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 = internal::traits<MatrixType>::Options , MaxSize = MatrixType::MaxRowsAtCompileTime ,   MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1) }
typedef MatrixType_	MatrixType
	Synonym for the template parameter MatrixType_. More...
typedef MatrixType::Scalar	Scalar
typedef NumTraits< Scalar >::Real	RealScalar
typedef Eigen::Index	Index
typedef Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >	CoeffVectorType
typedef internal::plain_col_type< MatrixType, RealScalar >::type	DiagonalType
typedef Matrix< RealScalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >	SubDiagonalType
typedef internal::remove_all_t< typename MatrixType::RealReturnType >	MatrixTypeRealView
typedef internal::TridiagonalizationMatrixTReturnType< MatrixTypeRealView >	MatrixTReturnType
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, internal::add_const_on_value_type_t< typename Diagonal< const MatrixType >::RealReturnType >, const Diagonal< const MatrixType > >	DiagonalReturnType
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, internal::add_const_on_value_type_t< typename Diagonal< const MatrixType, -1 >::RealReturnType >, const Diagonal< const MatrixType, -1 > >	SubDiagonalReturnType
typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename CoeffVectorType::ConjugateReturnType > >	HouseholderSequenceType
	Return type of matrixQ() More...

Public Member Functions
	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...
template<typename InputType >
Tridiagonalization &	compute (const EigenBase< InputType > &matrix)
	Computes tridiagonal decomposition of given matrix. More...
CoeffVectorType	householderCoefficients () const
	Returns the Householder coefficients. More...
const MatrixType &	packedMatrix () const
	Returns the internal representation of the decomposition. 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...
DiagonalReturnType	diagonal () const
	Returns the diagonal of the tridiagonal matrix T in the decomposition. More...
SubDiagonalReturnType	subDiagonal () const
	Returns the subdiagonal of the tridiagonal matrix T in the decomposition. More...

Protected Attributes
MatrixType	m_matrix
CoeffVectorType	m_hCoeffs
bool	m_isInitialized

Detailed Description

template<typename MatrixType_>
class Eigen::Tridiagonalization< MatrixType_ >

Tridiagonal decomposition of a selfadjoint matrix.

\eigenvalues_module

Template Parameters

MatrixType_

the 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

Member Typedef Documentation

CoeffVectorType

template<typename MatrixType_ >

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

DiagonalReturnType

template<typename MatrixType_ >

typedef std::conditional_t<NumTraits<Scalar>::IsComplex, internal::add_const_on_value_type_t<typename Diagonal<const MatrixType>::RealReturnType>, const Diagonal<const MatrixType> > Eigen::Tridiagonalization< MatrixType_ >::DiagonalReturnType

DiagonalType

template<typename MatrixType_ >

typedef internal::plain_col_type<MatrixType, RealScalar>::type Eigen::Tridiagonalization< MatrixType_ >::DiagonalType

HouseholderSequenceType

template<typename MatrixType_ >

typedef HouseholderSequence<MatrixType, internal::remove_all_t<typename CoeffVectorType::ConjugateReturnType> > Eigen::Tridiagonalization< MatrixType_ >::HouseholderSequenceType

Return type of matrixQ()

Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::Tridiagonalization< MatrixType_ >::Index

Deprecated:

since Eigen 3.3

MatrixTReturnType

template<typename MatrixType_ >

typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> Eigen::Tridiagonalization< MatrixType_ >::MatrixTReturnType

MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::Tridiagonalization< MatrixType_ >::MatrixType

Synonym for the template parameter MatrixType_.

MatrixTypeRealView

template<typename MatrixType_ >

typedef internal::remove_all_t<typename MatrixType::RealReturnType> Eigen::Tridiagonalization< MatrixType_ >::MatrixTypeRealView

RealScalar

template<typename MatrixType_ >

typedef NumTraits<Scalar>::Real Eigen::Tridiagonalization< MatrixType_ >::RealScalar

Scalar

template<typename MatrixType_ >

typedef MatrixType::Scalar Eigen::Tridiagonalization< MatrixType_ >::Scalar

SubDiagonalReturnType

template<typename MatrixType_ >

typedef std::conditional_t< NumTraits<Scalar>::IsComplex, internal::add_const_on_value_type_t<typename Diagonal<const MatrixType, -1>::RealReturnType>, const Diagonal<const MatrixType, -1> > Eigen::Tridiagonalization< MatrixType_ >::SubDiagonalReturnType

SubDiagonalType

template<typename MatrixType_ >

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

Member Enumeration Documentation

anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
Size
SizeMinusOne
Options
MaxSize
MaxSizeMinusOne

       {
    Size = MatrixType::RowsAtCompileTime,
    SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
    Options = internal::traits<MatrixType>::Options,
    MaxSize = MatrixType::MaxRowsAtCompileTime,
    MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
  };

Constructor & Destructor Documentation

Tridiagonalization() [1/2]

template<typename MatrixType_ >

Eigen::Tridiagonalization< MatrixType_ >::Tridiagonalization ( Index  size = Size == Dynamic ? 2 : Size )

inlineexplicit

Default constructor.

