codac2 Namespace Reference

Classes
struct	Color
	Color structure, in RGBA or HSVA format. More...
struct	ColorMap
	Represents a set of RGB or HSV values. More...
class	ConvexPolygon
	Represents a convex polygon defined by vertices enclosed in IntervalVectors. More...
class	CtcDist
	Implements the distance constraint on \(\mathbf{a}\in\mathbb{R}^2\), \(\mathbf{b}\in\mathbb{R}^2\) and \(d\in\mathbb{R}\) such that: More...
class	CtcGaussElim
	Contractor for a linear system of interval equations, based on the classical Gauss elimination procedure. More...
class	CtcGaussSeidel
	Contractor for a linear system of interval equations, using a fixed-point approach based on the Gauss Seidel method. More...
class	CtcLinearPrecond
	Contractor for a linear system of interval equations, using a preconditioning method before calling some provided contractor. More...
class	CtcPolar
	Implements the polar constraint on \(x\in\mathbb{R}\), \(y\in\mathbb{R}\), \(\rho\in\mathbb{R}^+\) and \(\theta\in\mathbb{R}\) such that: More...
class	DefaultFigure
	Default view class, used to manage the default figure. More...
class	Ellipsoid
	Ellipsoid representation. More...
class	ExprBase
	Abstract base class for representing an expression. More...
class	ExprID
	A class representing a unique identifier for expressions. More...
class	Figure2D
	Figure2D class, used for 2D display. More...
class	Figure2D_IPE
	IPE output class. More...
class	Figure2D_VIBes
	VIBes output class. More...
class	Figure2DInterface
	Interface for 2D figures. More...
class	Figure3D
	Figure3D class, used for 3D figures. More...
class	FunctionArgsList
	A container class to manage a collection of function arguments. More...
class	FunctionBase
	A base class for functions (either analytic functions, or set functions). More...
class	Interval
	Interval class, for representing closed and connected subsets of \(\mathbb{R}\). More...
class	IntvFullPivLU
	Full pivot LU decomposition for matrix of interval, based on Eigen decomposition. The decomposition is of the form M = P{-1} [L][U] Q{-1} where P and Q are permutation matrices, and [L] and [U] are lower and upper interval matrices ([L] diagonal is 1). More...
class	OperationExprBase
	A base class for expressions representing operations with multiple operands. More...
class	OutputFigure2D
	Output figure class, used to manage the output figures. More...
class	Polygon
	Represents a polygon (convex or non-convex) defined by its vertices enclosed in IntervalVectors. More...
class	Segment
	Represents a geometric segment defined by two points enclosed in IntervalVectors. More...
class	SepChi
	A separator on the Chi constraint If [x] in Sa, then Sb, else Sc. More...
struct	StyleProperties
	Style properties structure, to specify the style of a shape. More...
class	VarBase
	Abstract base class for representing variables in analytic or set functions. More...

Typedefs
using	IntervalMatrix = Eigen::Matrix<Interval,-1,-1>
	Alias for a dynamic-size matrix of intervals.
using	IntervalRow = Eigen::Matrix<Interval,1,-1>
	Alias for a dynamic-size row vector of intervals.
using	IntervalVector = Eigen::Matrix<Interval,-1,1>
	Alias for a dynamic-size column vector of intervals.
using	Matrix = Eigen::Matrix<double,-1,-1>
	Alias for a dynamic-size matrix of doubles.
using	Row = Eigen::Matrix<double,1,-1>
	Alias for a dynamically-sized row vector of doubles.
using	Vector = Eigen::Matrix<double,-1,1>
	Alias for a dynamically-sized column vector of doubles.

Enumerations
enum class	BoolInterval { }
	Enumeration representing a boolean interval. More...
enum class	OrientationInterval { }
	Enumeration representing feasible orientations. Can be used to assess an oriented angle, or the alignement of three points. More...
enum	Model
	Color model (RGB or HSV) More...

