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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:
1.36 -0.816 0.521 1.43 -0.144
-0.816 -0.659 0.794 -0.173 -0.406
0.521 0.794 -0.541 0.461 0.179
1.43 -0.173 0.461 -1.43 0.822
-0.144 -0.406 0.179 0.822 -1.37
The orthogonal matrix Q is:
1 0 0 0 0
0 -0.471 0.127 -0.671 -0.558
0 0.301 -0.195 0.437 -0.825
0 0.825 0.0459 -0.563 -0.00872
0 -0.0832 -0.971 -0.202 0.0922
The tridiagonal matrix T is:
1.36 1.73 0 0 0
1.73 -1.2 -0.966 0 0
0 -0.966 -1.28 0.214 0
0 0 0.214 -1.69 0.345
0 0 0 0.345 0.164
Q * T * Q^T =
1.36 -0.816 0.521 1.43 -0.144
-0.816 -0.659 0.794 -0.173 -0.406
0.521 0.794 -0.541 0.461 0.179
1.43 -0.173 0.461 -1.43 0.822
-0.144 -0.406 0.179 0.822 -1.37
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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:
1.36 -0.704 0 0
-0.704 0.0147 1.71 0
0 1.71 0.856 0.641
0 0 0.641 -0.506
The matrix T in the tridiagonal decomposition of 2A is:
2.72 -1.41 0 0
-1.41 0.0294 3.43 0
0 3.43 1.71 1.28
0 0 1.28 -1.01
| 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:
(-0.422,0) (0.705,-1.01) (-0.17,-0.552) (0.338,-0.357)
(0.705,1.01) (0.515,0) (0.241,-0.446) (0.05,-1.64)
(-0.17,0.552) (0.241,0.446) (-1.03,0) (0.0449,1.72)
(0.338,0.357) (0.05,1.64) (0.0449,-1.72) (1.36,0)
The tridiagonal matrix T is:
-0.422 -1.45 0 0
-1.45 1.01 -1.42 0
0 -1.42 1.8 -1.2
0 0 -1.2 -1.96
We can also extract the diagonals of T directly ...
The diagonal is:
-0.422
1.01
1.8
-1.96
The subdiagonal is:
-1.45
-1.42
-1.2
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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: 1.36 0.612 0.122 0.326 0.612 -1.21 -0.222 0.563 0.122 -0.222 -0.0904 1.16 0.326 0.563 1.16 1.66 The vector of Householder coefficients is: 1.87 1.24 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:
1.36 0.612 0.122 0.326
0.612 -1.21 -0.222 0.563
0.122 -0.222 -0.0904 1.16
0.326 0.563 1.16 1.66
The packed matrix M is:
1.36 0.612 0.122 0.326
-0.704 0.0147 -0.222 0.563
0.0925 1.71 0.856 1.16
0.248 0.785 0.641 -0.506
The diagonal and subdiagonal corresponds to the matrix T, which is:
1.36 -0.704 0 0
-0.704 0.0147 1.71 0
0 1.71 0.856 0.641
0 0 0.641 -0.506
| Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::Tridiagonalization< MatrixType >::subDiagonal | ( | ) | const |
Returns the subdiagonal of the tridiagonal matrix T in the decomposition.