CApproximateBases1< T > Class Template Reference

Class implementing virtual class CApproximateDirections. More...

#include <hyperbolic.h>

Inheritance diagram for CApproximateBases1< T >:

List of all members.

Public Member Functions
DERIVATIVE_MAP< 2, interval< T > >	find (const MAP< 2 > &F, const MAP< 2 > &F_1, const DERIVATIVE< 2, T > &Df, const Coords< 2, T > &sides, int steps) const
	Function which finds stable and unstable directions.
Private Member Functions
map< Coords< 2, int >, Coords < 2, int > >	oneValued (const MAP< 2 > &m) const
	Function constructing simplified, one valued versions of cubic representation map.
DERIVATIVE_MAP< 2, T >	inverseDerivative (const map< Coords< 2, int >, Coords< 2, int > > &f_1, const DERIVATIVE< 2, T > &Df, const Coords< 2, T > &sides, int scale) const
	Function which constructs inverse derivative representation.
CVector< T >	maxColumn (const CMatrix< T > &m) const
	Function returning column, which has the maximal euclidean norm.
CVector< T >	expandingVector (const CMatrix< T > &m, int iterations) const
	Function returning most expanding vector.
CVector< T >	direction (const vector< CMatrix< T > > &A, const vector< CMatrix< T > > &B) const
	Function returning normalized most expanding vector for system of mappings.
T	renormalization (const vector< T > &norms, T epsilon) const
	Function which computes the normalization constant.

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Detailed Description

template<class T>
class CApproximateBases1< T >

Class implementing virtual class CApproximateDirections.

Is responsible for constructing the base consist of stable and unstable directions for each cube. It is simplified since the dimenstion is 2.

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Member Function Documentation

template<class T>

map< Coords< 2, int >, Coords< 2, int > > CApproximateBases1< T >::oneValued

(

const MAP< 2 > &

m

)

const [inline, private]

Function constructing simplified, one valued versions of cubic representation map.

template<class T>

DERIVATIVE_MAP< 2, T > CApproximateBases1< T >::inverseDerivative	(	const map< Coords< 2, int >, Coords< 2, int > > &	f_1,
		const DERIVATIVE< 2, T > &	Df,
		const Coords< 2, T > &	sides,
		int	scale
	)			const [inline, private]

Function which constructs inverse derivative representation.

Parameters:

	f_1	one valued cubic representation of inverse function.
	Df	derivative of the function, for which inverse derivative representation is made.
	sides	is a Coords<_DIM, T> which represents rescaling coordinates.
	scale	determines how many cube divisions to perform.

template<class T>

CVector< T > CApproximateBases1< T >::maxColumn

(

const CMatrix< T > &

m

)

const [inline, private]

Function returning column, which has the maximal euclidean norm.

template<class T>

CVector< T > CApproximateBases1< T >::expandingVector	(	const CMatrix< T > &	m,
		int	iterations
	)			const [inline, private]

Function returning most expanding vector.

The simple power method is applied.

Parameters:

	m	matrix for which expanding vector is constructed.
	iterations	describes how many iterations of powet method to perform.

template<class T>

CVector< T > CApproximateBases1< T >::direction	(	const vector< CMatrix< T > > &	A,
		const vector< CMatrix< T > > &	B
	)			const [inline, private]

Function returning normalized most expanding vector for system of mappings.

System of mappings is assumed to be as follows: A[0]->A[1]->...->B[0]->B[1]->...

template<class T>

T CApproximateBases1< T >::renormalization	(	const vector< T > &	norms,
		T	epsilon
	)			const [inline, private]

Function which computes the normalization constant.

It purpose is to rescale the directions found by direction method to assess semi-hyperbolicity proporty as long as possible.

Parameters:

	norms	vector of norms of given direction on the part of trajectory.
	epsilon	small constant introduced to minimize errors conimg from finite length of trajectory considered and computational errors.

template<class T>

DERIVATIVE_MAP< 2, interval< T > > CApproximateBases1< T >::find	(	const MAP< 2 > &	F,
		const MAP< 2 > &	F_1,
		const DERIVATIVE< 2, T > &	Df,
		const Coords< 2, T > &	sides,
		int	steps
	)			const [inline, virtual]

Function which finds stable and unstable directions.

Value returned as DERIVATIVE_MAP. which maps cubes into bases consisted of stable and unstable directions.

Implements CApproximateBases< 2, T >.

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

• hyperbolic.h