CAPD interface for interval library cxsc. More...

#include <capd/cxsc/Interval.h>

Public Types
typedef ::cxsc::interval	BaseInterval

typedef double	BoundType

typedef capd::intervals::IntervalError< BoundType >	IntervalError

Public Member Functions
	Interval ()

	Interval (const Interval &A_iv)
	copying constructor More...

	Interval (const BoundType &A_scalar)
	constructor from any class that can be coverted to BoundType More...

	Interval (const BoundType &A_left, const BoundType &A_right)
	constructor from any class that can be coverted to BoundType More...

	Interval (const BaseInterval &interval)

	Interval (const char left[], const char right[])

	Interval (const std::string &left, const std::string &right)

BaseInterval &	getBaseInterval ()
	returns reference to base (cxsc) interval More...

const BaseInterval &	getBaseInterval () const
	returns const reference to base (cxsc) interval More...

BoundType	inf () const
	< returns the left end of the interval More...

BoundType	sup () const
	< returns the right end of the interval More...

BoundType	leftBound () const
	< returns the left end of the interval More...

BoundType	rightBound () const
	< returns the right end of the interval More...

void	setLeftBound (const BoundType &A_left)

void	setRightBound (const BoundType &A_right)

Interval	left () const
	< returns interval containing left end More...

Interval	right () const
	< returns interval containing right end More...

template<typename T_Scalar >
bool	contains (const T_Scalar &A_X) const
	< checks if interval contains given point X More...

bool	contains (const Interval &A_iv) const
	< checks if interval contains given interval iv More...

template<typename T_Scalar >
bool	containsInInterior (const T_Scalar &A_X) const
	< checks if interval contains in interior given point X More...

bool	containsInInterior (const Interval &A_iv) const
	< checks if interval contains in interior given interval iv More...

bool	subset (const Interval &A_iv) const
	< checks if interval is subset of iv More...

bool	subsetInterior (const Interval &A_iv) const
	< checks if interval is subset of interior of iv More...

BoundType	midPoint () const
	< returns middle point of interval More...

Interval	mid () const
	< returns middle point of interval More...

Interval	abs () const
	<Returns the interval of absolute values of this interval, i.e. More...

void	split (Interval &A_rMid, Interval &A_rRemainder) const
	Splits interval into the form mid + remainder, where mid - is middle point. More...

void	split (BoundType &A_rMid, Interval &A_rRemainder) const

void	split (Interval &r)

Interval &	operator= (const Interval &A_iv)

Interval &	operator= (const BoundType &A_x)

Interval &	operator+= (const Interval &A_iv)

Interval &	operator-= (const Interval &A_iv)

Interval &	operator*= (const Interval &A_iv)

Interval &	operator/= (const Interval &A_iv)

Static Public Member Functions
static Interval	pi ()
	returns pi constant More...

static Interval	euler ()
	returns euler constant More...

Protected Types
typedef ::cxsc::real	Real

Protected Member Functions
	Interval (const Real &A_scalar)

	Interval (const Real &left, const Real &right)

Protected Attributes
BaseInterval	m_interval

Friends
BoundType	leftBound (const Interval &x)

BoundType	rightBound (const Interval &x)

BoundType	inf (const Interval &x)

BoundType	sup (const Interval &x)

Interval	left (const Interval &x)

Interval	right (const Interval &x)

bool	subset (const Interval &A_iv1, const Interval &A_iv2)

bool	subsetInterior (const Interval &A_iv1, const Interval &A_iv2)

bool	operator== (const Interval &A_iv1, const Interval &A_iv2)

bool	operator<= (const Interval &A_iv1, const Interval &A_iv2)

bool	operator>= (const Interval &A_iv1, const Interval &A_iv2)

bool	operator< (const Interval &A_iv1, const Interval &A_iv2)

bool	operator> (const Interval &A_iv1, const Interval &A_iv2)

bool	operator!= (const Interval &A_iv1, const Interval &A_iv2)

bool	operator== (const Interval &A_iVal1, const BoundType &A_Val2)
	operator == (interval, scalar) More...