Functions
BoolInterval	operator~ (BoolInterval x)
	Returns the complementary of a BoolInterval.
std::ostream &	operator<< (std::ostream &os, const BoolInterval &x)
	Streams out a BoolInterval.
Ellipsoid	operator+ (const Ellipsoid &e1, const Ellipsoid &e2)
	Compute the Minkowski sum of two ellipsoids.
Ellipsoid	linear_mapping (const Ellipsoid &e, const Matrix &A, const Vector &b)
	Guaranteed linear evaluation A*e+b, considering the rounding errors.
Ellipsoid	unreliable_linear_mapping (const Ellipsoid &e, const Matrix &A, const Vector &b)
	Nonrigorous linear evaluation A*e+b.
Ellipsoid	nonlinear_mapping (const Ellipsoid &e, const AnalyticFunction< VectorType > &f)
	(Rigorous?) non-linear evaluation f(e)
Ellipsoid	nonlinear_mapping (const Ellipsoid &e, const AnalyticFunction< VectorType > &f, const Vector &trig, const Vector &q)
	(Rigorous?) non-linear evaluation f(e), with parameters
Matrix	nonlinear_mapping_base (const Matrix &G, const Matrix &J, const IntervalMatrix &J_box, const Vector &trig, const Vector &q)
	(Rigorous?) non-linear evaluation f(e), from Jacobian information
std::ostream &	operator<< (std::ostream &os, const Ellipsoid &e)
	Streams out an Ellipsoid.
BoolInterval	stability_analysis (const AnalyticFunction< VectorType > &f, unsigned int alpha_max, Ellipsoid &e, Ellipsoid &e_out, bool verbose=false)
	...
Matrix	solve_discrete_lyapunov (const Matrix &A, const Matrix &Q)
	...
std::ostream &	operator<< (std::ostream &os, const Interval &x)
	Streams out this.
Interval	operator""_i (long double x)
	Codac defined literals allowing to produce an interval from a double.
double	previous_float (double x)
	Returns the previous representable double-precision floating-point value before x.
double	next_float (double x)
	Returns the next representable double-precision floating-point value after x.
Interval	operator& (const Interval &x, const Interval &y)
	Returns the intersection of two intervals: \([x]\cap[y]\).
Interval	operator| (const Interval &x, double y)
	Returns the squared-union of an interval and a real: \([x]\sqcup\{y\}\).
Interval	operator| (const Interval &x, const Interval &y)
	Returns the squared-union of two intervals: \([x]\sqcup[y]\).
const Interval &	operator+ (const Interval &x)
	Returns this.
Interval	operator+ (const Interval &x, double y)
	Returns \([x]+y\) with \(y\in\mathbb{R}\).
Interval	operator+ (double x, const Interval &y)
	Returns \(x+[y]\) with \(x\in\mathbb{R}\).
Interval	operator+ (const Interval &x, const Interval &y)
	Returns \([x]+[y]\).
Interval	operator- (const Interval &x, double y)
	Returns \([x]-y\) with \(y\in\mathbb{R}\).
Interval	operator- (double x, const Interval &y)
	Returns \(x-[y]\) with \(x\in\mathbb{R}\).
Interval	operator- (const Interval &x, const Interval &y)
	Returns \([x]-[y]\).
Interval	operator* (const Interval &x, double y)
	Returns \([x]*y\) with \(y\in\mathbb{R}\).
Interval	operator* (double x, const Interval &y)
	Returns \(x*[y]\) with \(x\in\mathbb{R}\).
Interval	operator* (const Interval &x, const Interval &y)
	Returns \([x]*[y]\).
Interval	operator/ (const Interval &x, double y)
	Returns \([x]/y\) with \(y\in\mathbb{R}\).
Interval	operator/ (double x, const Interval &y)
	Returns \(x/[y]\) with \(x\in\mathbb{R}\).
Interval	operator/ (const Interval &x, const Interval &y)
	Returns \([x]/[y]\).
Interval	sqr (const Interval &x)
	Returns \([x]^2\).
Interval	sqrt (const Interval &x)
	Returns \(\sqrt{[x]}\).
Interval	pow (const Interval &x, int n)
	Returns \([x]^n\), \(n\in\mathbb{Z}\).
Interval	pow (const Interval &x, double p)
	Returns \([x]^p\), \(p\in\mathbb{R}\).
Interval	pow (const Interval &x, const Interval &p)
	Returns \([x]^{[p]}\), \(p\in\mathbb{IR}\).
Interval	root (const Interval &x, int p)
	Returns the p-th root: \(\sqrt[p]{[x]}\).
Interval	exp (const Interval &x)
	Returns \(\exp([x])\).
Interval	log (const Interval &x)
	Returns \(\log([x])\).
Interval	cos (const Interval &x)
	Returns \(\cos([x])\).
Interval	sin (const Interval &x)
	Returns \(\sin([x])\).
Interval	tan (const Interval &x)
	Returns \(\tan([x])\).
Interval	acos (const Interval &x)
	Returns \(\acos([x])\).
Interval	asin (const Interval &x)
	Returns \(\asin([x])\).
Interval	atan (const Interval &x)
	Returns \(\atan([x])\).
Interval	atan2 (const Interval &y, const Interval &x)
	Returns \(\mathrm{arctan2}([y],[x])\).
Interval	cosh (const Interval &x)
	Returns \(\cosh([x])\).
Interval	sinh (const Interval &x)
	Returns \(\sinh([x])\).
Interval	tanh (const Interval &x)
	Returns \(\tanh([x])\).
Interval	acosh (const Interval &x)
	Returns \(\acosh([x])\).
Interval	asinh (const Interval &x)
	Returns \(\asinh([x])\).
Interval	atanh (const Interval &x)
	Returns \(\atanh([x])\).
Interval	abs (const Interval &x)
	Returns \(\mid[x]\mid = \left\{\mid x \mid, x\in[x]\right\}\).
Interval	min (const Interval &x, const Interval &y)
	Returns \(\min([x],[y])=\left\{\min(x,y), x\in[x], y\in[y]\right\}\).
Interval	max (const Interval &x, const Interval &y)
	Returns \(\max([x],[y])=\left\{\max(x,y), x\in[x], y\in[y]\right\}\).
Interval	sign (const Interval &x)
	Returns \(\sign([x])=\left[\left\{\sign(x), x\in[x]\right\}\right]\).
Interval	integer (const Interval &x)
	Returns the largest integer interval included in \([x]\).
Interval	floor (const Interval &x)
	Returns floor of \([x]\).
Interval	ceil (const Interval &x)
	Returns ceil of \([x]\).
Interval	chi (const Interval &x, const Interval &y, const Interval &z)
	Return \([y]\) if \(x^+ <= 0\), \([z]\) if \(x^- > 0\), \([y]\sqcup[z]\) else.
std::ostream &	operator<< (std::ostream &os, const IntervalMatrix &x)
	Stream output operator for IntervalMatrix objects.
std::ostream &	operator<< (std::ostream &os, const IntervalRow &x)
	Stream output operator for IntervalRow objects.
std::ostream &	operator<< (std::ostream &os, const IntervalVector &x)
	Stream output operator for IntervalVector objects.
ConvexPolygon	operator& (const ConvexPolygon &p1, const ConvexPolygon &p2)
	Computes the intersection of two convex polygons.
std::ostream &	operator<< (std::ostream &os, const OrientationInterval &x)
	Streams out a OrientationInterval.
OrientationInterval	orientation (const IntervalVector &p1, const IntervalVector &p2, const IntervalVector &p3)
	Computes the orientation of an ordered triplet of 2D points.
BoolInterval	aligned (const IntervalVector &p1, const IntervalVector &p2, const IntervalVector &p3)
	Checks whether three 2D points are aligned (colinear).
std::vector< IntervalVector >	convex_hull (std::vector< IntervalVector > pts)
	Computes the convex hull of a set of 2d points.
IntervalVector	rotate (const IntervalVector &p, const Interval &a, const IntervalVector &c=Vector::zero(2))
	Rotates a 2D interval vector around a point by a given angle.
template<typename T>
T	rotate (const T &x, const Interval &a, const IntervalVector &c=Vector::zero(2))
	Rotates a 2D object (Segment, Polygon, etc) around a point by a given angle.
std::ostream &	operator<< (std::ostream &str, const Polygon &p)
	Stream output operator for Polygon.
IntervalVector	operator& (const Segment &e1, const Segment &e2)
	Computes the intersection of two segments.
IntervalVector	proj_intersection (const Segment &e1, const Segment &e2)
	Computes the projected intersection of two segments.
BoolInterval	colinear (const Segment &e1, const Segment &e2)
	Checks if two segments are colinear.
std::ostream &	operator<< (std::ostream &str, const Segment &e)
	Stream output operator for Segment.
template<LeftOrRightInv O = LEFT_INV, typename OtherDerived, typename OtherDerived_>
IntervalMatrix	inverse_correction (const Eigen::MatrixBase< OtherDerived > &A, const Eigen::MatrixBase< OtherDerived_ > &B)
	Correct the approximation of the inverse \(\mathbf{B}\approx\mathbf{A}^{-1}\) of a square matrix \(\mathbf{A}\) by providing a reliable enclosure \([\mathbf{A}^{-1}]\).
template<typename OtherDerived>
IntervalMatrix	inverse_enclosure (const Eigen::MatrixBase< OtherDerived > &A)
	Enclosure of the inverse of a (non-singular) matrix expression, possibly an interval matrix.
IntervalMatrix	infinite_sum_enclosure (const IntervalMatrix &A, double &mrad)
	Compute an upper bound of \(\left([\mathbf{A}]+[\mathbf{A}]^2+[\mathbf{A}]^3+\dots\right)\), with \([\mathbf{A}]\) a matrix of intervals as an "error term" (uses only bounds on coefficients).
template<typename OtherDerived>
auto	abs (const Eigen::MatrixBase< OtherDerived > &x)
	Compute the element-wise absolute value of a matrix.
template<typename OtherDerived> requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)
auto	floor (const Eigen::MatrixBase< OtherDerived > &x)
	Compute the element-wise floor of a matrix.
template<typename OtherDerived> requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)
auto	ceil (const Eigen::MatrixBase< OtherDerived > &x)
	Compute the element-wise ceiling of a matrix.
template<typename OtherDerived> requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)
auto	round (const Eigen::MatrixBase< OtherDerived > &x)
	Compute the element-wise rounding of a matrix.
Eigen::IOFormat	codac_row_fmt ()
	Provides an Eigen IOFormat for formatting row vectors.
Eigen::IOFormat	codac_vector_fmt ()
	Provides an Eigen IOFormat for formatting column vectors.
Eigen::IOFormat	codac_matrix_fmt ()
	Provides an Eigen IOFormat for formatting matrices.
std::ostream &	operator<< (std::ostream &os, const Matrix &x)
	Stream output operator for Matrix objects.
std::ostream &	operator<< (std::ostream &os, const Row &x)
	Stream output operator for Row objects.
std::ostream &	operator<< (std::ostream &os, const Vector &x)
	Stream output operator for Vector objects.
template<typename T> requires (std::is_trivially_copyable_v<T>)
void	serialize (std::ostream &f, const T &x)
	Writes the binary representation of a trivially copyable object to the given output stream.
template<typename T> requires (std::is_trivially_copyable_v<T>)
void	deserialize (std::istream &f, T &x)
	Reads the binary representation of a trivially copyable object from the given input stream.
void	serialize (std::ostream &f, const Interval &x)
	Writes the binary representation of an Interval object to the given output stream.
void	deserialize (std::istream &f, Interval &x)
	Creates an Interval object from the binary representation given in an input stream.
template<typename T, int R = -1, int C = -1>
void	serialize (std::ostream &f, const Eigen::Matrix< T, R, C > &x)
	Writes the binary representation of an Eigen::Matrix to the given output stream. The structure can be a matrix, a row, a vector, with double or Interval components.
template<typename T, int R = -1, int C = -1>
void	deserialize (std::istream &f, Eigen::Matrix< T, R, C > &x)
	Reads the binary representation of an Eigen::Matrix from the given input stream. The structure can be a matrix, a row, a vector, with double or Interval components.
template<typename T>
void	serialize (std::ostream &f, const SampledTraj< T > &x)
	Writes the binary representation of a SampledTraj object to the given output stream.
template<typename T>
void	deserialize (std::istream &f, SampledTraj< T > &x)
	Reads the binary representation of a SampledTraj object from the given input stream.
template<typename T>
const SampledTraj< T > &	operator+ (const SampledTraj< T > &x1)
	\(x1(\cdot)\)
template<typename T>
SampledTraj< T >	operator+ (const SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_binary_traj_traj(operator_add< T >)
	\(x_1(\cdot)+x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T >	operator+ (const SampledTraj< T > &x1, const Q &x2) macro_binary_traj_real(operator_add< T >)
	\(x_1(\cdot)+x_2\)
template<typename T, typename Q>
SampledTraj< T >	operator+ (const Q &x1, const SampledTraj< T > &x2) macro_binary_real_traj(operator_add< T >)
	\(x+x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T > &	operator+= (SampledTraj< T > &x1, const Q &x2) macro_member_binary_traj_real(operator_add< T >)
	Operates +=.
template<typename T>
SampledTraj< T > &	operator+= (SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_member_binary_traj_traj(operator_add< T >)
	Operates +=.
template<typename T>
SampledTraj< T >	operator- (const SampledTraj< T > &x1)
	\(-x_1(\cdot)\)
template<typename T>
SampledTraj< T >	operator- (const SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_binary_traj_traj(operator_sub< T >)
	\(x_1(\cdot)-x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T >	operator- (const SampledTraj< T > &x1, const Q &x2) macro_binary_traj_real(operator_sub< T >)
	\(x_1(\cdot)-x_2\)
template<typename T, typename Q>
SampledTraj< T >	operator- (const Q &x1, const SampledTraj< T > &x2) macro_binary_real_traj(operator_sub< T >)
	\(x-x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T > &	operator-= (SampledTraj< T > &x1, const Q &x2) macro_member_binary_traj_real(operator_sub< T >)
	Operates -=.
template<typename T>
SampledTraj< T > &	operator-= (SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_member_binary_traj_traj(operator_sub< T >)
	Operates -=.
template<typename T> requires (!std::is_same_v<T,double>)
SampledTraj< T >	operator* (double x1, const SampledTraj< T > &x2) macro_binary_real_traj(operator_mul_scal< T >)
	\(x_1\cdot x_2(\cdot)\)
template<typename T> requires (!std::is_same_v<T,double>)
SampledTraj< T >	operator* (const SampledTraj< T > &x1, double x2) macro_binary_traj_real(operator_mul_scal< T >)
	\(x_1(\cdot)\cdot x_2\)
template<typename T>
SampledTraj< T >	operator* (const SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_binary_traj_traj(operator_mul< T >)
	\(x_1(\cdot)\cdot x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T >	operator* (const SampledTraj< T > &x1, const Q &x2) macro_binary_traj_real(operator_mul< T >)
	\(x_1(\cdot)\cdot x_2\)
template<typename T, typename Q>
SampledTraj< T >	operator* (const Q &x1, const SampledTraj< T > &x2) macro_binary_real_traj(operator_mul< T >)
	\(x\cdot x_2(\cdot)\)
SampledTraj< Vector >	operator* (const SampledTraj< Matrix > &x1, const SampledTraj< Vector > &x2) macro_binary_traj_traj(operator_mul_vec)
	\(x_1(\cdot)\cdot x_2\)
template<typename T, typename Q>
SampledTraj< T > &	operator*= (SampledTraj< T > &x1, const Q &x2) macro_member_binary_traj_real(operator_mul< T >)
	Operates *=.
template<typename T>
SampledTraj< T > &	operator*= (SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_member_binary_traj_traj(operator_mul< T >)
	Operates *=.
template<typename T> requires (!std::is_same_v<T,double>)
SampledTraj< T >	operator/ (const SampledTraj< T > &x1, double x2) macro_binary_traj_real(operator_div_scal< T >)
	\(x_2(\cdot)/x_1\)
template<typename T>
SampledTraj< T >	operator/ (const SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_binary_traj_traj(operator_div< T >)
	\(x_1(\cdot)/x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T >	operator/ (const SampledTraj< T > &x1, const Q &x2) macro_binary_traj_real(operator_div< T >)
	\(x_1(\cdot)/x_2\)
template<typename T, typename Q>
SampledTraj< T >	operator/ (const Q &x1, const SampledTraj< T > &x2) macro_binary_real_traj(operator_div< T >)
	\(x/x_2(\cdot)\)
template<typename T, typename Q>
SampledTraj< T > &	operator/= (SampledTraj< T > &x1, const Q &x2) macro_member_binary_traj_real(operator_div< T >)
	Operates /=.
template<typename T>
SampledTraj< T > &	operator/= (SampledTraj< T > &x1, const SampledTraj< T > &x2) macro_member_binary_traj_traj(operator_div< T >)
	Operates /=.
SampledTraj< double >	sqr (const SampledTraj< double > &x1)
	\(x^2(\cdot)\)
SampledTraj< double >	sqrt (const SampledTraj< double > &x1)
	\(\sqrt{x_1(\cdot)}\)
SampledTraj< double >	pow (const SampledTraj< double > &x1, int x2)
	\(x^x_2(\cdot)\)
SampledTraj< double >	pow (const SampledTraj< double > &x1, double x2)
	\(x^x_2(\cdot)\)
SampledTraj< double >	root (const SampledTraj< double > &x1, int x2)
	\(x^x_2(\cdot)\)
SampledTraj< double >	exp (const SampledTraj< double > &x1)
	\(\exp(x_1(\cdot))\)
SampledTraj< double >	log (const SampledTraj< double > &x1)
	\(\log(x_1(\cdot))\)
SampledTraj< double >	cos (const SampledTraj< double > &x1)
	\(\cos(x_1(\cdot))\)
SampledTraj< double >	sin (const SampledTraj< double > &x1)
	\(\sin(x_1(\cdot))\)
SampledTraj< double >	tan (const SampledTraj< double > &x1)
	\(\tan(x_1(\cdot))\)
SampledTraj< double >	acos (const SampledTraj< double > &x1)
	\(\arccos(x_1(\cdot))\)
SampledTraj< double >	asin (const SampledTraj< double > &x1)
	\(\arcsin(x_1(\cdot))\)
SampledTraj< double >	atan (const SampledTraj< double > &x1)
	\(\arctan(x_1(\cdot))\)
SampledTraj< double >	atan2 (const SampledTraj< double > &x1, const SampledTraj< double > &x2)
	\(\mathrm{arctan2}(x_1(\cdot),x_2(\cdot))\)
SampledTraj< double >	atan2 (const SampledTraj< double > &x1, double x2)
	\(\mathrm{arctan2}(x_1(\cdot),x_2)\)
SampledTraj< double >	atan2 (double x1, const SampledTraj< double > &x2)
	\(\mathrm{arctan2}(x_1, x_2(\cdot))\)
SampledTraj< double >	cosh (const SampledTraj< double > &x1)
	\(\cosh(x_1(\cdot))\)
SampledTraj< double >	sinh (const SampledTraj< double > &x1)
	\(\sinh(x_1(\cdot))\)
SampledTraj< double >	tanh (const SampledTraj< double > &x1)
	\(\tanh(x_1(\cdot))\)
SampledTraj< double >	acosh (const SampledTraj< double > &x1)
	\(\mathrm{arccosh}(x_1(\cdot))\)
SampledTraj< double >	asinh (const SampledTraj< double > &x1)
	\(\mathrm{arcsinh}(x_1(\cdot))\)
SampledTraj< double >	atanh (const SampledTraj< double > &x1)
	\(\mathrm{arctanh}(x_1(\cdot))\)
SampledTraj< double >	abs (const SampledTraj< double > &x1)
	\(\mid x_1(\cdot)\mid\)
SampledTraj< double >	min (const SampledTraj< double > &x1, const SampledTraj< double > &x2)
	\(\min(x_1(\cdot),x_2(\cdot))\)
SampledTraj< double >	min (const SampledTraj< double > &x1, double x2)
	\(\min(x_1(\cdot),x_2)\)
SampledTraj< double >	min (double x1, const SampledTraj< double > &x2)
	\(\min(x_1, x_2(\cdot))\)
SampledTraj< double >	max (const SampledTraj< double > &x1, const SampledTraj< double > &x2)
	\(\max(x_1(\cdot),x_2(\cdot))\)
SampledTraj< double >	max (const SampledTraj< double > &x1, double x2)
	\(\max(x_1(\cdot),x_2)\)
SampledTraj< double >	max (double x1, const SampledTraj< double > &x2)
	\(\max(x_1, x_2(\cdot))\)
capd::Interval	to_capd (const codac2::Interval &x)
	Converts a Codac Interval object into an CAPD Interval object.
codac2::Interval	to_codac (const capd::Interval &x)
	Converts an CAPD Interval object into a Codac Interval object.
capd::IVector	to_capd (const codac2::IntervalVector &x)
	Converts a Codac IntervalVector object into an CAPD IVector object.
codac2::IntervalVector	to_codac (const capd::IVector &x)
	Converts an CAPD IVector object into a Codac IntervalVector object.
capd::IMatrix	to_capd (const codac2::IntervalMatrix &x)
	Converts a Codac IntervalMatrix object into an CAPD IMatrix object.
codac2::IntervalMatrix	to_codac (const capd::IMatrix &x)
	Converts an CAPD IMatrix object into a Codac IntervalMatrix object.
void	draw_while_paving (const IntervalVector &x0, std::shared_ptr< const CtcBase< IntervalVector > > c, double eps, std::shared_ptr< Figure2D > fig=nullptr)
	Draws the paving while it is being computed.
void	draw_while_paving (const IntervalVector &x0, const CtcBase< IntervalVector > &c, double eps, std::shared_ptr< Figure2D > fig=nullptr)
	Draws the paving while it is being computed.
void	draw_while_paving (const IntervalVector &x0, std::shared_ptr< const SepBase > s, double eps, std::shared_ptr< Figure2D > fig=nullptr)
	Draws the paving while it is being computed.
void	draw_while_paving (const IntervalVector &x0, const SepBase &s, double eps, std::shared_ptr< Figure2D > fig=nullptr)
	Draws the paving while it is being computed.

