#include <MatrixFunction.h>

Public Member Functions
template<typename ResultType >
void	compute (ResultType &result)
	Compute the matrix function.
	MatrixFunction (const MatrixType &A, AtomicType &atomic)
	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::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.
AtomicType &	m_atomic
	Class for computing matrix function of atomic blocks.
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
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

Detailed Description

template<typename MatrixType, typename AtomicType>
class Eigen::MatrixFunction< MatrixType, AtomicType, 1 >

Definition at line 123 of file MatrixFunction.h.

Member Typedef Documentation

template<typename MatrixType , typename AtomicType >

typedef std::list<Scalar> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::Cluster [private]

Definition at line 137 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef Matrix<Index, Dynamic, 1> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::DynamicIntVectorType [private]

Definition at line 136 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::DynMatrixType [private]

Definition at line 139 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef MatrixType::Index Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::Index [private]

Definition at line 129 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef Matrix<Index, Traits::RowsAtCompileTime, 1> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::IntVectorType [private]

Definition at line 135 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef std::list<Cluster> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::ListOfClusters [private]

Definition at line 138 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef NumTraits<Scalar>::Real Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::RealScalar [private]

Definition at line 133 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef MatrixType::Scalar Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::Scalar [private]

Definition at line 128 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef internal::traits<MatrixType> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::Traits [private]

Definition at line 127 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

typedef Matrix<Scalar, Traits::RowsAtCompileTime, 1> Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::VectorType [private]

Definition at line 134 of file MatrixFunction.h.

Constructor & Destructor Documentation

template<typename MatrixType , typename AtomicType >

Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::MatrixFunction	(	const MatrixType &	A,
		AtomicType &	atomic
	)

Constructor.

Parameters:

[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

template<typename MatrixType , typename AtomicType >

Block< MatrixType > Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::block	(	MatrixType &	A,
		Index	i,
		Index	j
	)		`[private]`

Return block of matrix according to blocking given by m_blockStart.

Definition at line 378 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

template<typename ResultType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::compute ( ResultType & result )

Compute the matrix function.

Parameters:

[out] result the function f applied to A, as specified in the constructor.

Definition at line 202 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::computeBlockAtomic ( ) [private]

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.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::computeBlockStart ( ) [private]

Compute m_blockStart using m_clusterSize.

Definition at line 309 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::computeClusterSize ( ) [private]

Compute m_clusterSize and m_eivalToCluster using m_clusters.

Definition at line 287 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::computeOffDiagonal ( ) [private]

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.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::computeSchurDecomposition ( ) [private]

Store the Schur decomposition of m_A in m_T and m_U.

Definition at line 217 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::constructPermutation ( ) [private]

Compute m_permutation using m_eivalToCluster and m_blockStart.

Definition at line 320 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixFunction< MatrixType, AtomicType, 1 >::ListOfClusters::iterator Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::findCluster ( Scalar key ) [private]

Find cluster in m_clusters containing some value.

Parameters:

[in] key Value to find

Returns:: Iterator to cluster containing key, or m_clusters.end() if no cluster in m_clusters contains key.

Definition at line 274 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixFunction& Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::operator= ( const MatrixFunction< MatrixType, AtomicType, 1 > & ) [private]

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 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 236 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::permuteSchur ( ) [private]

Permute Schur decomposition in m_U and m_T according to m_permutation.

Definition at line 333 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

static const RealScalar Eigen::MatrixFunction< MatrixType, AtomicType, 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 178 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixFunction< MatrixType, AtomicType, 1 >::DynMatrixType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::solveTriangularSylvester	(	const DynMatrixType &	A,
		const DynMatrixType &	B,
		const DynMatrixType &	C
	)		`[private]`

Solve a triangular Sylvester equation AX + XB = C.

Parameters:

[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

$\sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}.$

This can be re-arranged to yield:

$X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij} - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr).$

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 $i=m,\ldots,1$ and $j=1,\ldots,n$ .

Definition at line 433 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

void Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::swapEntriesInSchur ( Index index ) [private]

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

template<typename MatrixType , typename AtomicType >

const int Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::ColsAtCompileTime = Traits::ColsAtCompileTime [static, private]

Definition at line 131 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

internal::nested<MatrixType>::type Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_A [private]

Reference to argument of matrix function.

Definition at line 161 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

AtomicType& Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_atomic [private]

Class for computing matrix function of atomic blocks.

Definition at line 162 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

DynamicIntVectorType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_blockStart [private]

Row index at which block corresponding to i-th cluster starts.

Definition at line 169 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

ListOfClusters Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_clusters [private]

Partition of eigenvalues into clusters of ei'vals "close" to each other.

Definition at line 166 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

DynamicIntVectorType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_clusterSize [private]

Number of eigenvalues in each clusters.

Definition at line 168 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

DynamicIntVectorType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_eivalToCluster [private]

m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster

Definition at line 167 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_fT [private]

Matrix function applied to m_T

Definition at line 165 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

IntVectorType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_permutation [private]

Permutation which groups ei'vals in the same cluster together.

Definition at line 170 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_T [private]

Triangular part of Schur decomposition.

Definition at line 163 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

MatrixType Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::m_U [private]

Unitary part of Schur decomposition.

Definition at line 164 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

const int Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::Options = MatrixType::Options [static, private]

Definition at line 132 of file MatrixFunction.h.

template<typename MatrixType , typename AtomicType >

const int Eigen::MatrixFunction< MatrixType, AtomicType, 1 >::RowsAtCompileTime = Traits::RowsAtCompileTime [static, private]

Definition at line 130 of file MatrixFunction.h.

The documentation for this class was generated from the following file:

MatrixFunction.h

Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

template<typename MatrixType, typename AtomicType> class Eigen::MatrixFunction< MatrixType, AtomicType, 1 >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename MatrixType, typename AtomicType>
class Eigen::MatrixFunction< MatrixType, AtomicType, 1 >