Detailed Description

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

Partial specialization of MatrixFunction for complex matrices.

Definition at line 131 of file MatrixFunctions.

Public Member Functions
	MatrixFunction (const MatrixType &A, StemFunction f)
	Constructor.
template<typename ResultType >
void	compute (ResultType &result)
	Compute the matrix function.
Private Types
typedef internal::traits < MatrixType >	Traits
typedef MatrixType::Scalar	Scalar
typedef MatrixType::Index	Index
typedef NumTraits< Scalar >::Real	RealScalar
typedef internal::stem_function < Scalar >::type	StemFunction
typedef Matrix< Scalar, Traits::RowsAtCompileTime, 1 >	VectorType
typedef Matrix< Index, Traits::RowsAtCompileTime, 1 >	IntVectorType
typedef Matrix< Index, Dynamic, 1 >	DynamicIntVectorType
typedef std::list< Scalar >	Cluster
typedef std::list< Cluster >	ListOfClusters
typedef Matrix< Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime >	DynMatrixType
Private Member Functions
void	computeSchurDecomposition ()
	Store the Schur decomposition of m_A in m_T and m_U.
void	partitionEigenvalues ()
	Partition eigenvalues in clusters of ei'vals close to each other.
ListOfClusters::iterator	findCluster (Scalar key)
	Find cluster in m_clusters containing some value.
void	computeClusterSize ()
	Compute m_clusterSize and m_eivalToCluster using m_clusters.
void	computeBlockStart ()
	Compute m_blockStart using m_clusterSize.
void	constructPermutation ()
	Compute m_permutation using m_eivalToCluster and m_blockStart.
void	permuteSchur ()
	Permute Schur decomposition in m_U and m_T according to m_permutation.
void	swapEntriesInSchur (Index index)
	Swap rows index and index+1 in Schur decomposition in m_U and m_T.
void	computeBlockAtomic ()
	Compute block diagonal part of m_fT.
Block< MatrixType >	block (MatrixType &A, Index i, Index j)
	Return block of matrix according to blocking given by m_blockStart.
void	computeOffDiagonal ()
	Compute part of m_fT above block diagonal.
DynMatrixType	solveTriangularSylvester (const DynMatrixType &A, const DynMatrixType &B, const DynMatrixType &C)
	Solve a triangular Sylvester equation AX + XB = C.
MatrixFunction &	operator= (const MatrixFunction &)
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.
StemFunction *	m_f
	Stem function for matrix function under consideration.
MatrixType	m_T
	Triangular part of Schur decomposition.
MatrixType	m_U
	Unitary part of Schur decomposition.
MatrixType	m_fT
	Matrix function applied to m_T
ListOfClusters	m_clusters
	Partition of eigenvalues into clusters of ei'vals "close" to each other.
DynamicIntVectorType	m_eivalToCluster
	m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster
DynamicIntVectorType	m_clusterSize
	Number of eigenvalues in each clusters.
DynamicIntVectorType	m_blockStart
	Row index at which block corresponding to i-th cluster starts.
IntVectorType	m_permutation
	Permutation which groups ei'vals in the same cluster together.
Static Private Attributes
static const int	RowsAtCompileTime = Traits::RowsAtCompileTime
static const int	ColsAtCompileTime = Traits::ColsAtCompileTime
static const int	Options = MatrixType::Options

Member Typedef Documentation

template<typename MatrixType >

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

Definition at line 146 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 145 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 148 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 137 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 144 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 147 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 141 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 136 of file MatrixFunctions.

template<typename MatrixType >

typedef internal::stem_function<Scalar>::type Eigen::MatrixFunction< MatrixType, 1 >::StemFunction [private]

Definition at line 142 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 135 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 143 of file MatrixFunctions.

Constructor & Destructor Documentation

template<typename MatrixType >

Eigen::MatrixFunction< MatrixType, 1 >::MatrixFunction	(	const MatrixType &	A,
		StemFunction	f
	)

Constructor.

Parameters:

[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 MatrixFunctions.

Member Function Documentation

template<typename MatrixType >

Block< MatrixType > Eigen::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 MatrixFunctions.

template<typename MatrixType >

template<typename ResultType >

void Eigen::MatrixFunction< MatrixType, 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 211 of file MatrixFunctions.

template<typename MatrixType >

void Eigen::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 MatrixFunctions.

template<typename MatrixType >

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

Compute m_blockStart using m_clusterSize.

Definition at line 317 of file MatrixFunctions.

template<typename MatrixType >

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

Compute m_clusterSize and m_eivalToCluster using m_clusters.

Definition at line 295 of file MatrixFunctions.

template<typename MatrixType >

void Eigen::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 MatrixFunctions.

template<typename MatrixType >

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

Store the Schur decomposition of m_A in m_T and m_U.

Definition at line 226 of file MatrixFunctions.

template<typename MatrixType >

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

Compute m_permutation using m_eivalToCluster and m_blockStart.

Definition at line 328 of file MatrixFunctions.

template<typename MatrixType >

MatrixFunction< MatrixType, 1 >::ListOfClusters::iterator Eigen::MatrixFunction< MatrixType, 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 282 of file MatrixFunctions.

template<typename MatrixType >

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

template<typename MatrixType >

void Eigen::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 MatrixFunctions.

template<typename MatrixType >

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

Permute Schur decomposition in m_U and m_T according to m_permutation.

Definition at line 341 of file MatrixFunctions.

template<typename MatrixType >

static const RealScalar Eigen::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 MatrixFunctions.

template<typename MatrixType >

MatrixFunction< MatrixType, 1 >::DynMatrixType Eigen::MatrixFunction< MatrixType, 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 443 of file MatrixFunctions.

template<typename MatrixType >

void Eigen::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 MatrixFunctions.

Member Data Documentation

template<typename MatrixType >

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

Definition at line 139 of file MatrixFunctions.

template<typename MatrixType >

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

Reference to argument of matrix function.

Definition at line 170 of file MatrixFunctions.

template<typename MatrixType >

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

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

Definition at line 178 of file MatrixFunctions.

template<typename MatrixType >

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

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

Definition at line 175 of file MatrixFunctions.

template<typename MatrixType >

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

Number of eigenvalues in each clusters.

Definition at line 177 of file MatrixFunctions.

template<typename MatrixType >

DynamicIntVectorType Eigen::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 MatrixFunctions.

template<typename MatrixType >

StemFunction* Eigen::MatrixFunction< MatrixType, 1 >::m_f [private]

Stem function for matrix function under consideration.

Definition at line 171 of file MatrixFunctions.

template<typename MatrixType >

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

Matrix function applied to m_T

Definition at line 174 of file MatrixFunctions.

template<typename MatrixType >

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

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

Definition at line 179 of file MatrixFunctions.

template<typename MatrixType >

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

Triangular part of Schur decomposition.

Definition at line 172 of file MatrixFunctions.

template<typename MatrixType >

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

Unitary part of Schur decomposition.

Definition at line 173 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 140 of file MatrixFunctions.

template<typename MatrixType >

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

Definition at line 138 of file MatrixFunctions.

Detailed Description

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

Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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