& Eigen::Tridiagonalization< _MatrixType >::compute ( const EigenBase< InputType > &  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 158 of file Tridiagonalization.h.

◆ diagonal()

template<typename MatrixType >
Tridiagonalization< MatrixType >::DiagonalReturnType Eigen::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 308 of file Tridiagonalization.h.

◆ householderCoefficients()

template<typename _MatrixType>
CoeffVectorType Eigen::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 183 of file Tridiagonalization.h.

◆ matrixQ()

template<typename _MatrixType>
HouseholderSequenceType Eigen::Tridiagonalization< _MatrixType >::matrixQ ( ) 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 241 of file Tridiagonalization.h.

◆ matrixT()

template<typename _MatrixType>
MatrixTReturnType Eigen::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 266 of file Tridiagonalization.h.

◆ packedMatrix()

template<typename _MatrixType>
const MatrixType& Eigen::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 220 of file Tridiagonalization.h.

◆ subDiagonal()

template<typename MatrixType >
Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::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 316 of file Tridiagonalization.h.

Member Data Documentation

◆ m_hCoeffs

template<typename _MatrixType>
CoeffVectorType Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs
protected

Definition at line 302 of file Tridiagonalization.h.

◆ m_isInitialized

template<typename _MatrixType>
bool Eigen::Tridiagonalization< _MatrixType >::m_isInitialized
protected

Definition at line 303 of file Tridiagonalization.h.

◆ m_matrix

template<typename _MatrixType>
MatrixType Eigen::Tridiagonalization< _MatrixType >::m_matrix
protected

Definition at line 301 of file Tridiagonalization.h.


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


gtsam
Author(s):
autogenerated on Tue Jul 4 2023 02:44:19