bool	operator== (const BoundType &A_Val1, const Interval &A_iVal2)
	operator == (scalar, interval) More...

bool	operator!= (const Interval &A_iVal1, const BoundType &A_Val2)
	operator != (interval, scalar) More...

bool	operator!= (const BoundType &A_Val1, const Interval &A_iVal2)
	operator != (scalar, interval) More...

bool	operator> (const Interval &A_iVal1, const BoundType &A_Val2)
	operator > (interval, scalar) More...

bool	operator> (const BoundType &A_Val1, const Interval &A_iVal2)
	operator > (scalar, interval) More...

bool	operator>= (const Interval &A_iVal1, const BoundType &A_Val2)
	operator >= (interval, scalar) More...

bool	operator>= (const BoundType &A_Val1, const Interval &A_iVal2)
	operator >= (scalar, interval) More...

bool	operator< (const Interval &A_iVal1, const BoundType &A_Val2)
	operator < (interval, scalar) More...

bool	operator< (const BoundType &A_Val1, const Interval &A_iVal2)
	operator < (scalar, interval) More...

bool	operator<= (const Interval &A_iVal1, const BoundType &A_Val2)
	operator <= (interval, scalar) More...

bool	operator<= (const BoundType &A_Val1, const Interval &A_iVal2)
	operator <= (scalar, interval) More...

Interval	operator- (const Interval &A_iv)

Interval	operator+ (const Interval &A_iv1, const Interval &A_iv2)

Interval	operator- (const Interval &A_iv1, const Interval &A_iv2)

Interval	operator* (const Interval &A_iv1, const Interval &A_iv2)

Interval	operator/ (const Interval &A_iv1, const Interval &A_iv2)

Interval	operator^ (const Interval &A_iv1, int i)

Interval	operator+ (const Interval &A_iVal, const BoundType &A_x)
	operator + (interval, scalar) More...

Interval	operator+ (const BoundType &A_x, const Interval &A_iVal)
	operator + (scalar, interval) More...

Interval	operator- (const Interval &A_iVal, const BoundType &A_x)
	operator - (interval, scalar) More...

Interval	operator- (const BoundType &A_x, const Interval &A_iVal)
	operator - (scalar, interval) More...

Interval	operator* (const Interval &A_iVal, const BoundType &A_x)
	operator * (interval, scalar) More...

Interval	operator* (const BoundType &A_x, const Interval &A_iVal)
	operator * (scalar, interval) More...

Interval	operator/ (const Interval &A_iVal, const BoundType &A_x)
	operator / (scalar, interval) More...

Interval	operator/ (const BoundType &A_x, const Interval &A_iVal)
	operator / (interval, scalar) More...

std::ostream &	operator<< (std::ostream &s, const Interval &A_iv)

std::istream &	operator>> (std::istream &s, Interval &A_iv)

Interval	diam (const Interval &ix)

BoundType	width (const Interval &ix)

Interval	mid (const Interval &A_iv)

bool	intersection (const Interval &A_iv1, const Interval &A_iv2, Interval &A_rIntersection)
	Intersection of two intervals. More...

Interval	intervalHull (const Interval &ix, const Interval &iy)
	returns an interval containing ix and iy More...

void	split (Interval &A_iv, Interval &A_rMid, BoundType &A_diam)

void	split (Interval &A_rIv, BoundType &A_diam)

bool	isSingular (const Interval &A_x)

Interval	power (const Interval &x, int n)

Interval	power (const Interval &a, const Interval &b)

Interval	sqrt (const Interval &x)

Interval	sin (const Interval &x)

Interval	cos (const Interval &x)

Interval	tan (const Interval &x)

Interval	cot (const Interval &x)

Interval	atan (const Interval &x)

Interval	asin (const Interval &x)

Interval	acos (const Interval &x)

Interval	sinh (const Interval &x)

Interval	cosh (const Interval &x)

Interval	tanh (const Interval &x)

Interval	coth (const Interval &x)

Interval	exp (const Interval &x)

Interval	log (const Interval &x)

Interval	sqr (const Interval &x)

Interval	nonnegativePart (const Interval &iv)
	returns nonnegative part of interval More...

