#include <MatrixFunction.h>
Public Member Functions | |
| template<typename ResultType > | |
| void | compute (ResultType &result) |
| Compute the matrix function. More... | |
| MatrixFunction (const MatrixType &A, AtomicType &atomic) | |
| Constructor. More... | |
Private Types | |
| typedef std::list< Scalar > | Cluster |
| typedef Matrix< Index, Dynamic, 1 > | DynamicIntVectorType |
| typedef Matrix< Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime > | DynMatrixType |
| typedef MatrixType::Index | Index |
| typedef Matrix< Index, Traits::RowsAtCompileTime, 1 > | IntVectorType |
| typedef std::list< Cluster > | ListOfClusters |
| typedef NumTraits< Scalar >::Real | RealScalar |
| typedef MatrixType::Scalar | Scalar |
| typedef internal::traits< MatrixType > | Traits |
| typedef Matrix< Scalar, Traits::RowsAtCompileTime, 1 > | VectorType |
Private Member Functions | |
| Block< MatrixType > | block (MatrixType &A, Index i, Index j) |
| Return block of matrix according to blocking given by m_blockStart. More... | |
| void | computeBlockAtomic () |
| Compute block diagonal part of m_fT. More... | |
| void | computeBlockStart () |
| Compute m_blockStart using m_clusterSize. More... | |
| void | computeClusterSize () |
| Compute m_clusterSize and m_eivalToCluster using m_clusters. More... | |
| void | computeOffDiagonal () |
| Compute part of m_fT above block diagonal. More... | |
| void | computeSchurDecomposition () |
| Store the Schur decomposition of m_A in m_T and m_U. More... | |
| void | constructPermutation () |
| Compute m_permutation using m_eivalToCluster and m_blockStart. More... | |
| ListOfClusters::iterator | findCluster (Scalar key) |
| Find cluster in m_clusters containing some value. More... | |
| MatrixFunction & | operator= (const MatrixFunction &) |
| void | partitionEigenvalues () |
| Partition eigenvalues in clusters of ei'vals close to each other. More... | |
| void | permuteSchur () |
| Permute Schur decomposition in m_U and m_T according to m_permutation. More... | |
| DynMatrixType | solveTriangularSylvester (const DynMatrixType &A, const DynMatrixType &B, const DynMatrixType &C) |
| Solve a triangular Sylvester equation AX + XB = C. More... | |
| void | swapEntriesInSchur (Index index) |
| Swap rows index and index+1 in Schur decomposition in m_U and m_T. More... | |
Static Private Member Functions | |
| static const RealScalar | separation () |
| Maximum distance allowed between eigenvalues to be considered "close". More... | |
Private Attributes | |
| internal::nested< MatrixType >::type | m_A |
| Reference to argument of matrix function. More... | |
| AtomicType & | m_atomic |
| Class for computing matrix function of atomic blocks. More... | |
| DynamicIntVectorType | m_blockStart |
| Row index at which block corresponding to i-th cluster starts. More... | |
| ListOfClusters | m_clusters |
| Partition of eigenvalues into clusters of ei'vals "close" to each other. More... | |
| DynamicIntVectorType | m_clusterSize |
| Number of eigenvalues in each clusters. More... | |
| DynamicIntVectorType | m_eivalToCluster |
| m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster More... | |
| MatrixType | m_fT |
| Matrix function applied to m_T More... | |
| IntVectorType | m_permutation |
| Permutation which groups ei'vals in the same cluster together. More... | |
| MatrixType | m_T |
| Triangular part of Schur decomposition. More... | |
| MatrixType | m_U |
| Unitary part of Schur decomposition. More... | |
Static Private Attributes | |
| static const int | ColsAtCompileTime = Traits::ColsAtCompileTime |
| static const int | Options = MatrixType::Options |
| static const int | RowsAtCompileTime = Traits::RowsAtCompileTime |
Detailed Description
template<typename MatrixType, typename AtomicType>
class Eigen::MatrixFunction< MatrixType, AtomicType, 1 >
Definition at line 123 of file MatrixFunction.h.
Member Typedef Documentation
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Definition at line 137 of file MatrixFunction.h.
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Definition at line 136 of file MatrixFunction.h.
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Definition at line 139 of file MatrixFunction.h.
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Definition at line 129 of file MatrixFunction.h.
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Definition at line 135 of file MatrixFunction.h.
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Definition at line 138 of file MatrixFunction.h.
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Definition at line 133 of file MatrixFunction.h.
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Definition at line 128 of file MatrixFunction.h.
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Definition at line 127 of file MatrixFunction.h.
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Definition at line 134 of file MatrixFunction.h.
Constructor & Destructor Documentation
| Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::MatrixFunction | ( | const MatrixType & | A, |
| AtomicType & | atomic | ||
| ) |
Constructor.
