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WPILibC++ 2023.4.3-108-ge5452e3
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WPILibC++ 2023.4.3-108-ge5452e3
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\eigenvalues_module More...
Public Types | |
| enum | { RowsAtCompileTime = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , Options = MatrixType::Options , MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime } |
| typedef _MatrixType | MatrixType |
Synonym for the template parameter _MatrixType. More... | |
| typedef MatrixType::Scalar | Scalar |
| Scalar type for matrices of type MatrixType. More... | |
| typedef NumTraits< Scalar >::Real | RealScalar |
| typedef Eigen::Index | Index |
| typedef std::complex< RealScalar > | ComplexScalar |
| Complex scalar type for MatrixType. More... | |
| typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > | VectorType |
| Type for vector of real scalar values eigenvalues as returned by betas(). More... | |
| typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > | ComplexVectorType |
| Type for vector of complex scalar values eigenvalues as returned by alphas(). More... | |
| typedef CwiseBinaryOp< internal::scalar_quotient_op< ComplexScalar, Scalar >, ComplexVectorType, VectorType > | EigenvalueType |
| Expression type for the eigenvalues as returned by eigenvalues(). More... | |
| typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime > | EigenvectorsType |
| Type for matrix of eigenvectors as returned by eigenvectors(). More... | |
Public Member Functions | |
| GeneralizedEigenSolver () | |
| Default constructor. More... | |
| GeneralizedEigenSolver (Index size) | |
| Default constructor with memory preallocation. More... | |
| GeneralizedEigenSolver (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true) | |
| Constructor; computes the generalized eigendecomposition of given matrix pair. More... | |
| EigenvectorsType | eigenvectors () const |
| EigenvalueType | eigenvalues () const |
| Returns an expression of the computed generalized eigenvalues. More... | |
| ComplexVectorType | alphas () const |
| VectorType | betas () const |
| GeneralizedEigenSolver & | compute (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true) |
| Computes generalized eigendecomposition of given matrix. More... | |
| ComputationInfo | info () const |
| GeneralizedEigenSolver & | setMaxIterations (Index maxIters) |
| Sets the maximal number of iterations allowed. More... | |
Static Protected Member Functions | |
| static void | check_template_parameters () |
Protected Attributes | |
| EigenvectorsType | m_eivec |
| ComplexVectorType | m_alphas |
| VectorType | m_betas |
| bool | m_valuesOkay |
| bool | m_vectorsOkay |
| RealQZ< MatrixType > | m_realQZ |
| ComplexVectorType | m_tmp |
\eigenvalues_module
Computes the generalized eigenvalues and eigenvectors of a pair of general matrices
| _MatrixType | the type of the matrices of which we are computing the eigen-decomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported. |
The generalized eigenvalues and eigenvectors of a matrix pair \( A \) and \( B \) are scalars \( \lambda \) and vectors \( v \) such that \( Av = \lambda Bv \). If \( D \) is a diagonal matrix with the eigenvalues on the diagonal, and \( V \) is a matrix with the eigenvectors as its columns, then \( A V = B V D \). The matrix \( V \) is almost always invertible, in which case we have \( A = B V D V^{-1} \). This is called the generalized eigen-decomposition.
The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \( \alpha \) and real \( \beta \) such that: \( \lambda_i = \alpha_i / \beta_i \). If \( \beta_i \) is (nearly) zero, then one can consider the well defined left eigenvalue \( \mu = \beta_i / \alpha_i\) such that: \( \mu_i A v_i = B v_i \), or even \( \mu_i u_i^T A = u_i^T B \) where \( u_i \) is called the left eigenvector.
Call the function compute() to compute the generalized eigenvalues and eigenvectors of a given matrix pair. Alternatively, you can use the GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues() and eigenvectors() functions.
Here is an usage example of this class: Example:
Output:
| typedef std::complex<RealScalar> Eigen::GeneralizedEigenSolver< _MatrixType >::ComplexScalar |
Complex scalar type for MatrixType.
This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.
| typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::ComplexVectorType |
Type for vector of complex scalar values eigenvalues as returned by alphas().
This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.
| typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> Eigen::GeneralizedEigenSolver< _MatrixType >::EigenvalueType |
Expression type for the eigenvalues as returned by eigenvalues().
| typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::GeneralizedEigenSolver< _MatrixType >::EigenvectorsType |
Type for matrix of eigenvectors as returned by eigenvectors().
This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.
| typedef Eigen::Index Eigen::GeneralizedEigenSolver< _MatrixType >::Index |
| typedef _MatrixType Eigen::GeneralizedEigenSolver< _MatrixType >::MatrixType |
Synonym for the template parameter _MatrixType.
| typedef NumTraits<Scalar>::Real Eigen::GeneralizedEigenSolver< _MatrixType >::RealScalar |
| typedef MatrixType::Scalar Eigen::GeneralizedEigenSolver< _MatrixType >::Scalar |
Scalar type for matrices of type MatrixType.
| typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::VectorType |
Type for vector of real scalar values eigenvalues as returned by betas().
This is a column vector with entries of type Scalar. The length of the vector is the size of MatrixType.
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Default constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).
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Default constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
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Constructor; computes the generalized eigendecomposition of given matrix pair.
| [in] | A | Square matrix whose eigendecomposition is to be computed. |
| [in] | B | Square matrix whose eigendecomposition is to be computed. |
| [in] | computeEigenvectors | If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed. |
This constructor calls compute() to compute the generalized eigenvalues and eigenvectors.
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This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
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This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
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| GeneralizedEigenSolver< MatrixType > & Eigen::GeneralizedEigenSolver< MatrixType >::compute | ( | const MatrixType & | A, |
| const MatrixType & | B, | ||
| bool | computeEigenvectors = true |
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Computes generalized eigendecomposition of given matrix.
| [in] | A | Square matrix whose eigendecomposition is to be computed. |
| [in] | B | Square matrix whose eigendecomposition is to be computed. |
| [in] | computeEigenvectors | If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed. |
*this This function computes the eigenvalues of the real matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().
The matrix is first reduced to real generalized Schur form using the RealQZ class. The generalized Schur decomposition is then used to compute the eigenvalues and eigenvectors.
The cost of the computation is dominated by the cost of the generalized Schur decomposition.
This method reuses of the allocated data in the GeneralizedEigenSolver object.
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Returns an expression of the computed generalized eigenvalues.
It is a shortcut for
Not that betas might contain zeros. It is therefore not recommended to use this function, but rather directly deal with the alphas and betas vectors.
The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.
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Sets the maximal number of iterations allowed.
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1.9.4