Interval	ball (const Interval &iv, const Interval &r)
	Ball with center iv and radius r. More...

Interval	ball (const Interval &iv, const BoundType &r)
	Ball with center iv and radius r. More...

Interval	solveAffineInclusion (const Interval &a, const Interval &p, const Interval &c)
	solves inclusion a+[0,t]*p\subset c for t More...

Interval	solveAffineInclusion (const Interval &a, const Interval &p, const Interval &c, int &dir)
	solves inclusion a+[0,t]*p\subset c for t More...

Detailed Description

CAPD interface for interval library cxsc.

It works as an adapter for the other CAPD routines. It has exactly the same interface as native CAPD interval and it just calls the corresponding functions from the cxsc.

To prevent misusing and enable easy interchange of a different interval implementation the cxsc interval T is not a base class of the Interval class and can be accesed only by getBaseInterval() method.

Member Typedef Documentation

◆ BaseInterval

typedef ::cxsc::interval capd::cxsc::Interval::BaseInterval

◆ BoundType

typedef double capd::cxsc::Interval::BoundType

◆ IntervalError

typedef capd::intervals::IntervalError<BoundType> capd::cxsc::Interval::IntervalError

◆ Real

typedef ::cxsc::real capd::cxsc::Interval::Real

protected

Constructor & Destructor Documentation

◆ Interval() [1/9]

capd::cxsc::Interval::Interval ( const Real & A_scalar )

inlineprotected

◆ Interval() [2/9]

capd::cxsc::Interval::Interval	(	const Real &	left,
		const Real &	right
	)

inlineprotected

◆ Interval() [3/9]

capd::cxsc::Interval::Interval ( )

inline

◆ Interval() [4/9]

capd::cxsc::Interval::Interval ( const Interval & A_iv )

inline

copying constructor

◆ Interval() [5/9]

capd::cxsc::Interval::Interval ( const BoundType & A_scalar )

inline

constructor from any class that can be coverted to BoundType

◆ Interval() [6/9]

capd::cxsc::Interval::Interval	(	const BoundType &	A_left,
		const BoundType &	A_right
	)

inline

constructor from any class that can be coverted to BoundType

◆ Interval() [7/9]

capd::cxsc::Interval::Interval ( const BaseInterval & interval )

inline

◆ Interval() [8/9]

capd::cxsc::Interval::Interval	(	const char	left[],
		const char	right[]
	)

inline

◆ Interval() [9/9]

capd::cxsc::Interval::Interval	(	const std::string &	left,
		const std::string &	right
	)

inline

Member Function Documentation

◆ abs()

Interval capd::cxsc::Interval::abs ( ) const

inline

<Returns the interval of absolute values of this interval, i.e.

◆ contains() [1/2]

bool capd::cxsc::Interval::contains ( const Interval & A_iv ) const

inline

< checks if interval contains given interval iv

◆ contains() [2/2]

template<typename T_Scalar >

bool capd::cxsc::Interval::contains ( const T_Scalar & A_X ) const

inline

< checks if interval contains given point X

◆ containsInInterior() [1/2]

bool capd::cxsc::Interval::containsInInterior ( const Interval & A_iv ) const

inline

< checks if interval contains in interior given interval iv

◆ containsInInterior() [2/2]

template<typename T_Scalar >

bool capd::cxsc::Interval::containsInInterior ( const T_Scalar & A_X ) const

inline

< checks if interval contains in interior given point X

◆ euler()

Interval capd::cxsc::Interval::euler ( )

inlinestatic

returns euler constant

◆ getBaseInterval() [1/2]