- Parameters
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[in] A argument of matrix function, should be a square matrix. [in] atomic class for computing matrix function of atomic blocks.
Definition at line 189 of file MatrixFunction.h.
Member Function Documentation
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Return block of matrix according to blocking given by m_blockStart.
Definition at line 378 of file MatrixFunction.h.
| void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::compute | ( | ResultType & | result | ) |
Compute the matrix function.
- Parameters
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[out] result the function fapplied toA, as specified in the constructor.
Definition at line 202 of file MatrixFunction.h.
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Compute block diagonal part of m_fT.
This routine computes the matrix function applied to the block diagonal part of m_T, with the blocking given by m_blockStart. The matrix function of each diagonal block is computed by m_atomic. The off-diagonal parts of m_fT are set to zero.
Definition at line 367 of file MatrixFunction.h.
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Compute m_blockStart using m_clusterSize.
Definition at line 309 of file MatrixFunction.h.
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Compute m_clusterSize and m_eivalToCluster using m_clusters.
Definition at line 287 of file MatrixFunction.h.
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Compute part of m_fT above block diagonal.
This routine assumes that the block diagonal part of m_fT (which equals the matrix function applied to m_T) has already been computed and computes the part above the block diagonal. The part below the diagonal is zero, because m_T is upper triangular.
Definition at line 391 of file MatrixFunction.h.
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Store the Schur decomposition of m_A in m_T and m_U.
Definition at line 217 of file MatrixFunction.h.
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Compute m_permutation using m_eivalToCluster and m_blockStart.
Definition at line 320 of file MatrixFunction.h.
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Find cluster in m_clusters containing some value.
- Parameters
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[in] key Value to find
- Returns
- Iterator to cluster containing
key, orm_clusters.end()if no cluster in m_clusters containskey.
Definition at line 274 of file MatrixFunction.h.
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Partition eigenvalues in clusters of ei'vals close to each other.
This function computes m_clusters. This is a partition of the eigenvalues of m_T in clusters, such that
Any eigenvalue in a certain cluster is at most separation() away
from another eigenvalue in the same cluster.
The distance between two eigenvalues in different clusters is
more than separation(). The implementation follows Algorithm 4.1 in the paper of Davies and Higham.
Definition at line 236 of file MatrixFunction.h.
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Permute Schur decomposition in m_U and m_T according to m_permutation.
Definition at line 333 of file MatrixFunction.h.
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Maximum distance allowed between eigenvalues to be considered "close".
This is morally a static const Scalar, but only integers can be static constant class members in C++. The separation constant is set to 0.1, a value taken from the paper by Davies and Higham.
Definition at line 178 of file MatrixFunction.h.
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Solve a triangular Sylvester equation AX + XB = C.
- Parameters
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[in] A the matrix A; should be square and upper triangular [in] B the matrix B; should be square and upper triangular [in] C the matrix C; should have correct size.
- Returns
- the solution X.
If A is m-by-m and B is n-by-n, then both C and X are m-by-n. The (i,j)-th component of the Sylvester equation is
![\[ \sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}. \]](form_260.png)
This can be re-arranged to yield:
![\[ X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij} - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr). \]](form_261.png)
It is assumed that A and B are such that the numerator is never zero (otherwise the Sylvester equation does not have a unique solution). In that case, these equations can be evaluated in the order 
and 
.
Definition at line 433 of file MatrixFunction.h.
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Swap rows index and index+1 in Schur decomposition in m_U and m_T.
Definition at line 351 of file MatrixFunction.h.
Member Data Documentation
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Definition at line 131 of file MatrixFunction.h.
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Reference to argument of matrix function.
Definition at line 161 of file MatrixFunction.h.
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Class for computing matrix function of atomic blocks.
Definition at line 162 of file MatrixFunction.h.
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Row index at which block corresponding to i-th cluster starts.
Definition at line 169 of file MatrixFunction.h.
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Partition of eigenvalues into clusters of ei'vals "close" to each other.
Definition at line 166 of file MatrixFunction.h.
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Number of eigenvalues in each clusters.
Definition at line 168 of file MatrixFunction.h.
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m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster
Definition at line 167 of file MatrixFunction.h.
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Matrix function applied to m_T
Definition at line 165 of file MatrixFunction.h.
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Permutation which groups ei'vals in the same cluster together.
Definition at line 170 of file MatrixFunction.h.
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Triangular part of Schur decomposition.
Definition at line 163 of file MatrixFunction.h.
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Unitary part of Schur decomposition.
Definition at line 164 of file MatrixFunction.h.
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Definition at line 132 of file MatrixFunction.h.
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Definition at line 130 of file MatrixFunction.h.
The documentation for this class was generated from the following file: