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

Detailed Description

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

Partial specialization of MatrixFunction for complex matrices.

Definition at line 131 of file MatrixFunction.h.

Member Typedef Documentation

template<typename MatrixType >

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

Definition at line 146 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 145 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 148 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 137 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 144 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 147 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 141 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 136 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 142 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 135 of file MatrixFunction.h.

template<typename MatrixType >

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

Definition at line 143 of file MatrixFunction.h.

Constructor & Destructor Documentation

template<typename MatrixType >

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 MatrixFunction.h.

Member Function Documentation

template<typename MatrixType >

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.

template<typename MatrixType >

template<typename ResultType >

void 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 MatrixFunction.h.

template<typename MatrixType >

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.

template<typename MatrixType >

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

Compute m_blockStart using m_clusterSize.

Definition at line 317 of file MatrixFunction.h.

template<typename MatrixType >

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

Compute m_clusterSize and m_eivalToCluster using m_clusters.

Definition at line 295 of file MatrixFunction.h.

template<typename MatrixType >

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.

template<typename MatrixType >

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.

template<typename MatrixType >

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

Compute m_permutation using m_eivalToCluster and m_blockStart.

Definition at line 328 of file MatrixFunction.h.

template<typename MatrixType >

MatrixFunction< MatrixType, 1 >::ListOfClusters::iterator 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 MatrixFunction.h.

template<typename MatrixType >

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

template<typename MatrixType >

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.

template<typename MatrixType >

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.

template<typename MatrixType >

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.

template<typename MatrixType >

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.

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 $j=1,\ldots,n$ .