BaseInterval& capd::cxsc::Interval::getBaseInterval ( )

inline

returns reference to base (cxsc) interval

◆ getBaseInterval() [2/2]

const BaseInterval& capd::cxsc::Interval::getBaseInterval ( ) const

inline

returns const reference to base (cxsc) interval

◆ inf()

BoundType capd::cxsc::Interval::inf ( ) const

inline

< returns the left end of the interval

◆ left()

Interval capd::cxsc::Interval::left ( ) const

inline

< returns interval containing left end

◆ leftBound()

BoundType capd::cxsc::Interval::leftBound ( ) const

inline

< returns the left end of the interval

◆ mid()

Interval capd::cxsc::Interval::mid ( ) const

inline

< returns middle point of interval

◆ midPoint()

BoundType capd::cxsc::Interval::midPoint ( ) const

inline

< returns middle point of interval

◆ operator*=()

Interval& capd::cxsc::Interval::operator*= ( const Interval & A_iv )

inline

◆ operator+=()

Interval& capd::cxsc::Interval::operator+= ( const Interval & A_iv )

inline

◆ operator-=()

Interval& capd::cxsc::Interval::operator-= ( const Interval & A_iv )

inline

◆ operator/=()

Interval& capd::cxsc::Interval::operator/= ( const Interval & A_iv )

inline

◆ operator=() [1/2]

Interval& capd::cxsc::Interval::operator= ( const BoundType & A_x )

inline

◆ operator=() [2/2]

Interval& capd::cxsc::Interval::operator= ( const Interval & A_iv )

inline

◆ pi()

Interval capd::cxsc::Interval::pi ( )

inlinestatic

returns pi constant

◆ right()

Interval capd::cxsc::Interval::right ( ) const

inline

< returns interval containing right end

◆ rightBound()

BoundType capd::cxsc::Interval::rightBound ( ) const

inline

< returns the right end of the interval

◆ setLeftBound()

void capd::cxsc::Interval::setLeftBound ( const BoundType & A_left )

inline

◆ setRightBound()

void capd::cxsc::Interval::setRightBound ( const BoundType & A_right )

inline

◆ split() [1/3]

void capd::cxsc::Interval::split	(	BoundType &	A_rMid,
		Interval &	A_rRemainder
	)		const

inline

◆ split() [2/3]

void capd::cxsc::Interval::split	(	Interval &	A_rMid,
		Interval &	A_rRemainder
	)		const

inline

Splits interval into the form mid + remainder, where mid - is middle point.

◆ split() [3/3]

void capd::cxsc::Interval::split ( Interval & r )

inline

◆ subset()

bool capd::cxsc::Interval::subset ( const Interval & A_iv ) const

inline

< checks if interval is subset of iv

◆ subsetInterior()

bool capd::cxsc::Interval::subsetInterior ( const Interval & A_iv ) const

inline

< checks if interval is subset of interior of iv

◆ sup()

BoundType capd::cxsc::Interval::sup ( ) const

inline

< returns the right end of the interval

Friends And Related Function Documentation

◆ acos

Interval acos ( const Interval & x )

friend

◆ asin

Interval asin ( const Interval & x )

friend

◆ atan

Interval atan ( const Interval & x )

friend

◆ ball [1/2]

Interval ball	(	const Interval &	iv,
		const BoundType &	r
	)

friend

Ball with center iv and radius r.

◆ ball [2/2]

Interval ball	(	const Interval &	iv,
		const Interval &	r
	)

friend

Ball with center iv and radius r.

◆ cos

Interval cos ( const Interval & x )

friend

◆ cosh

Interval cosh ( const Interval & x )

friend

◆ cot

Interval cot ( const Interval & x )

friend

◆ coth

Interval coth ( const Interval & x )

friend

◆ diam

Interval diam ( const Interval & ix )

friend

◆ exp

Interval exp ( const Interval & x )

friend

◆ inf

BoundType inf ( const Interval & x )

friend

Parameters

x	returns the left end of the interval

◆ intersection

bool intersection	(	const Interval &	A_iv1,
		const Interval &	A_iv2,
		Interval &	A_rIntersection
	)

friend

Intersection of two intervals.

