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Eigen::Tridiagonalization< MatrixType_ > Class Template Reference
Dense linear problems and decompositions » Reference » Eigenvalues module

#include <Eigen/src/Eigenvalues/Tridiagonalization.h>

Detailed Description

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

Tridiagonal decomposition of a selfadjoint matrix.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>
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

Public Types

typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename CoeffVectorType::ConjugateReturnType > > HouseholderSequenceType
 Return type of matrixQ()
 
typedef Eigen::Index Index
 
typedef MatrixType_ MatrixType
 Synonym for the template parameter MatrixType_.
 

Public Member Functions

template<typename InputType>
Tridiagonalization & compute (const EigenBase< InputType > &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.
 
template<typename InputType>
 Tridiagonalization (const EigenBase< InputType > &matrix)
 Constructor; computes tridiagonal decomposition of given matrix.
 
 Tridiagonalization (Index size=Size==Dynamic ? 2 :Size)
 Default constructor.
 

Member Typedef Documentation

◆ Index

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

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.

◆ 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;
Eigen::Tridiagonalization::Tridiagonalization
Tridiagonalization(Index size=Size==Dynamic ? 2 :Size)
Default constructor.
Definition Tridiagonalization.h:116
Eigen::MatrixXd
Matrix< double, Dynamic, Dynamic > MatrixXd
Dynamic×Dynamic matrix of type double.
Definition Matrix.h:479

Output: 

Here is a random symmetric 5x5 matrix:
   1.39    1.13  0.0173  -0.164  -0.686
   1.13    0.89  -0.679  -0.703   0.925
 0.0173  -0.679   0.268   0.581   0.825
 -0.164  -0.703   0.581   -1.51 -0.0932
 -0.686   0.925   0.825 -0.0932   -1.56

The orthogonal matrix Q is:
     1      0      0      0      0
     0 -0.849  0.401 -0.326  0.109
     0 -0.013 -0.549 -0.396  0.736
     0  0.123 -0.215 -0.778 -0.577
     0  0.514  0.701 -0.363  0.337
The tridiagonal matrix T is:
  1.39  -1.33      0      0      0
 -1.33 -0.492  -1.93      0      0
     0  -1.93 -0.138  0.171      0
     0      0  0.171 -0.755  -1.14
     0      0      0  -1.14 -0.526

Q * T * Q^T = 
   1.39    1.13  0.0173  -0.164  -0.686
   1.13    0.89  -0.679  -0.703   0.925
 0.0173  -0.679   0.268   0.581   0.825
 -0.164  -0.703   0.581   -1.51 -0.0932
 -0.686   0.925   0.825 -0.0932   -1.56

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;
Eigen::Tridiagonalization::matrixT
MatrixTReturnType matrixT() const
Returns an expression of the tridiagonal matrix T in the decomposition.
Definition Tridiagonalization.h:256
Eigen::Tridiagonalization::compute
Tridiagonalization & compute(const EigenBase< InputType > &matrix)
Computes tridiagonal decomposition of given matrix.
Definition Tridiagonalization.h:154
Eigen::MatrixXf
Matrix< float, Dynamic, Dynamic > MatrixXf
Dynamic×Dynamic matrix of type float.
Definition Matrix.h:478

Output: 

The matrix T in the tridiagonal decomposition of A is: 
 -1.65  -1.16      0      0
 -1.16 -0.814   1.45      0
     0   1.45 -0.842   1.19
     0      0   1.19  0.986
The matrix T in the tridiagonal decomposition of 2A is: 
 -3.3 -2.32     0     0
-2.32 -1.63  2.89     0
    0  2.89 -1.68  2.38
    0     0  2.38  1.97

◆ 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;
Eigen::MatrixXcd
Matrix< std::complex< double >, Dynamic, Dynamic > MatrixXcd
Dynamic×Dynamic matrix of type std::complex<double>.
Definition Matrix.h:481
Eigen::VectorXd
Matrix< double, Dynamic, 1 > VectorXd
Dynamic×1 vector of type double.
Definition Matrix.h:479

Output: 

Here is a random self-adjoint 4x4 matrix:
        (1.39,0)  (-0.391,-0.262)    (-1.2,-0.187)  (-0.333,-0.125)
  (-0.391,0.262)        (0.864,0)     (-0.623,0.9)  (-0.721,0.0309)
    (-1.2,0.187)    (-0.623,-0.9)       (-0.433,0)     (0.842,1.05)
  (-0.333,0.125) (-0.721,-0.0309)    (0.842,-1.05)         (1.72,0)

The tridiagonal matrix T is:
  1.39   1.35      0      0
  1.35 -0.355   1.63      0
     0   1.63   1.83 -0.928
     0      0 -0.928  0.673

We can also extract the diagonals of T directly ...
The diagonal is:
  1.39
-0.355
  1.83
 0.673
The subdiagonal is:
  1.35
  1.63
-0.928
See also
matrixT(), subDiagonal()

◆ 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;
Eigen::Vector3d
Matrix< double, 3, 1 > Vector3d
3×1 vector of type double.
Definition Matrix.h:479
Eigen::Matrix4d
Matrix< double, 4, 4 > Matrix4d
4×4 matrix of type double.
Definition Matrix.h:479

Output: 

Here is a random symmetric 4x4 matrix:
   1.39  -0.264  -0.391   0.468
 -0.264    1.86   0.517  -0.792
 -0.391   0.517   0.864 -0.0559
  0.468  -0.792 -0.0559  -0.996
The vector of Householder coefficients is:
 1.4
1.17
   0
See also
packedMatrix(), Householder module

◆ 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

◆ 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()

◆ 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: 

Here is a random symmetric 4x4 matrix:
   1.39  -0.264  -0.391   0.468
 -0.264    1.86   0.517  -0.792
 -0.391   0.517   0.864 -0.0559
  0.468  -0.792 -0.0559  -0.996
The packed matrix M is:
   1.39  -0.264  -0.391   0.468
  0.665   0.829   0.517  -0.792
  0.421    -1.6   0.238 -0.0559
 -0.504  -0.844  -0.581   0.656
The diagonal and subdiagonal corresponds to the matrix T, which is:
  1.39  0.665      0      0
 0.665  0.829   -1.6      0
     0   -1.6  0.238 -0.581
     0      0 -0.581  0.656
See also
householderCoefficients()

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