Detailed Description

codac2_ibex_impl.h

Date

2024

Author

Gilles Chabert, (Simon Rohou)

Copyright

Copyright 2024 Codac Team

License: GNU Lesser General Public License (LGPL)

Typedef Documentation

IntervalMatrix

using codac2::IntervalMatrix = Eigen::Matrix<Interval,-1,-1>

Alias for a dynamic-size matrix of intervals.

Represents a matrix with a dynamic number of rows and columns, where each element is an Interval object.

This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<Interval,-1,-1>.

IntervalRow

using codac2::IntervalRow = Eigen::Matrix<Interval,1,-1>

Alias for a dynamic-size row vector of intervals.

Represents a row vector with a dynamic number of columns, where each element is an Interval object.

This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<Interval,1,-1>.

IntervalVector

using codac2::IntervalVector = Eigen::Matrix<Interval,-1,1>

Alias for a dynamic-size column vector of intervals.

Represents a column vector with a dynamic number of rows, where each element is an Interval object.

This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<Interval,-1,1>.

Matrix

using codac2::Matrix = Eigen::Matrix<double,-1,-1>

Alias for a dynamic-size matrix of doubles.

Represents a matrix with dynamic number of rows and columns, where each element is a double precision floating point number.

This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<double,-1,-1>.

See also

Vector, Row, Eigen::Matrix

Row

using codac2::Row = Eigen::Matrix<double,1,-1>

Alias for a dynamically-sized row vector of doubles.

Defines a convenient shorthand for representing row vectors with dynamic size. This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<double,1,-1>.

See also

Eigen::Matrix

Vector

using codac2::Vector = Eigen::Matrix<double,-1,1>

Alias for a dynamically-sized column vector of doubles.

Defines a convenient shorthand for representing column vectors with dynamic size. This type alias is based on Eigen's matrix template and corresponds to Eigen::Matrix<double,-1,1>.

See also

Eigen::Matrix

Enumeration Type Documentation

BoolInterval

enum class codac2::BoolInterval

strong

Enumeration representing a boolean interval.

The logical operators & and | can be used to combine BoolInterval values.

Enumerator
EMPTY	EMPTY is equivalent to the operation TRUE & FALSE.
UNKNOWN	UNKNOWN is equivalent to the operation TRUE | FALSE.

  {
    FALSE = 0x01,
    TRUE = 0x02,
    EMPTY = 0x00,
    UNKNOWN = 0x01 | 0x02
  };

OrientationInterval

enum class codac2::OrientationInterval

strong

Enumeration representing feasible orientations. Can be used to assess an oriented angle, or the alignement of three points.

The logical operator | can be used to combine OrientationInterval values.

Enumerator
UNKNOWN	UNKNOWN: the orientation can be either clockwise, counter-clockwise, or aligned.

  {
    COLINEAR = 0x01,
    CLOCKWISE = 0x02,
    COUNTERCLOCKWISE = 0x04,
    EMPTY = 0x00,
    UNKNOWN = 0x01 | 0x02 | 0x04
  };

Model

enum codac2::Model

Color model (RGB or HSV)

This enum is used to specify the color model of a color, either RGB or HSV.

{ RGB, HSV };

Function Documentation

operator~()

BoolInterval codac2::operator~ ( BoolInterval x )

inline

Returns the complementary of a BoolInterval.

Parameters

x	the boolean interval

Returns

the complementary

  {
    switch(x)
    {
      case BoolInterval::FALSE:
        return BoolInterval::TRUE;
      case BoolInterval::TRUE:
        return BoolInterval::FALSE;
      default:
        return BoolInterval::UNKNOWN;
    }
  }

operator<<() [1/12]

std::ostream & codac2::operator<< ( std::ostream & os, const BoolInterval & x )

inline

Streams out a BoolInterval.

Parameters

os	the stream to be updated
x	the boolean interval to stream out

Returns

a reference to the updated stream

  {
    switch(x)
    {
      case BoolInterval::EMPTY:
        os << "[ empty ]";
        break;
      case BoolInterval::FALSE:
        os << "[ false ]";
        break;
      case BoolInterval::TRUE:
        os << "[ true ]";
        break;
      case BoolInterval::UNKNOWN:
        os << "[ true, false ]";
        break;
    }
    return os;
  }

operator+() [1/9]

Ellipsoid codac2::operator+	(	const Ellipsoid &	e1,
		const Ellipsoid &	e2 )

Compute the Minkowski sum of two ellipsoids.

Parameters

e1	first ellipsoid
e2	second ellipsoid

Returns

the Minkowski sum

linear_mapping()

Ellipsoid codac2::linear_mapping	(	const Ellipsoid &	e,
		const Matrix &	A,
		const Vector &	b )

Guaranteed linear evaluation A*e+b, considering the rounding errors.

Parameters

e	input ellipsoid
A	matrix
b	vector

Returns

a rigorous outer enclosure of the linear mapping

unreliable_linear_mapping()

Ellipsoid codac2::unreliable_linear_mapping	(	const Ellipsoid &	e,
		const Matrix &	A,
		const Vector &	b )

Nonrigorous linear evaluation A*e+b.

Note

This function is used in linear_mapping() and provides a faster output than its guaranteed counterpart.

