Partial specialization of MatrixFunction for complex matrices. More...
#include <MatrixFunction.h>
Public Member Functions | |
template<typename ResultType > | |
void | compute (ResultType &result) |
Compute the matrix function. | |
MatrixFunction (const MatrixType &A, StemFunction f) | |
Constructor. | |
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::stem_function < Scalar >::type | StemFunction |
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. | |
void | computeBlockAtomic () |
Compute block diagonal part of m_fT. | |
void | computeBlockStart () |
Compute m_blockStart using m_clusterSize. | |
void | computeClusterSize () |
Compute m_clusterSize and m_eivalToCluster using m_clusters. | |
void | computeOffDiagonal () |
Compute part of m_fT above block diagonal. | |
void | computeSchurDecomposition () |
Store the Schur decomposition of m_A in m_T and m_U. | |
void | constructPermutation () |
Compute m_permutation using m_eivalToCluster and m_blockStart. | |
ListOfClusters::iterator | findCluster (Scalar key) |
Find cluster in m_clusters containing some value. | |
MatrixFunction & | operator= (const MatrixFunction &) |
void | partitionEigenvalues () |
Partition eigenvalues in clusters of ei'vals close to each other. | |
void | permuteSchur () |
Permute Schur decomposition in m_U and m_T according to m_permutation. | |
DynMatrixType | solveTriangularSylvester (const DynMatrixType &A, const DynMatrixType &B, const DynMatrixType &C) |
Solve a triangular Sylvester equation AX + XB = C. | |
void | swapEntriesInSchur (Index index) |
Swap rows index and index+1 in Schur decomposition in m_U and m_T. | |
Static Private Member Functions | |
static const RealScalar | separation () |
Maximum distance allowed between eigenvalues to be considered "close". | |
Private Attributes | |
internal::nested< MatrixType > ::type | m_A |
Reference to argument of matrix function. | |
DynamicIntVectorType | m_blockStart |
Row index at which block corresponding to i-th cluster starts. | |
ListOfClusters | m_clusters |
Partition of eigenvalues into clusters of ei'vals "close" to each other. | |
DynamicIntVectorType | m_clusterSize |
Number of eigenvalues in each clusters. | |
DynamicIntVectorType | m_eivalToCluster |
m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster | |
StemFunction * | m_f |
Stem function for matrix function under consideration. | |
MatrixType | m_fT |
Matrix function applied to m_T | |
IntVectorType | m_permutation |
Permutation which groups ei'vals in the same cluster together. | |
MatrixType | m_T |
Triangular part of Schur decomposition. | |
MatrixType | m_U |
Unitary part of Schur decomposition. | |
Static Private Attributes | |
static const int | ColsAtCompileTime = Traits::ColsAtCompileTime |
static const int | Options = MatrixType::Options |
static const int | RowsAtCompileTime = Traits::RowsAtCompileTime |
Partial specialization of MatrixFunction for complex matrices.
Definition at line 131 of file MatrixFunction.h.
typedef std::list<Scalar> MatrixFunction< MatrixType, 1 >::Cluster [private] |
Definition at line 146 of file MatrixFunction.h.
typedef Matrix<Index, Dynamic, 1> MatrixFunction< MatrixType, 1 >::DynamicIntVectorType [private] |
Definition at line 145 of file MatrixFunction.h.
typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> MatrixFunction< MatrixType, 1 >::DynMatrixType [private] |
Definition at line 148 of file MatrixFunction.h.
typedef MatrixType::Index MatrixFunction< MatrixType, 1 >::Index [private] |
Definition at line 137 of file MatrixFunction.h.
typedef Matrix<Index, Traits::RowsAtCompileTime, 1> MatrixFunction< MatrixType, 1 >::IntVectorType [private] |
Definition at line 144 of file MatrixFunction.h.
typedef std::list<Cluster> MatrixFunction< MatrixType, 1 >::ListOfClusters [private] |
Definition at line 147 of file MatrixFunction.h.
typedef NumTraits<Scalar>::Real MatrixFunction< MatrixType, 1 >::RealScalar [private] |
Definition at line 141 of file MatrixFunction.h.
typedef MatrixType::Scalar MatrixFunction< MatrixType, 1 >::Scalar [private] |
Definition at line 136 of file MatrixFunction.h.
typedef internal::stem_function<Scalar>::type MatrixFunction< MatrixType, 1 >::StemFunction [private] |
Definition at line 142 of file MatrixFunction.h.
typedef internal::traits<MatrixType> MatrixFunction< MatrixType, 1 >::Traits [private] |
Definition at line 135 of file MatrixFunction.h.
typedef Matrix<Scalar, Traits::RowsAtCompileTime, 1> MatrixFunction< MatrixType, 1 >::VectorType [private] |
Definition at line 143 of file MatrixFunction.h.
MatrixFunction< MatrixType, 1 >::MatrixFunction | ( | const MatrixType & | A, |
StemFunction | f | ||
) |
Constructor.
[in] | A | argument of matrix function, should be a square matrix. |
[in] | f | an entire function; f(x,n) should compute the n-th derivative of f at x. |
Definition at line 198 of file MatrixFunction.h.
Block< MatrixType > MatrixFunction< MatrixType, 1 >::block | ( | MatrixType & | A, |
Index | i, | ||
Index | j | ||
) | [private] |
Return block of matrix according to blocking given by m_blockStart.
