Inversion of sliced tubes

Functions
Interval	codac2::SlicedTube< T >::invert (const T &y, const Interval &t) const
	Returns the interval inversion \(\left[[x]^{-1}([y])\right]\).
void	codac2::SlicedTube< T >::invert (const T &y, std::vector< Interval > &v_t, const Interval &t) const
	Computes the set of continuous values of the inversion \([x]^{-1}([y])\).
Interval	codac2::SlicedTube< T >::invert (const T &y, const SlicedTube< T > &v, const Interval &t) const
	Returns the optimal interval inversion \(\left[[x]^{-1}([y])\right]\).
void	codac2::SlicedTube< T >::invert (const T &y, std::vector< Interval > &v_t, const SlicedTube< T > &v, const Interval &t) const
	Computes the set of continuous values of the optimal inversion \([x]^{-1}([y])\).

Detailed Description

Function Documentation

invert() [1/4]

template<typename T>

Interval codac2::SlicedTube< T >::invert ( const T & y, const Interval & t ) const

inline

Returns the interval inversion \(\left[[x]^{-1}([y])\right]\).

If the inversion results in several pre-images, their union is returned.

Parameters

y	interval codomain
t	(optional) temporal domain on which the inversion will be performed

Returns

the hull of \([x]^{-1}([y])\)

      {
        return invert_common(y, t,
          [this,&y](auto it, const Interval& t_) {
            return slice(it)->invert(y, t_);
          });
      }

invert() [2/4]

template<typename T>

void codac2::SlicedTube< T >::invert ( const T & y, std::vector< Interval > & v_t, const Interval & t ) const

inline

Computes the set of continuous values of the inversion \([x]^{-1}([y])\).

Parameters

y	interval codomain
v_t	vector of the sub-tdomains \([t_k]\) for which \(\forall t\in[t_k] \mid x(t)\in[y], x(\cdot)\in[x](\cdot)\)
t	(optional) temporal domain on which the inversion will be performed

      {
        return invert_common_subsets(y, v_t, t,
          [this,&y](auto it, const Interval& t_) {
            return slice(it)->invert(y, t_);
          });
      }

invert() [3/4]

template<typename T>

Interval codac2::SlicedTube< T >::invert ( const T & y, const SlicedTube< T > & v, const Interval & t ) const

inline

Returns the optimal interval inversion \(\left[[x]^{-1}([y])\right]\).

The knowledge of the derivative tube \([v](\cdot)\) allows a finer inversion. If the inversion results in several pre-images, their union is returned.

Parameters

y	interval codomain
v	derivative tube such that \(\dot{x}(\cdot)\in[v](\cdot)\)
t	(optional) temporal domain on which the inversion will be performed

Returns

hull of \([x]^{-1}([y])\)

      {
        return invert_common(y, t,
          [this,&v,&y](auto it, const Interval& t_) {
            return slice(it)->invert(y, *v.slice(it), t_);
          });
      }

invert() [4/4]

template<typename T>

void codac2::SlicedTube< T >::invert ( const T & y, std::vector< Interval > & v_t, const SlicedTube< T > & v, const Interval & t ) const

inline

Computes the set of continuous values of the optimal inversion \([x]^{-1}([y])\).

The knowledge of the derivative tube \([v](\cdot)\) allows finer inversions.

Parameters

y	interval codomain
v_t	vector of the sub-tdomains \([t_k]\) for which \(\forall t\in[t_k] \mid x(t)\in[y], x(\cdot)\in[x](\cdot), \dot{x}(\cdot)\in[v](\cdot)\)
v	derivative tube such that \(\dot{x}(\cdot)\in[v](\cdot)\)
t	(optional) temporal domain on which the inversion will be performed

      {
        return invert_common_subsets(y, v_t, t,
          [this,&v,&y](auto it, const Interval& t_) {
            return slice(it)->invert(y, *v.slice(it), t_);
          });
      }