Parameters

e	input ellipsoid
A	matrix
b	vector

Returns

a nonrigorous approximation of the linear mapping

nonlinear_mapping() [1/2]

Ellipsoid codac2::nonlinear_mapping	(	const Ellipsoid &	e,
		const AnalyticFunction< VectorType > &	f )

(Rigorous?) non-linear evaluation f(e)

Parameters

e	input ellipsoid
f	non-linear analytical function

Returns

a (rigorous?) outer enclosure of the non-linear mapping

nonlinear_mapping() [2/2]

Ellipsoid codac2::nonlinear_mapping	(	const Ellipsoid &	e,
		const AnalyticFunction< VectorType > &	f,
		const Vector &	trig,
		const Vector &	q )

(Rigorous?) non-linear evaluation f(e), with parameters

Parameters

e	input ellipsoid
f	non-linear analytical function
trig	(?)
q	(?)

Returns

a (rigorous?) outer enclosure of the non-linear mapping

nonlinear_mapping_base()

Matrix codac2::nonlinear_mapping_base	(	const Matrix &	G,
		const Matrix &	J,
		const IntervalMatrix &	J_box,
		const Vector &	trig,
		const Vector &	q )

(Rigorous?) non-linear evaluation f(e), from Jacobian information

Parameters

G	(?)
J	approximated Jacobian matrix of f
J_box	reliable enclosure of the Jacobian matrix of f
trig	(?)
q	(?)

Returns

a (rigorous?) outer enclosure of the non-linear mapping

operator<<() [2/12]

std::ostream & codac2::operator<<	(	std::ostream &	os,
		const Ellipsoid &	e )

Streams out an Ellipsoid.

Parameters

os	the stream to be updated
e	the ellipsoid stream out

Returns

a reference to the updated stream

stability_analysis()

BoolInterval codac2::stability_analysis	(	const AnalyticFunction< VectorType > &	f,
		unsigned int	alpha_max,
		Ellipsoid &	e,
		Ellipsoid &	e_out,
		bool	verbose = false )

...

Parameters

f	...
alpha_max	...
e	...
e_out	...
verbose	...

Returns

...

solve_discrete_lyapunov()

Matrix codac2::solve_discrete_lyapunov	(	const Matrix &	A,
		const Matrix &	Q )

...

Parameters

A	...
Q	...

Returns

...

operator<<() [3/12]

std::ostream & codac2::operator<< ( std::ostream & os, const Interval & x )

inline

Streams out this.

Parameters

os	the stream to be updated
x	the interval to stream out

Returns

a reference to the updated stream

  {
    gaol::interval::precision(os.precision());
    gaol::operator<<(os,x);
    return os;
  }

operator""_i()

Interval codac2::operator""_i ( long double x )

inline

Codac defined literals allowing to produce an interval from a double.

Parameters

x	value to convert into an interval object

Returns

the interval \([x]\)

  {
    return Interval(x);
  }

previous_float()

double codac2::previous_float ( double x )

inline

Returns the previous representable double-precision floating-point value before x.

This function computes the largest double value that is strictly less than the input value x, effectively moving one step down in the floating-point representation.

See also

For obtaining the next representable double value greater than x, use next_float().

Parameters

x	The input double value.

Returns

The previous representable double value before x.

  {
    return gaol::previous_float(x);
  }

next_float()

double codac2::next_float ( double x )

inline

Returns the next representable double-precision floating-point value after x.

This function computes the smallest double value that is strictly greater than the input value x, effectively moving one step up in the floating-point representation.

See also

For obtaining the previous representable double value less than x, use previous_float().

Parameters

x	The input double value.

Returns

The next representable double value after x.

  {
    return gaol::next_float(x);
  }

operator&() [1/3]

Interval codac2::operator& ( const Interval & x, const Interval & y )

inline

Returns the intersection of two intervals: \([x]\cap[y]\).

Note

Returns an empty interval if there is no intersection.

Parameters

x	interval value
y	interval value

Returns

intersection result

  {
    if(x.is_empty() || y.is_empty() || x.ub() < y.lb())
      return Interval::empty();
    
    else
      return gaol::operator&(x,y);
  }

operator|() [1/2]

Interval codac2::operator| ( const Interval & x, double y )

inline

Returns the squared-union of an interval and a real: \([x]\sqcup\{y\}\).

Note

The squared-union is defined as: \([x]\sqcup[y]=\left[[x]\cup[y]\right]\)

Parameters

x	interval value
y	real value

Returns

squared-union result

  {
    return gaol::operator|(x,Interval(y));
  }

operator|() [2/2]

Interval codac2::operator| ( const Interval & x, const Interval & y )

inline

Returns the squared-union of two intervals: \([x]\sqcup[y]\).

Note

The squared-union is defined as: \([x]\sqcup[y]=\left[[x]\cup[y]\right]\)

Parameters

x	interval value
y	interval value

Returns

squared-union result

  {
    return gaol::operator|(x,y);
  }

operator+() [2/9]

const Interval & codac2::operator+ ( const Interval & x )

inline

Returns this.

Note

This operator is only provided for consistency purposes.

Parameters

x	interval value

Returns

the same interval

  {
    return x;
  }

operator+() [3/9]

Interval codac2::operator+ ( const Interval & x, double y )

inline

Returns \([x]+y\) with \(y\in\mathbb{R}\).

Parameters

x	interval value
y	real value

Returns

the addition result

  {
    if(y == -oo || y == oo)
      return Interval::empty();
 
    else
      return gaol::operator+(x,y);
  }

operator+() [4/9]

Interval codac2::operator+ ( double x, const Interval & y )

inline

Returns \(x+[y]\) with \(x\in\mathbb{R}\).

Parameters

x	real value
y	interval value

Returns

the addition result

  {
    if(x == -oo || x == oo)
      return Interval::empty();
 
    else
      return gaol::operator+(x,y);
  }

operator+() [5/9]

Interval codac2::operator+ ( const Interval & x, const Interval & y )

inline

Returns \([x]+[y]\).

Parameters

x	interval value
y	interval value

Returns

the addition result

  {
    return gaol::operator+(x,y);
  }

operator-() [1/7]

Interval codac2::operator- ( const Interval & x, double y )

inline

Returns \([x]-y\) with \(y\in\mathbb{R}\).

Parameters

x	interval value
y	real value

Returns

the substraction result

  {
    if(y == -oo || y == oo)
      return Interval::empty();
 
    else
      return gaol::operator-(x, y);
  }

operator-() [2/7]

Interval codac2::operator- ( double x, const Interval & y )

inline

Returns \(x-[y]\) with \(x\in\mathbb{R}\).

Parameters

x	real value
y	interval value

Returns

the substraction result

  {
    if(x == -oo || x == oo)
      return Interval::empty();
 
    else
      return gaol::operator-(x, y);
  }

operator-() [3/7]

Interval codac2::operator- ( const Interval & x, const Interval & y )

inline

Returns \([x]-[y]\).

Parameters

x	interval value
y	interval value

Returns

the substraction result

  {
    return gaol::operator-(x, y);
  }

operator*() [1/9]

Interval codac2::operator* ( const Interval & x, double y )

inline

Returns \([x]*y\) with \(y\in\mathbb{R}\).

Parameters

x	interval value
y	real value

Returns

the multiplication result

  {
    if(y == -oo || y == oo)
      return Interval::empty();
 
    else
      return gaol::operator*(x,y);
  }

operator*() [2/9]

Interval codac2::operator* ( double x, const Interval & y )

inline

Returns \(x*[y]\) with \(x\in\mathbb{R}\).

Parameters

x	real value
y	interval value

Returns

the multiplication result

  {
    if(x == -oo || x == oo)
      return Interval::empty();
 
    else
      return gaol::operator*(x,y);
  }

operator*() [3/9]

Interval codac2::operator* ( const Interval & x, const Interval & y )

inline

Returns \([x]*[y]\).

Parameters

x	interval value
y	interval value

Returns

the multiplication result

  {
    return gaol::operator*(x,y);
  }

operator/() [1/7]

Interval codac2::operator/ ( const Interval & x, double y )

inline

Returns \([x]/y\) with \(y\in\mathbb{R}\).

Parameters

x	interval value
y	real value

Returns

the division result

  {
    if(y == -oo || y == oo)
      return Interval::empty();
 
    else
      return gaol::operator/(x,y);
  }

operator/() [2/7]

Interval codac2::operator/ ( double x, const Interval & y )

inline

Returns \(x/[y]\) with \(x\in\mathbb{R}\).

Parameters

x	real value
y	interval value

Returns

the division result

  {
    if(x == -oo || x == oo)
      return Interval::empty();
 
    else
      return gaol::operator/(x,y);
  }

operator/() [3/7]

Interval codac2::operator/ ( const Interval & x, const Interval & y )

inline

Returns \([x]/[y]\).

Parameters

x	interval value
y	interval value

Returns

the division result

  {
    return gaol::operator/(x,y);
  }

sqr() [1/2]

Interval codac2::sqr ( const Interval & x )

inline

Returns \([x]^2\).

Parameters

x	interval value

Returns

the operation result

  {
    return gaol::sqr(x);
  }

sqrt() [1/2]

Interval codac2::sqrt ( const Interval & x )

inline

Returns \(\sqrt{[x]}\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::sqrt(x);
    gaol::round_upward();
    return y;
  }

pow() [1/5]

Interval codac2::pow ( const Interval & x, int n )

inline

Returns \([x]^n\), \(n\in\mathbb{Z}\).

Parameters

x	interval value
n	integer power value

Returns

the operation result

  {
    Interval y = gaol::pow(x,p);
    //gaol::round_upward(); // not necessary?
    return y;
  }

pow() [2/5]

Interval codac2::pow ( const Interval & x, double p )

inline

Returns \([x]^p\), \(p\in\mathbb{R}\).

Parameters

x	interval value
p	real power value

Returns

the operation result

  {
    if(p == -oo || p == oo)
      return Interval::empty();
 
    else
    {
      Interval y = gaol::pow(x,p);
      gaol::round_upward();
      return y;
    }
  }

pow() [3/5]

Interval codac2::pow ( const Interval & x, const Interval & p )

inline

Returns \([x]^{[p]}\), \(p\in\mathbb{IR}\).

Parameters

x	interval value
p	interval power value

Returns

the operation result

  {
    Interval y = gaol::pow(x,p);
    gaol::round_upward();
    return y;
  }

root() [1/2]

Interval codac2::root ( const Interval & x, int p )

inline

Returns the p-th root: \(\sqrt[p]{[x]}\).

