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Eigen
3.4.0
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Tridiagonal decomposition of a selfadjoint matrix.
This is defined in the Eigenvalues module.
_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.
Public Types | |
typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType >::type > | 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. | |
typedef Eigen::Index Eigen::Tridiagonalization< _MatrixType >::Index |
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inlineexplicit |
Default constructor.
[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.
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inlineexplicit |
Constructor; computes tridiagonal decomposition of given matrix.
[in] | matrix | Selfadjoint matrix whose tridiagonal decomposition is to be computed. |
This constructor calls compute() to compute the tridiagonal decomposition.
Example:
Output:
Here is a random symmetric 5x5 matrix: 0.0555 -1.31 -1.21 0.14 1.54 -1.31 0.809 -0.768 0.521 -1.45 -1.21 -0.768 -0.891 0.0291 1.22 0.14 0.521 0.0291 1.07 0.19 1.54 -1.45 1.22 0.19 -0.612 The orthogonal matrix Q is: 1 0 0 0 0 0 -0.553 0.65 0.443 0.276 0 -0.515 -0.757 0.303 0.264 0 0.0592 0.0461 -0.523 0.849 0 0.653 -0.051 0.662 0.365 The tridiagonal matrix T is: 0.0555 2.36 0 0 0 2.36 -0.478 -1.89 0 0 0 -1.89 0.805 -0.823 0 0 0 -0.823 -0.846 -0.706 0 0 0 -0.706 0.894 Q * T * Q^T = 0.0555 -1.31 -1.21 0.14 1.54 -1.31 0.809 -0.768 0.521 -1.45 -1.21 -0.768 -0.891 0.0291 1.22 0.14 0.521 0.0291 1.07 0.19 1.54 -1.45 1.22 0.19 -0.612
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inline |
Computes tridiagonal decomposition of given matrix.
[in] | matrix | Selfadjoint matrix whose tridiagonal decomposition is to be computed. |
*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:
Output:
The matrix T in the tridiagonal decomposition of A is: 0.0555 -0.598 0 0 -0.598 0.34 0.996 0 0 0.996 -1.48 0.0846 0 0 0.0846 -1.7 The matrix T in the tridiagonal decomposition of 2A is: 0.111 -1.2 0 0 -1.2 0.681 1.99 0 0 1.99 -2.95 0.169 0 0 0.169 -3.4
Tridiagonalization< MatrixType >::DiagonalReturnType Eigen::Tridiagonalization< MatrixType >::diagonal | ( | ) | const |
Returns the diagonal of the tridiagonal matrix T in the decomposition.
Example:
Output:
Here is a random self-adjoint 4x4 matrix: (-1.3,0) (-0.681,0.372) (-0.364,-0.364) (0.287,-0.711) (-0.681,-0.372) (-0.441,0) (0.297,0.98) (-0.127,-0.927) (-0.364,0.364) (0.297,-0.98) (-1.39,0) (0.2,-0.0374) (0.287,0.711) (-0.127,0.927) (0.2,0.0374) (1.48,0) The tridiagonal matrix T is: -1.3 1.21 0 0 1.21 0.346 -0.883 0 0 -0.883 0.946 1.22 0 0 1.22 -1.65 We can also extract the diagonals of T directly ... The diagonal is: -1.3 0.346 0.946 -1.65 The subdiagonal is: 1.21 -0.883 1.22
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inline |
Returns the Householder coefficients.
The Householder coefficients allow the reconstruction of the matrix \( Q \) in the tridiagonal decomposition from the packed data.
Example:
Output:
Here is a random symmetric 4x4 matrix: 0.0555 0.247 -0.393 -0.376 0.247 -1.31 -0.346 -0.811 -0.393 -0.346 -1.66 0.746 -0.376 -0.811 0.746 0.142 The vector of Householder coefficients is: 1.41 1.73 0
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inline |
Returns the unitary matrix Q in the decomposition.
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.
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inline |
Returns an expression of the tridiagonal matrix T in the decomposition.
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.
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inline |
Returns the internal representation of the decomposition.
The returned matrix contains the following information:
See LAPACK for further details on this packed storage.
Example:
Output:
Here is a random symmetric 4x4 matrix: 0.0555 0.247 -0.393 -0.376 0.247 -1.31 -0.346 -0.811 -0.393 -0.346 -1.66 0.746 -0.376 -0.811 0.746 0.142 The packed matrix M is: 0.0555 0.247 -0.393 -0.376 -0.598 0.34 -0.346 -0.811 -0.466 0.996 -1.48 0.746 -0.446 -0.391 0.0846 -1.7 The diagonal and subdiagonal corresponds to the matrix T, which is: 0.0555 -0.598 0 0 -0.598 0.34 0.996 0 0 0.996 -1.48 0.0846 0 0 0.0846 -1.7
Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::Tridiagonalization< MatrixType >::subDiagonal | ( | ) | const |
Returns the subdiagonal of the tridiagonal matrix T in the decomposition.