Parameters

[in]

size

Positive 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.

                                                               : Size)
      : m_matrix(size, size), m_hCoeffs(size > 1 ? size - 1 : 1), m_isInitialized(false) {}

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]

matrix

Selfadjoint 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:

      : m_matrix(matrix.derived()), m_hCoeffs(matrix.cols() > 1 ? matrix.cols() - 1 : 1), m_isInitialized(false) {
    internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
    m_isInitialized = true;
  }

References Eigen::Tridiagonalization< MatrixType_ >::m_hCoeffs, Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized, Eigen::Tridiagonalization< MatrixType_ >::m_matrix, and Eigen::internal::tridiagonalization_inplace().

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]

matrix

Selfadjoint 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:

                                                                  {
    m_matrix = matrix.derived();
    m_hCoeffs.resize(matrix.rows() - 1, 1);
    internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
    m_isInitialized = true;
    return *this;
  }

References Eigen::Tridiagonalization< MatrixType_ >::m_hCoeffs, Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized, Eigen::Tridiagonalization< MatrixType_ >::m_matrix, matrix(), Eigen::PlainObjectBase< Derived >::resize(), and Eigen::internal::tridiagonalization_inplace().

Referenced by ctms_decompositions().

diagonal()

template<typename MatrixType >

Tridiagonalization< MatrixType >::DiagonalReturnType Eigen::Tridiagonalization< MatrixType >::diagonal

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()

                                                                                                         {
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  return m_matrix.diagonal().real();
}

References eigen_assert.

Referenced by selfadjointeigensolver().

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

                                                         {
    eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
    return m_hCoeffs;
  }

References eigen_assert, Eigen::Tridiagonalization< MatrixType_ >::m_hCoeffs, and Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized.

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

                                          {
    eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
    return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate()).setLength(m_matrix.rows() - 1).setShift(1);
  }

References eigen_assert, Eigen::Tridiagonalization< MatrixType_ >::m_hCoeffs, Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized, and Eigen::Tridiagonalization< MatrixType_ >::m_matrix.

Referenced by selfadjointeigensolver().

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()

                                    {
    eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
    return MatrixTReturnType(m_matrix.real());
  }

References eigen_assert, Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized, and Eigen::Tridiagonalization< MatrixType_ >::m_matrix.

Referenced by selfadjointeigensolver().

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()

                                                {
    eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
    return m_matrix;
  }

References eigen_assert, Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized, and Eigen::Tridiagonalization< MatrixType_ >::m_matrix.

subDiagonal()

template<typename MatrixType >

Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::Tridiagonalization< MatrixType >::subDiagonal

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()

                                                                                                               {
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  return m_matrix.template diagonal<-1>().real();
}

References diagonal(), eigen_assert, and Eigen::real().

Referenced by selfadjointeigensolver().

Member Data Documentation

m_hCoeffs

template<typename MatrixType_ >

CoeffVectorType Eigen::Tridiagonalization< MatrixType_ >::m_hCoeffs

protected

Referenced by Eigen::Tridiagonalization< MatrixType_ >::compute(), Eigen::Tridiagonalization< MatrixType_ >::householderCoefficients(), Eigen::Tridiagonalization< MatrixType_ >::matrixQ(), and Eigen::Tridiagonalization< MatrixType_ >::Tridiagonalization().

m_isInitialized

template<typename MatrixType_ >

bool Eigen::Tridiagonalization< MatrixType_ >::m_isInitialized

protected

Referenced by Eigen::Tridiagonalization< MatrixType_ >::compute(), Eigen::Tridiagonalization< MatrixType_ >::householderCoefficients(), Eigen::Tridiagonalization< MatrixType_ >::matrixQ(), Eigen::Tridiagonalization< MatrixType_ >::matrixT(), Eigen::Tridiagonalization< MatrixType_ >::packedMatrix(), and Eigen::Tridiagonalization< MatrixType_ >::Tridiagonalization().

m_matrix

template<typename MatrixType_ >

MatrixType Eigen::Tridiagonalization< MatrixType_ >::m_matrix

protected

Referenced by Eigen::Tridiagonalization< MatrixType_ >::compute(), Eigen::Tridiagonalization< MatrixType_ >::matrixQ(), Eigen::Tridiagonalization< MatrixType_ >::matrixT(), Eigen::Tridiagonalization< MatrixType_ >::packedMatrix(), and Eigen::Tridiagonalization< MatrixType_ >::Tridiagonalization().

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

• Tridiagonalization.h