Parameters

x	interval value
p	integer root

Returns

the operation result

  {
    // Get the root of the positive part (gaol does
    // not consider negative values to be in the definition
    // domain of the root function)
 
    gaol::interval y = gaol::nth_root(x, p>=0 ? p : -p);
 
    if(p%2 == 1 && x.lb() < 0)
      y |= -gaol::nth_root(-x, p >= 0 ? p : -p);
 
    if(p < 0)
      y = 1.0/y;
 
    gaol::round_upward();
    return y;
  }

exp() [1/2]

Interval codac2::exp ( const Interval & x )

inline

Returns \(\exp([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::exp(x);
    gaol::round_upward();
    return y;
  }

log() [1/2]

Interval codac2::log ( const Interval & x )

inline

Returns \(\log([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    if(x.ub() <= 0) // gaol returns (-oo,-DBL_MAX) if x.ub()==0, instead of empty set
      return Interval::empty();
 
    else
    {
      Interval y = gaol::log(x);
      gaol::round_upward();
      return y;
    }
  }

cos() [1/2]

Interval codac2::cos ( const Interval & x )

inline

Returns \(\cos([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::cos(x);
    gaol::round_upward();
    return y;
  }

sin() [1/2]

Interval codac2::sin ( const Interval & x )

inline

Returns \(\sin([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::sin(x);
    gaol::round_upward();
    return y;
  }

tan() [1/2]

Interval codac2::tan ( const Interval & x )

inline

Returns \(\tan([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::tan(x);
    gaol::round_upward();
    return y;
  }

acos() [1/2]

Interval codac2::acos ( const Interval & x )

inline

Returns \(\acos([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::acos(x);
    gaol::round_upward();
    return y;
  }

asin() [1/2]

Interval codac2::asin ( const Interval & x )

inline

Returns \(\asin([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::asin(x);
    gaol::round_upward();
    return y;
  }

atan() [1/2]

Interval codac2::atan ( const Interval & x )

inline

Returns \(\atan([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::atan(x);
    gaol::round_upward();
    return y;
  }

atan2() [1/4]

Interval codac2::atan2 ( const Interval & y, const Interval & x )

inline

Returns \(\mathrm{arctan2}([y],[x])\).

Parameters

y	interval value
x	interval value

Returns

the operation result

  {
    if(y.is_empty() || x.is_empty())
      return Interval::empty();
 
    // We handle the special case x=[0,0] separately
    else if(x == Interval::zero())
    {
      if(y.lb() >= 0)
      {
        if(y.ub() == 0)
          return Interval::empty(); // atan2(0,0) is undefined
        else
          return Interval::half_pi();
      }
 
      else if(y.ub() <= 0)
        return -Interval::half_pi();
 
      else
        return Interval(-1,1)*Interval::half_pi();
    }
 
    else if(x.lb() >= 0)
      return atan(y/x); // now, x.ub()>0 -> atan does not give an empty set
 
    else if(x.ub() <= 0)
    {
      if(y.lb() >= 0)
        return atan(y/x) + Interval::pi(); // x.lb()<0
      else if(y.ub() < 0)
        return atan(y/x) - Interval::pi();
      else
        return Interval(-1,1)*Interval::pi();
    }
 
    else
    {
      if(y.lb() >= 0)
        return atan(y/x.ub()) | (atan(y/x.lb()) + Interval::pi());
 
      else if(y.ub() <= 0)
      {
        if(x.lb() != -oo)
        {
          if(x.ub() != oo)
            return (atan(y/x.lb())-Interval::pi()) | atan(y/x.ub());
          else
            return (atan(y/x.lb())-Interval::pi()) | Interval::zero();
        }
 
        else
        {
          if(x.ub() != oo)
            return (-Interval::pi()) | atan(y/x.ub());
          else
            return -Interval::pi() | Interval::zero();
        }
      }
 
      else
        return Interval(-1,1)*Interval::pi();
    }
  }

cosh() [1/2]

Interval codac2::cosh ( const Interval & x )

inline

Returns \(\cosh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y;
    if(x.is_unbounded()) 
      y = Interval(gaol::cosh(x).left(),oo);
    else
      y = gaol::cosh(x);
    gaol::round_upward();
    return y;
  }

sinh() [1/2]

Interval codac2::sinh ( const Interval & x )

inline

Returns \(\sinh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::sinh(x);
    gaol::round_upward();
    return y;
  }

tanh() [1/2]

Interval codac2::tanh ( const Interval & x )

inline

Returns \(\tanh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::tanh(x);
    gaol::round_upward();
    return y;
  }

acosh() [1/2]

Interval codac2::acosh ( const Interval & x )

inline

Returns \(\acosh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::acosh(x);
    gaol::round_upward();
    return y;
  }

asinh() [1/2]

Interval codac2::asinh ( const Interval & x )

inline

Returns \(\asinh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    if(x.is_empty())
      return Interval::empty();
 
    else if(x.lb() >= 0)
      return gaol::asinh(x);
 
    else if(x.ub() <= 0)
      return -gaol::asinh(-x);
 
    else
      return {
        -gaol::asinh(gaol::interval(0,-x.lb())).right(),
        gaol::asinh(gaol::interval(0,x.ub())).right()
      };
 
    // no round_upward?
  }

atanh() [1/2]

Interval codac2::atanh ( const Interval & x )

inline

Returns \(\atanh([x])\).

Parameters

x	interval value

Returns

the operation result

  {
    Interval y = gaol::atanh(x);
    gaol::round_upward();
    return y;
  }

abs() [1/3]

Interval codac2::abs ( const Interval & x )

inline

Returns \(\mid[x]\mid = \left\{\mid x \mid, x\in[x]\right\}\).

Parameters

x	interval value

Returns

the operation result

  {
    return gaol::abs(x);
  }

min() [1/4]

Interval codac2::min ( const Interval & x, const Interval & y )

inline

Returns \(\min([x],[y])=\left\{\min(x,y), x\in[x], y\in[y]\right\}\).

Parameters

x	interval value
y	interval value

Returns

the operation result

  {
    return gaol::min(x,y);
  }

max() [1/4]

Interval codac2::max ( const Interval & x, const Interval & y )

inline

Returns \(\max([x],[y])=\left\{\max(x,y), x\in[x], y\in[y]\right\}\).

Parameters

x	interval value
y	interval value

Returns

the operation result

  {
    return gaol::max(x,y);
  }

sign()

Interval codac2::sign ( const Interval & x )

inline

Returns \(\sign([x])=\left[\left\{\sign(x), x\in[x]\right\}\right]\).

Note

By convention, \( 0\in[x] \Longrightarrow \sign([x])=[-1,1]\).

Parameters

x	interval value

Returns

the operation result

  {
    return x.ub() < 0 ? -1. : x.lb() > 0 ? 1. : Interval(-1.,1.);
  }

integer()

Interval codac2::integer ( const Interval & x )

inline

Returns the largest integer interval included in \([x]\).

The result is a subset interval of x.

Parameters

x	interval value

Returns

the operation result

  {
    return gaol::integer(x);
  }

floor() [1/2]

Interval codac2::floor ( const Interval & x )

inline

Returns floor of \([x]\).

Parameters

x	interval value

Returns

the operation result

  {
    return gaol::floor(x);
  }

ceil() [1/2]

Interval codac2::ceil ( const Interval & x )

inline

Returns ceil of \([x]\).

Parameters

x	interval value

Returns

the operation result

  {
    return gaol::ceil(x);
  }

chi()

Interval codac2::chi ( const Interval & x, const Interval & y, const Interval & z )

inline

Return \([y]\) if \(x^+ <= 0\), \([z]\) if \(x^- > 0\), \([y]\sqcup[z]\) else.

Parameters

x	interval value
y	interval value
z	interval value

Returns

the operation result

  {
    if(x.ub() <= 0)
      return y;
    
    else if(x.lb() > 0)
      return z;
 
    else
      return y | z;
  }

operator<<() [4/12]

std::ostream & codac2::operator<< ( std::ostream & os, const IntervalMatrix & x )

inline

Stream output operator for IntervalMatrix objects.

Parameters

os	The output stream to write to.
x	The interval matrix whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    if(x.is_empty())
      return os << "[ empty " << x.rows() << "x" << x.cols() << " mat ]";
 
    else
    {
      os << x.format(codac_matrix_fmt());
      return os;
    }
  }

operator<<() [5/12]

std::ostream & codac2::operator<< ( std::ostream & os, const IntervalRow & x )

inline

Stream output operator for IntervalRow objects.

Parameters

os	The output stream to write to.
x	The interval row whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    if(x.is_empty())
      return os << "[ empty row ]";
 
    else
    {
      os << x.format(codac_row_fmt());
      return os;
    }
  }

operator<<() [6/12]

std::ostream & codac2::operator<< ( std::ostream & os, const IntervalVector & x )

inline

Stream output operator for IntervalVector objects.

Parameters

os	The output stream to write to.
x	The interval vector whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    if(x.is_empty())
      return os << "[ empty vector ]";
 
    else
    {
      os << x.format(codac_vector_fmt());
      return os;
    }
  }

operator&() [2/3]

ConvexPolygon codac2::operator&	(	const ConvexPolygon &	p1,
		const ConvexPolygon &	p2 )

Computes the intersection of two convex polygons.

The result is a new ConvexPolygon enclosing the intersection area. If the polygons do not intersect, the result is an empty convex polygon.

Parameters

p1	The first convex polygon.
p2	The second convex polygon.

Returns

A ConvexPolygon enclosing the intersection area.

operator<<() [7/12]

std::ostream & codac2::operator<< ( std::ostream & os, const OrientationInterval & x )

inline

Streams out a OrientationInterval.

Parameters

os	The stream to be updated
x	The orientation interval to stream out

Returns

A Reference to the updated stream

  {
    if(x == OrientationInterval::EMPTY)
      os << "[ empty ]";
    else if(x == OrientationInterval::COLINEAR)
      os << "[ col ]";
    else if(x == OrientationInterval::CLOCKWISE)
      os << "[ cw ]";
    else if(x == OrientationInterval::COUNTERCLOCKWISE)
      os << "[ ccw ]";
    else if(x == (OrientationInterval::COLINEAR | OrientationInterval::CLOCKWISE))
      os << "[ col, cw ]";
    else if(x == (OrientationInterval::COLINEAR | OrientationInterval::COUNTERCLOCKWISE))
      os << "[ col, ccw ]";
    else if(x == OrientationInterval::UNKNOWN)
      os << "[ col, cw, ccw ]";
    return os;
  }

orientation()

OrientationInterval codac2::orientation	(	const IntervalVector &	p1,
		const IntervalVector &	p2,
		const IntervalVector &	p3 )

Computes the orientation of an ordered triplet of 2D points.

Determines whether the oriented angle \(\widehat{p_1 p_2 p_3}\) is positive (counterclockwise), negative (clockwise), or if the points are colinear (flat or 0 angle). Depending on floating point uncertainties, the test may not be able to conclude (a UNKNOWN value would then be returned).

