# 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.
renormalization (const vector< T > &norms, T epsilon) const
Function which computes the normalization constant.

## 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.

## 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->A->...->B->B->...

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 >.

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

Generated on Sat May 24 19:47:31 2008 for Semi-hyperbolicity by 1.5.5