◆ intervalHull

Interval intervalHull	(	const Interval &	ix,
		const Interval &	iy
	)

friend

returns an interval containing ix and iy

◆ isSingular

bool isSingular ( const Interval & A_x )

friend

◆ left

Interval left ( const Interval & x )

friend

Parameters

x	returns interval containing left end

◆ leftBound

BoundType leftBound ( const Interval & x )

friend

Parameters

x	returns the left end of the interval

◆ log

Interval log ( const Interval & x )

friend

◆ mid

Interval mid ( const Interval & A_iv )

friend

◆ nonnegativePart

Interval nonnegativePart ( const Interval & iv )

friend

returns nonnegative part of interval

Remarks: if nonnegative part is empty throws exception

◆ operator!= [1/3]

bool operator!=	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator != (scalar, interval)

◆ operator!= [2/3]

bool operator!=	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator!= [3/3]

bool operator!=	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator != (interval, scalar)

◆ operator* [1/3]

Interval operator*	(	const BoundType &	A_x,
		const Interval &	A_iVal
	)

friend

operator * (scalar, interval)

◆ operator* [2/3]

Interval operator*	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator* [3/3]

Interval operator*	(	const Interval &	A_iVal,
		const BoundType &	A_x
	)

friend

operator * (interval, scalar)

◆ operator+ [1/3]

Interval operator+	(	const BoundType &	A_x,
		const Interval &	A_iVal
	)

friend

operator + (scalar, interval)

◆ operator+ [2/3]

Interval operator+	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator+ [3/3]

Interval operator+	(	const Interval &	A_iVal,
		const BoundType &	A_x
	)

friend

operator + (interval, scalar)

◆ operator- [1/4]

Interval operator-	(	const BoundType &	A_x,
		const Interval &	A_iVal
	)

friend

operator - (scalar, interval)

◆ operator- [2/4]

Interval operator- ( const Interval & A_iv )

friend

◆ operator- [3/4]

Interval operator-	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator- [4/4]

Interval operator-	(	const Interval &	A_iVal,
		const BoundType &	A_x
	)

friend

operator - (interval, scalar)

◆ operator/ [1/3]

Interval operator/	(	const BoundType &	A_x,
		const Interval &	A_iVal
	)

friend

operator / (interval, scalar)

◆ operator/ [2/3]

Interval operator/	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator/ [3/3]

Interval operator/	(	const Interval &	A_iVal,
		const BoundType &	A_x
	)

friend

operator / (scalar, interval)

◆ operator< [1/3]

bool operator<	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator < (scalar, interval)

◆ operator< [2/3]

bool operator<	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator< [3/3]

bool operator<	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator < (interval, scalar)

◆ operator<<

std::ostream& operator<<	(	std::ostream &	s,
		const Interval &	A_iv
	)

friend

◆ operator<= [1/3]

bool operator<=	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator <= (scalar, interval)

◆ operator<= [2/3]

bool operator<=	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator<= [3/3]

bool operator<=	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator <= (interval, scalar)

◆ operator== [1/3]

bool operator==	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator == (scalar, interval)

◆ operator== [2/3]

bool operator==	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator== [3/3]

bool operator==	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator == (interval, scalar)

◆ operator> [1/3]

bool operator>	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator > (scalar, interval)

◆ operator> [2/3]

bool operator>	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator> [3/3]

bool operator>	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator > (interval, scalar)

◆ operator>= [1/3]

bool operator>=	(	const BoundType &	A_Val1,
		const Interval &	A_iVal2
	)

friend

operator >= (scalar, interval)

◆ operator>= [2/3]

bool operator>=	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ operator>= [3/3]

bool operator>=	(	const Interval &	A_iVal1,
		const BoundType &	A_Val2
	)

friend

operator >= (interval, scalar)