Parameters

p1	First point (2d IntervalVector) of the triplet.
p2	Second point (2d IntervalVector) of the triplet (vertex of the angle).
p3	Third point (2d IntervalVector) of the triplet.

Returns

An orientation of type OrientationInterval

aligned()

BoolInterval codac2::aligned	(	const IntervalVector &	p1,
		const IntervalVector &	p2,
		const IntervalVector &	p3 )

Checks whether three 2D points are aligned (colinear).

Determines if the points lie on the same straight line using an orientation test (cross product). Depending on floating point uncertainties, the test may not be able to conclude (a UNKNOWN value would then be returned).

Parameters

p1	First point (2d IntervalVector) of the triplet.
p2	Second point (2d IntervalVector) of the triplet.
p3	Third point (2d IntervalVector) of the triplet.

Returns

A BooleanInterval

convex_hull()

std::vector< IntervalVector > codac2::convex_hull

(

std::vector< IntervalVector >

pts

)

Computes the convex hull of a set of 2d points.

Given a set of 2d points enclosed in tiny boxes, the function computes their convex hull. The method is based on a Graham scan algorithm. The output list of the algorithm is a subset of the input list, with same uncertainties and a possible different order.

Parameters

pts	2d points in any order

Returns

Points on the convex hull in counterclockwise order.

rotate() [1/2]

IntervalVector codac2::rotate	(	const IntervalVector &	p,
		const Interval &	a,
		const IntervalVector &	c = Vector::zero(2) )

Rotates a 2D interval vector around a point by a given angle.

This function performs a 2D rotation of the input interval vector p by an interval angle a, optionally around a center of rotation c (default is the origin). This operation propagates uncertainties through interval arithmetic.

Parameters

p	The input 2D interval vector to be rotated.
a	The rotation angle as an interval (in radians).
c	The center of rotation as an interval vector (default is the origin (0,0)).

Returns

The rotated interval vector.

rotate() [2/2]

template<typename T>

T codac2::rotate ( const T & x, const Interval & a, const IntervalVector & c = Vector::zero(2) )

inline

Rotates a 2D object (Segment, Polygon, etc) around a point by a given angle.

This function performs a 2D rotation of the input object x by an interval angle a, optionally around a center of rotation c (default is the origin). This operation propagates uncertainties through interval arithmetic.

Parameters

x	The input 2D object to be rotated.
a	The rotation angle as an interval (in radians).
c	The center of rotation as an interval vector (default is the origin (0,0)).

Returns

The rotated object.

  {
    T x_(x);
    for(auto& ei : x_)
      ei = rotate(ei,a,c);
    return x_;
  }

operator<<() [8/12]

std::ostream & codac2::operator<<	(	std::ostream &	str,
		const Polygon &	p )

Stream output operator for Polygon.

Parameters

str	Output stream.
p	The polygon to print.

Returns

The output stream with the polygon information.

operator&() [3/3]

IntervalVector codac2::operator&	(	const Segment &	e1,
		const Segment &	e2 )

Computes the intersection of two segments.

If the segments do not intersect, an empty IntervalVector is returned. If the segments are colinear, the set of intersection points is returned as a box.

Parameters

e1	The first segment.
e2	The second segment.

Returns

An IntervalVector enclosing the intersection point.

proj_intersection()

IntervalVector codac2::proj_intersection	(	const Segment &	e1,
		const Segment &	e2 )

Computes the projected intersection of two segments.

This corresponds to the intersection of the two lines related to the two segments. Therefore, the intersection point may not belong to the segments.

If the segments are parallel but not colinear, an empty IntervalVector is returned. If the segments are colinear, the set of intersection points is returned as a box.

Parameters

e1	The first segment.
e2	The second segment.

Returns

An IntervalVector enclosing the intersection point.

colinear()

BoolInterval codac2::colinear	(	const Segment &	e1,
		const Segment &	e2 )

Checks if two segments are colinear.

Parameters

e1	The first segment.
e2	The second segment.

Returns

A BoolInterval indicating possible colinearity.

operator<<() [9/12]

std::ostream & codac2::operator<<	(	std::ostream &	str,
		const Segment &	e )

Stream output operator for Segment.

Parameters

str	Output stream.
e	The segment to print.

Returns

The output stream with the segment information.

inverse_correction()

template<LeftOrRightInv O = LEFT_INV, typename OtherDerived, typename OtherDerived_>

IntervalMatrix codac2::inverse_correction ( const Eigen::MatrixBase< OtherDerived > & A, const Eigen::MatrixBase< OtherDerived_ > & B )

inline

Correct the approximation of the inverse \(\mathbf{B}\approx\mathbf{A}^{-1}\) of a square matrix \(\mathbf{A}\) by providing a reliable enclosure \([\mathbf{A}^{-1}]\).

Precondition

\(\mathbf{A}\) and \(\mathbf{B}\) are square matrices, possibly interval matrices.

Template Parameters

O	If LEFT_INV, use the inverse of \(\mathbf{BA}\) (otherwise use the inverse of \(\mathbf{AB}\), left inverse is normally better). In Python/Matlab, this template parameter is provided as a last boolean argument, and is left_inv = True by default.

Parameters

A	A matrix expression, possibly interval.
B	An (almost punctual) approximation of its inverse.

Returns

The enclosure of the inverse.

  {
    assert_release(A.is_squared());
    assert_release(B.is_squared());
 
    auto A_ = A.template cast<Interval>();
    auto B_ = B.template cast<Interval>();
 
    Index N = A_.rows();
    assert_release(N==B_.rows());
 
    auto Id = IntervalMatrix::Identity(N,N);
    auto erMat = [&]() { if constexpr(O == LEFT_INV) return -B_*A_+Id; else return -A_*B_+Id; }();
    
    double mrad=0.0;
    IntervalMatrix E = infinite_sum_enclosure(erMat,mrad);
    IntervalMatrix Ep = Id+erMat*(Id+E); 
        /* one could use several iterations here, either
           using mrad, or directly */
 
    auto res = (O == LEFT_INV) ? IntervalMatrix(Ep*B_) : IntervalMatrix(B_*Ep);
 
    // We want the presence of non-invertible matrices to
    // result in coefficients of the form [-oo,+oo].
    if (mrad==oo) {
       for (Index c=0;c<N;c++) {
         for (Index r=0;r<N;r++) {
           if (Ep(r,c).is_unbounded()) {
              for (Index k=0;k<N;k++) {
                   if constexpr(O == LEFT_INV)
                       res(r,k) = Interval();
                   else 
                       res(k,c) = Interval();
              }
           }
         }
       }
    }
    return  res;
  }

inverse_enclosure()

template<typename OtherDerived>

IntervalMatrix codac2::inverse_enclosure ( const Eigen::MatrixBase< OtherDerived > & A )

inline

Enclosure of the inverse of a (non-singular) matrix expression, possibly an interval matrix.

Precondition

\(\mathbf{A}\) is a square matrix.

Parameters

A	A matrix expression, possibly interval.

Returns

The enclosure of the inverse. Can have \((-\infty,\infty)\) coefficients if \(\mathbf{A}\) is singular or almost singular, if the inversion "failed".

  {
    assert_release(A.is_squared());
    Index N = A.rows();
 
    if constexpr(std::is_same_v<typename OtherDerived::Scalar,Interval>)
      return inverse_correction<LEFT_INV>(A, 
        (A.mid()).fullPivLu().solve(Matrix::Identity(N,N)));
 
    else
      return inverse_correction<LEFT_INV>(A, 
        A.fullPivLu().solve(Matrix::Identity(N,N)));
  }

infinite_sum_enclosure()

IntervalMatrix codac2::infinite_sum_enclosure	(	const IntervalMatrix &	A,
		double &	mrad )

Compute an upper bound of \(\left([\mathbf{A}]+[\mathbf{A}]^2+[\mathbf{A}]^3+\dots\right)\), with \([\mathbf{A}]\) a matrix of intervals as an "error term" (uses only bounds on coefficients).

The function also returns mrad, which gives an idea of the magnification of the matrix during calculation. In particular, if mrad = \(\infty\), then the inversion calculation (e.g., performed by Eigen) has somehow failed and some coefficients of the output interval matrix are \([-\infty,\infty]\).

Precondition

\([\mathbf{A}]\) is a square matrix.

Parameters

A	A matrix of intervals (supposed around \(\mathbf{0}\)).
mrad	The maximum radius of the result added (output argument).

Returns

The sum enclosure. May be unbounded.

abs() [2/3]

template<typename OtherDerived>

auto codac2::abs ( const Eigen::MatrixBase< OtherDerived > & x )

inline

Compute the element-wise absolute value of a matrix.

This function takes an Eigen matrix expression and returns a matrix where each element is replaced by its absolute value.

For scalar type double, the standard library fabs() is used. For other scalar types, it uses the generic abs() function.

Parameters

x	Input matrix expression.

Returns

A new Eigen matrix with the absolute values of the elements of x.

  {
    using M = Eigen::MatrixBase<OtherDerived>;
    Eigen::Matrix<typename M::Scalar,M::RowsAtCompileTime,M::ColsAtCompileTime> a(x.rows(),x.cols());
 
    for(Index i = 0 ; i < x.rows() ; i++)
      for(Index j = 0 ; j < x.cols() ; j++)
      {
        if constexpr(std::is_same_v<typename M::Scalar,double>)
          a(i,j) = fabs(x(i,j));
        else
          a(i,j) = abs(x(i,j));
      }
 
    return a;
  }

floor() [2/2]

template<typename OtherDerived>
requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)

auto codac2::floor ( const Eigen::MatrixBase< OtherDerived > & x )

inline

Compute the element-wise floor of a matrix.

This function returns a matrix where each element is replaced by the largest integer not greater than that element.

Disabled for interval matrices.

Parameters

x	Input matrix expression.

Returns

A new Eigen matrix with floored elements.

  {
    return x.array().floor().matrix();
  }

ceil() [2/2]

template<typename OtherDerived>
requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)

auto codac2::ceil ( const Eigen::MatrixBase< OtherDerived > & x )

inline

Compute the element-wise ceiling of a matrix.

This function returns a matrix where each element is replaced by the smallest integer not less than that element.

Disabled for interval matrices.

Parameters

x	Input matrix expression.

Returns

A new Eigen matrix with ceiled elements.

  {
    return x.array().ceil().matrix();
  }

round()

template<typename OtherDerived>
requires (!Eigen::IsIntervalDomain<typename OtherDerived::Scalar>)

auto codac2::round ( const Eigen::MatrixBase< OtherDerived > & x )

inline

Compute the element-wise rounding of a matrix.