Definition at line 388 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::compute | ( | ResultType & | result | ) |
Compute the matrix function.
[out] | result | the function f applied to A , as specified in the constructor. |
Definition at line 211 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::computeBlockAtomic | ( | ) | [private] |
Compute block diagonal part of m_fT.
This routine computes the matrix function m_f applied to the block diagonal part of m_T, with the blocking given by m_blockStart. The result is stored in m_fT. The off-diagonal parts of m_fT are set to zero.
Definition at line 376 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::computeBlockStart | ( | ) | [private] |
Compute m_blockStart using m_clusterSize.
Definition at line 317 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::computeClusterSize | ( | ) | [private] |
Compute m_clusterSize and m_eivalToCluster using m_clusters.
Definition at line 295 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::computeOffDiagonal | ( | ) | [private] |
Compute part of m_fT above block diagonal.
This routine assumes that the block diagonal part of m_fT (which equals m_f 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 401 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::computeSchurDecomposition | ( | ) | [private] |
Store the Schur decomposition of m_A in m_T and m_U.
Definition at line 226 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::constructPermutation | ( | ) | [private] |
Compute m_permutation using m_eivalToCluster and m_blockStart.
Definition at line 328 of file MatrixFunction.h.
MatrixFunction< MatrixType, 1 >::ListOfClusters::iterator MatrixFunction< MatrixType, 1 >::findCluster | ( | Scalar | key | ) | [private] |
Find cluster in m_clusters containing some value.
[in] | key | Value to find |
key
, or m_clusters.end()
if no cluster in m_clusters contains key
. Definition at line 282 of file MatrixFunction.h.
MatrixFunction& MatrixFunction< MatrixType, 1 >::operator= | ( | const MatrixFunction< MatrixType, 1 > & | ) | [private] |
void MatrixFunction< MatrixType, 1 >::partitionEigenvalues | ( | ) | [private] |
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 245 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::permuteSchur | ( | ) | [private] |
Permute Schur decomposition in m_U and m_T according to m_permutation.
Definition at line 341 of file MatrixFunction.h.
static const RealScalar MatrixFunction< MatrixType, 1 >::separation | ( | ) | [inline, static, private] |
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 187 of file MatrixFunction.h.
MatrixFunction< MatrixType, 1 >::DynMatrixType MatrixFunction< MatrixType, 1 >::solveTriangularSylvester | ( | const DynMatrixType & | A, |
const DynMatrixType & | B, | ||
const DynMatrixType & | C | ||
) | [private] |
Solve a triangular Sylvester equation AX + XB = C.
[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. |
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
This can be re-arranged to yield:
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 443 of file MatrixFunction.h.
void MatrixFunction< MatrixType, 1 >::swapEntriesInSchur | ( | Index | index | ) | [private] |
Swap rows index and index+1 in Schur decomposition in m_U and m_T.
Definition at line 359 of file MatrixFunction.h.
const int MatrixFunction< MatrixType, 1 >::ColsAtCompileTime = Traits::ColsAtCompileTime [static, private] |
Definition at line 139 of file MatrixFunction.h.
internal::nested<MatrixType>::type MatrixFunction< MatrixType, 1 >::m_A [private] |
Reference to argument of matrix function.
Definition at line 170 of file MatrixFunction.h.
DynamicIntVectorType MatrixFunction< MatrixType, 1 >::m_blockStart [private] |
Row index at which block corresponding to i-th cluster starts.
Definition at line 178 of file MatrixFunction.h.
ListOfClusters MatrixFunction< MatrixType, 1 >::m_clusters [private] |
Partition of eigenvalues into clusters of ei'vals "close" to each other.
Definition at line 175 of file MatrixFunction.h.
DynamicIntVectorType MatrixFunction< MatrixType, 1 >::m_clusterSize [private] |
Number of eigenvalues in each clusters.
Definition at line 177 of file MatrixFunction.h.
DynamicIntVectorType MatrixFunction< MatrixType, 1 >::m_eivalToCluster [private] |
m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster
Definition at line 176 of file MatrixFunction.h.
StemFunction* MatrixFunction< MatrixType, 1 >::m_f [private] |
Stem function for matrix function under consideration.
Definition at line 171 of file MatrixFunction.h.
MatrixType MatrixFunction< MatrixType, 1 >::m_fT [private] |
Matrix function applied to m_T
Definition at line 174 of file MatrixFunction.h.
IntVectorType MatrixFunction< MatrixType, 1 >::m_permutation [private] |
Permutation which groups ei'vals in the same cluster together.
Definition at line 179 of file MatrixFunction.h.
MatrixType MatrixFunction< MatrixType, 1 >::m_T [private] |
Triangular part of Schur decomposition.
Definition at line 172 of file MatrixFunction.h.
MatrixType MatrixFunction< MatrixType, 1 >::m_U [private] |
Unitary part of Schur decomposition.
Definition at line 173 of file MatrixFunction.h.
const int MatrixFunction< MatrixType, 1 >::Options = MatrixType::Options [static, private] |
Definition at line 140 of file MatrixFunction.h.
const int MatrixFunction< MatrixType, 1 >::RowsAtCompileTime = Traits::RowsAtCompileTime [static, private] |
Definition at line 138 of file MatrixFunction.h.