◆ operator>>

std::istream& operator>>	(	std::istream &	s,
		Interval &	A_iv
	)

friend

◆ operator^

Interval operator^	(	const Interval &	A_iv1,
		int	i
	)

friend

◆ power [1/2]

Interval power	(	const Interval &	a,
		const Interval &	b
	)

friend

◆ power [2/2]

Interval power	(	const Interval &	x,
		int	n
	)

friend

◆ right

Interval right ( const Interval & x )

friend

Parameters

x	returns interval containing right end

◆ rightBound

BoundType rightBound ( const Interval & x )

friend

Parameters

x	returns the right end of the interval

◆ sin

Interval sin ( const Interval & x )

friend

◆ sinh

Interval sinh ( const Interval & x )

friend

◆ solveAffineInclusion [1/2]

Interval solveAffineInclusion	(	const Interval &	a,
		const Interval &	p,
		const Interval &	c
	)

friend

solves inclusion a+[0,t]*p\subset c for t

◆ solveAffineInclusion [2/2]

Interval solveAffineInclusion	(	const Interval &	a,
		const Interval &	p,
		const Interval &	c,
		int &	dir
	)

friend

solves inclusion a+[0,t]*p\subset c for t

◆ split [1/2]

void split	(	Interval &	A_iv,
		Interval &	A_rMid,
		BoundType &	A_diam
	)

friend

TODO : correct to not switch rounding

◆ split [2/2]

void split	(	Interval &	A_rIv,
		BoundType &	A_diam
	)

friend

◆ sqr

Interval sqr ( const Interval & x )

friend

◆ sqrt

Interval sqrt ( const Interval & x )

friend

◆ subset

bool subset	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ subsetInterior

bool subsetInterior	(	const Interval &	A_iv1,
		const Interval &	A_iv2
	)

friend

◆ sup

BoundType sup ( const Interval & x )

friend

Parameters

x	returns the right end of the interval

◆ tan

Interval tan ( const Interval & x )

friend

◆ tanh

Interval tanh ( const Interval & x )

friend

◆ width

BoundType width ( const Interval & ix )

friend

Member Data Documentation

◆ m_interval

BaseInterval capd::cxsc::Interval::m_interval

protected

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

capd/cxsc/Interval.h

Public Types

Public Member Functions

Static Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ BaseInterval

◆ BoundType

◆ IntervalError

◆ Real

Constructor & Destructor Documentation

◆ Interval() [1/9]

◆ Interval() [2/9]

◆ Interval() [3/9]

◆ Interval() [4/9]

◆ Interval() [5/9]

◆ Interval() [6/9]

◆ Interval() [7/9]

◆ Interval() [8/9]

◆ Interval() [9/9]

Member Function Documentation

◆ abs()

◆ contains() [1/2]

◆ contains() [2/2]

◆ containsInInterior() [1/2]

◆ containsInInterior() [2/2]

◆ euler()

◆ getBaseInterval() [1/2]

◆ getBaseInterval() [2/2]

◆ inf()

◆ left()

◆ leftBound()

◆ mid()

◆ midPoint()

◆ operator*=()

◆ operator+=()

◆ operator-=()

◆ operator/=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ pi()

◆ right()

◆ rightBound()

◆ setLeftBound()

◆ setRightBound()

◆ split() [1/3]

◆ split() [2/3]

◆ split() [3/3]

◆ subset()

◆ subsetInterior()

◆ sup()

Friends And Related Function Documentation

◆ acos

◆ asin

◆ atan

◆ ball [1/2]

◆ ball [2/2]

◆ cos

◆ cosh

◆ cot

◆ coth

◆ diam

◆ exp

◆ inf

◆ intersection

◆ intervalHull

◆ isSingular

◆ left

◆ leftBound

◆ log

◆ mid

◆ nonnegativePart

◆ operator!= [1/3]

◆ operator!= [2/3]

◆ operator!= [3/3]

◆ operator* [1/3]

◆ operator* [2/3]