This function returns a matrix where each element is replaced by the nearest integer to that element.

Disabled for interval matrices.

Parameters

x	Input matrix expression.

Returns

A new Eigen matrix with rounded elements.

  {
    return x.array().round().matrix();
  }

codac_row_fmt()

Eigen::IOFormat codac2::codac_row_fmt ( )

inline

Provides an Eigen IOFormat for formatting row vectors.

This format prints elements separated by spaces, with brackets around the entire row vector.

Returns

An Eigen::IOFormat configured for row vector formatting.

  {
    return Eigen::IOFormat(Eigen::StreamPrecision, Eigen::DontAlignCols, " ", "", "", "", "[ ", " ]");
  }

codac_vector_fmt()

Eigen::IOFormat codac2::codac_vector_fmt ( )

inline

Provides an Eigen IOFormat for formatting column vectors.

This format prints elements separated by semicolons, with brackets around the entire vector.

Returns

An Eigen::IOFormat configured for column vector formatting.

  {
    return Eigen::IOFormat(Eigen::StreamPrecision, Eigen::DontAlignCols, "", " ; ", "", "", "[ ", " ]");
  }

codac_matrix_fmt()

Eigen::IOFormat codac2::codac_matrix_fmt ( )

inline

Provides an Eigen IOFormat for formatting matrices.

This format prints elements separated by commas, rows separated by new lines, and brackets around the entire matrix.

Returns

An Eigen::IOFormat configured for matrix formatting.

  {
    return Eigen::IOFormat(Eigen::StreamPrecision, 0, " , ", "\n", "[ ", " ]", "[", "]");
  }

operator<<() [10/12]

std::ostream & codac2::operator<< ( std::ostream & os, const Matrix & x )

inline

Stream output operator for Matrix objects.

Parameters

os	The output stream to write to.
x	The matrix whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    os << x.format(codac_matrix_fmt());
    return os;
  }

operator<<() [11/12]

std::ostream & codac2::operator<< ( std::ostream & os, const Row & x )

inline

Stream output operator for Row objects.

Parameters

os	The output stream to write to.
x	The row whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    os << x.format(codac_row_fmt());
    return os;
  }

operator<<() [12/12]

std::ostream & codac2::operator<< ( std::ostream & os, const Vector & x )

inline

Stream output operator for Vector objects.

Parameters

os	The output stream to write to.
x	The vector whose contents are to be printed.

Returns

A reference to the modified output stream.

  {
    os << x.format(codac_vector_fmt());
    return os;
  }

serialize() [1/4]

template<typename T>
requires (std::is_trivially_copyable_v<T>)

void codac2::serialize ( std::ostream & f, const T & x )

inline

Writes the binary representation of a trivially copyable object to the given output stream.

Binary structure:
[raw memory of x]

Parameters

f	output stream
x	object/variable to be serialized

    {
      f.write(reinterpret_cast<const char*>(&x), sizeof(T));
    }

deserialize() [1/4]

template<typename T>
requires (std::is_trivially_copyable_v<T>)

void codac2::deserialize ( std::istream & f, T & x )

inline

Reads the binary representation of a trivially copyable object from the given input stream.

Binary structure:
[raw memory of x]

Parameters

f	input stream
x	object to be deserialized

    {
      f.read(reinterpret_cast<char*>(&x), sizeof(T));
    }

serialize() [2/4]

void codac2::serialize ( std::ostream & f, const Interval & x )

inline

Writes the binary representation of an Interval object to the given output stream.

Interval binary structure:
[double_lb][double_ub]

Parameters

f	output stream
x	Interval object to be serialized

    {
      serialize(f,x.lb());
      serialize(f,x.ub());
    }

deserialize() [2/4]

void codac2::deserialize ( std::istream & f, Interval & x )

inline

Creates an Interval object from the binary representation given in an input stream.

Interval binary structure:
[double_lb][double_ub]

Parameters

f	input stream
x	Interval object to be deserialized

    {
      double lb, ub;
      deserialize(f,lb);
      deserialize(f,ub);
      x = Interval(lb,ub);
    }

serialize() [3/4]

template<typename T, int R = -1, int C = -1>

void codac2::serialize ( std::ostream & f, const Eigen::Matrix< T, R, C > & x )

inline

Writes the binary representation of an Eigen::Matrix to the given output stream. The structure can be a matrix, a row, a vector, with double or Interval components.

Matrix binary structure:
[Index_rows][Index_cols][element_00][element_01]...[element_rc]

Parameters

f	output stream
x	matrix structure to be serialized

    {
      Index r = x.rows(), c = x.cols();
      f.write(reinterpret_cast<const char*>(&r), sizeof(Index));
      f.write(reinterpret_cast<const char*>(&c), sizeof(Index));
      for(Index i = 0 ; i < r ; i++)
        for(Index j = 0 ; j < c ; j++)
          serialize(f,x(i,j));
    }

deserialize() [3/4]

template<typename T, int R = -1, int C = -1>

void codac2::deserialize ( std::istream & f, Eigen::Matrix< T, R, C > & x )

inline

Reads the binary representation of an Eigen::Matrix from the given input stream. The structure can be a matrix, a row, a vector, with double or Interval components.

Matrix binary structure:
[Index_rows][Index_cols][element_00][element_01]...[element_rc]

Parameters

f	input stream
x	matrix structure to be deserialized

    {
      Index r, c;
      f.read(reinterpret_cast<char*>(&r), sizeof(Index));
      f.read(reinterpret_cast<char*>(&c), sizeof(Index));
 
      if constexpr(R == -1 && C == -1)
        x.resize(r,c);
      else if constexpr(R == -1 || C == -1)
        x.resize(std::max(r,c));
      else
        assert_release(R == r && C == c);
 
      for(Index i = 0 ; i < r ; i++)
        for(Index j = 0 ; j < c ; j++)
          deserialize(f,x(i,j));
    }

serialize() [4/4]

template<typename T>

void codac2::serialize ( std::ostream & f, const SampledTraj< T > & x )

inline

Writes the binary representation of a SampledTraj object to the given output stream.

SampledTraj binary structure:
[nb_samples][t0][x0][t1][x1]...[tn][xn]

Parameters

f	output stream
x	SampledTraj object to be serialized

    {
      Index size = x.nb_samples();
      f.write(reinterpret_cast<const char*>(&size), sizeof(size));
 
      for(const auto& [ti, xi] : x)
      {
        serialize(f, ti);
        serialize(f, xi);
      }
    }

deserialize() [4/4]

template<typename T>

void codac2::deserialize ( std::istream & f, SampledTraj< T > & x )

inline

Reads the binary representation of a SampledTraj object from the given input stream.

SampledTraj binary structure:
[nb_samples][t0][x0][t1][x1]...[tn][xn]

Parameters

f	input stream
x	SampledTraj object to be deserialized

    {
      Index size;
      x.clear();
      f.read(reinterpret_cast<char*>(&size), sizeof(size));
 
      for(Index i = 0 ; i < size ; i++)
      {
        double ti; deserialize(f,ti);
        T xi; deserialize(f,xi);
        x.set(xi,ti);
      }
    }

operator+() [6/9]

template<typename T>

const SampledTraj< T > & codac2::operator+ ( const SampledTraj< T > & x1 )

inline

\(x1(\cdot)\)

Parameters

x1

Returns

trajectory output

                                                                   {
    return x1;
  }

operator+() [7/9]

template<typename T>

SampledTraj< T > codac2::operator+ ( const SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

\(x_1(\cdot)+x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator+() [8/9]

template<typename T, typename Q>

SampledTraj< T > codac2::operator+ ( const SampledTraj< T > & x1, const Q & x2 )

inline

\(x_1(\cdot)+x_2\)

Parameters

x1
x2

Returns

trajectory output

operator+() [9/9]

template<typename T, typename Q>

SampledTraj< T > codac2::operator+ ( const Q & x1, const SampledTraj< T > & x2 )

inline

\(x+x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator+=() [1/2]

template<typename T, typename Q>

SampledTraj< T > & codac2::operator+= ( SampledTraj< T > & x1, const Q & x2 )

inline

Operates +=.

Parameters

x1
x2

Returns

updated output

operator+=() [2/2]

template<typename T>

SampledTraj< T > & codac2::operator+= ( SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

Operates +=.

Parameters

x1
x2

Returns

updated output

operator-() [4/7]

template<typename T>

SampledTraj< T > codac2::operator- ( const SampledTraj< T > & x1 )

inline

\(-x_1(\cdot)\)

Parameters

x1

Returns

trajectory output

                                                            {
    return -1.*x1;
  }

operator-() [5/7]

template<typename T>

SampledTraj< T > codac2::operator- ( const SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

\(x_1(\cdot)-x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator-() [6/7]

template<typename T, typename Q>

SampledTraj< T > codac2::operator- ( const SampledTraj< T > & x1, const Q & x2 )

inline

\(x_1(\cdot)-x_2\)

Parameters

x1
x2

Returns

trajectory output

operator-() [7/7]

template<typename T, typename Q>

SampledTraj< T > codac2::operator- ( const Q & x1, const SampledTraj< T > & x2 )

inline

\(x-x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator-=() [1/2]

template<typename T, typename Q>

SampledTraj< T > & codac2::operator-= ( SampledTraj< T > & x1, const Q & x2 )

inline

Operates -=.

Parameters

x1
x2

Returns

updated output

operator-=() [2/2]

template<typename T>

SampledTraj< T > & codac2::operator-= ( SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

Operates -=.

Parameters

x1
x2

Returns

updated output

operator*() [4/9]

template<typename T>
requires (!std::is_same_v<T,double>)

SampledTraj< T > codac2::operator* ( double x1, const SampledTraj< T > & x2 )

inline

\(x_1\cdot x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator*() [5/9]

template<typename T>
requires (!std::is_same_v<T,double>)

SampledTraj< T > codac2::operator* ( const SampledTraj< T > & x1, double x2 )

inline

\(x_1(\cdot)\cdot x_2\)

Parameters

x1
x2

Returns

trajectory output

operator*() [6/9]

template<typename T>

SampledTraj< T > codac2::operator* ( const SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

\(x_1(\cdot)\cdot x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator*() [7/9]

template<typename T, typename Q>

SampledTraj< T > codac2::operator* ( const SampledTraj< T > & x1, const Q & x2 )

inline

\(x_1(\cdot)\cdot x_2\)

Parameters

x1
x2

Returns

trajectory output

operator*() [8/9]

template<typename T, typename Q>

SampledTraj< T > codac2::operator* ( const Q & x1, const SampledTraj< T > & x2 )

inline

\(x\cdot x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator*() [9/9]

SampledTraj< Vector > codac2::operator* ( const SampledTraj< Matrix > & x1, const SampledTraj< Vector > & x2 )

inline

\(x_1(\cdot)\cdot x_2\)

Parameters

x1
x2

Returns

trajectory output

operator*=() [1/2]

template<typename T, typename Q>

SampledTraj< T > & codac2::operator*= ( SampledTraj< T > & x1, const Q & x2 )

inline

Operates *=.

Parameters

x1
x2

Returns

updated output

operator*=() [2/2]

template<typename T>

SampledTraj< T > & codac2::operator*= ( SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

Operates *=.

Parameters

x1
x2

Returns

updated output

operator/() [4/7]

template<typename T>
requires (!std::is_same_v<T,double>)

SampledTraj< T > codac2::operator/ ( const SampledTraj< T > & x1, double x2 )

inline

\(x_2(\cdot)/x_1\)

Parameters

x1
x2

Returns

trajectory output

operator/() [5/7]

template<typename T>

SampledTraj< T > codac2::operator/ ( const SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

\(x_1(\cdot)/x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator/() [6/7]

template<typename T, typename Q>

SampledTraj< T > codac2::operator/ ( const SampledTraj< T > & x1, const Q & x2 )

inline

\(x_1(\cdot)/x_2\)

Parameters

x1
x2

Returns

trajectory output

operator/() [7/7]

template<typename T, typename Q>

SampledTraj< T > codac2::operator/ ( const Q & x1, const SampledTraj< T > & x2 )

inline

\(x/x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

operator/=() [1/2]

template<typename T, typename Q>

SampledTraj< T > & codac2::operator/= ( SampledTraj< T > & x1, const Q & x2 )

inline

Operates /=.

Parameters

x1
x2

Returns

updated output

operator/=() [2/2]

template<typename T>

SampledTraj< T > & codac2::operator/= ( SampledTraj< T > & x1, const SampledTraj< T > & x2 )

inline

Operates /=.

Parameters

x1
x2

Returns

updated output

sqr() [2/2]

SampledTraj< double > codac2::sqr

(

const SampledTraj< double > &

x1

)

\(x^2(\cdot)\)

Parameters

x1

Returns

trajectory output

sqrt() [2/2]

SampledTraj< double > codac2::sqrt

(

const SampledTraj< double > &

x1

)

\(\sqrt{x_1(\cdot)}\)

Parameters

x1

Returns

trajectory output

pow() [4/5]

SampledTraj< double > codac2::pow	(	const SampledTraj< double > &	x1,
		int	x2 )

\(x^x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

pow() [5/5]

SampledTraj< double > codac2::pow	(	const SampledTraj< double > &	x1,
		double	x2 )

\(x^x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

root() [2/2]

SampledTraj< double > codac2::root	(	const SampledTraj< double > &	x1,
		int	x2 )

\(x^x_2(\cdot)\)

Parameters

x1
x2

Returns

trajectory output

exp() [2/2]

SampledTraj< double > codac2::exp

(

const SampledTraj< double > &

x1

)

\(\exp(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

log() [2/2]

SampledTraj< double > codac2::log

(

const SampledTraj< double > &

x1

)

\(\log(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

cos() [2/2]

SampledTraj< double > codac2::cos

(

const SampledTraj< double > &

x1

)

\(\cos(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

sin() [2/2]

SampledTraj< double > codac2::sin

(

const SampledTraj< double > &

x1

)

\(\sin(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

tan() [2/2]

SampledTraj< double > codac2::tan

(

const SampledTraj< double > &

x1

)

\(\tan(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

acos() [2/2]

SampledTraj< double > codac2::acos

(

const SampledTraj< double > &

x1

)

\(\arccos(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

asin() [2/2]

SampledTraj< double > codac2::asin

(

const SampledTraj< double > &

x1

)

\(\arcsin(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

atan() [2/2]

SampledTraj< double > codac2::atan

(

const SampledTraj< double > &

x1

)

\(\arctan(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

atan2() [2/4]

SampledTraj< double > codac2::atan2	(	const SampledTraj< double > &	x1,
		const SampledTraj< double > &	x2 )

\(\mathrm{arctan2}(x_1(\cdot),x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

atan2() [3/4]

SampledTraj< double > codac2::atan2	(	const SampledTraj< double > &	x1,
		double	x2 )

\(\mathrm{arctan2}(x_1(\cdot),x_2)\)

Parameters

x1
x2

Returns

trajectory output

atan2() [4/4]

SampledTraj< double > codac2::atan2	(	double	x1,
		const SampledTraj< double > &	x2 )

\(\mathrm{arctan2}(x_1, x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

cosh() [2/2]

SampledTraj< double > codac2::cosh

(

const SampledTraj< double > &

x1

)

\(\cosh(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

sinh() [2/2]

SampledTraj< double > codac2::sinh

(

const SampledTraj< double > &

x1

)

\(\sinh(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

tanh() [2/2]

SampledTraj< double > codac2::tanh

(

const SampledTraj< double > &

x1

)

\(\tanh(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

acosh() [2/2]

SampledTraj< double > codac2::acosh

(

const SampledTraj< double > &

x1

)

\(\mathrm{arccosh}(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

asinh() [2/2]

SampledTraj< double > codac2::asinh

(

const SampledTraj< double > &

x1

)

\(\mathrm{arcsinh}(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

atanh() [2/2]

SampledTraj< double > codac2::atanh

(

const SampledTraj< double > &

x1

)

\(\mathrm{arctanh}(x_1(\cdot))\)

Parameters

x1

Returns

trajectory output

abs() [3/3]

SampledTraj< double > codac2::abs

(

const SampledTraj< double > &

x1

)

\(\mid x_1(\cdot)\mid\)

Parameters

x1

Returns

trajectory output

min() [2/4]

SampledTraj< double > codac2::min	(	const SampledTraj< double > &	x1,
		const SampledTraj< double > &	x2 )

\(\min(x_1(\cdot),x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

min() [3/4]

SampledTraj< double > codac2::min	(	const SampledTraj< double > &	x1,
		double	x2 )

\(\min(x_1(\cdot),x_2)\)

Parameters

x1
x2

Returns

trajectory output

min() [4/4]

SampledTraj< double > codac2::min	(	double	x1,
		const SampledTraj< double > &	x2 )

\(\min(x_1, x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

max() [2/4]

SampledTraj< double > codac2::max	(	const SampledTraj< double > &	x1,
		const SampledTraj< double > &	x2 )

\(\max(x_1(\cdot),x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

max() [3/4]

SampledTraj< double > codac2::max	(	const SampledTraj< double > &	x1,
		double	x2 )

\(\max(x_1(\cdot),x_2)\)

Parameters

x1
x2

Returns

trajectory output

max() [4/4]

SampledTraj< double > codac2::max	(	double	x1,
		const SampledTraj< double > &	x2 )

\(\max(x_1, x_2(\cdot))\)

Parameters

x1
x2

Returns

trajectory output

to_capd() [1/3]

capd::Interval codac2::to_capd

(

const codac2::Interval &

x

)

Converts a Codac Interval object into an CAPD Interval object.

Parameters

x	the Codac Interval to be converted

Returns

the converted CAPD Interval

to_codac() [1/3]

codac2::Interval codac2::to_codac

(

const capd::Interval &

x

)

Converts an CAPD Interval object into a Codac Interval object.

Parameters

x	the CAPD Interval to be converted

Returns

the converted Codac Interval

to_capd() [2/3]

capd::IVector codac2::to_capd

(

const codac2::IntervalVector &

x

)

Converts a Codac IntervalVector object into an CAPD IVector object.

Parameters

x	the Codac IntervalVector to be converted

Returns

the converted CAPD IVector

to_codac() [2/3]

codac2::IntervalVector codac2::to_codac

(

const capd::IVector &

x

)

Converts an CAPD IVector object into a Codac IntervalVector object.

Parameters

x	the CAPD IVector to be converted

Returns

the converted Codac IntervalVector

to_capd() [3/3]

capd::IMatrix codac2::to_capd

(

const codac2::IntervalMatrix &

x

)

Converts a Codac IntervalMatrix object into an CAPD IMatrix object.

Parameters

x	the Codac IntervalMatrix to be converted

Returns

the converted CAPD IMatrix

to_codac() [3/3]

codac2::IntervalMatrix codac2::to_codac

(

const capd::IMatrix &

x

)

Converts an CAPD IMatrix object into a Codac IntervalMatrix object.

Parameters

x	the CAPD IMatrix to be converted

Returns

the converted Codac IntervalMatrix

draw_while_paving() [1/4]

void codac2::draw_while_paving	(	const IntervalVector &	x0,
		std::shared_ptr< const CtcBase< IntervalVector > >	c,
		double	eps,
		std::shared_ptr< Figure2D >	fig = nullptr )

Draws the paving while it is being computed.

Parameters

x0	Initial box
c	Contractor to use
eps	Accuracy of the paving algorithm (the undefined boxes will have their max_diam <= eps)
fig	Figure to draw on (optionnal, if not provided, the default figure will be used)

draw_while_paving() [2/4]

void codac2::draw_while_paving	(	const IntervalVector &	x0,
		const CtcBase< IntervalVector > &	c,
		double	eps,
		std::shared_ptr< Figure2D >	fig = nullptr )

Draws the paving while it is being computed.

Parameters

x0	Initial box
c	Contractor to use
eps	Accuracy of the paving algorithm (the undefined boxes will have their max_diam <= eps)
fig	Figure to draw on (optionnal, if not provided, the default figure will be used)

draw_while_paving() [3/4]

void codac2::draw_while_paving	(	const IntervalVector &	x0,
		std::shared_ptr< const SepBase >	s,
		double	eps,
		std::shared_ptr< Figure2D >	fig = nullptr )

Draws the paving while it is being computed.

Parameters

x0	Initial box
s	Separator to use
eps	Accuracy of the paving algorithm (the undefined boxes will have their max_diam <= eps)
fig	Figure to draw on (optionnal, if not provided, the default figure will be used)

draw_while_paving() [4/4]

void codac2::draw_while_paving	(	const IntervalVector &	x0,
		const SepBase &	s,
		double	eps,
		std::shared_ptr< Figure2D >	fig = nullptr )

Draws the paving while it is being computed.

Parameters

x0	Initial box
s	Separator to use
eps	Accuracy of the paving algorithm (the undefined boxes will have their max_diam <= eps)
fig	Figure to draw on (optionnal, if not provided, the default figure will be used)