Intervals of two reals. More...

#include <ia.h>

Public Member Functions
	Ia ()
	Empty.

	Ia (const double &)
	Creates an interval.

	Ia (const double &, const double &)
	Creates an interval.

	Ia (const double[], int)
	Create an interval by computing the minimum and maximum values.

	Ia (const Ia &, const Ia &)
	Creates an embedding interval.

constexpr double &	operator[] (int)
	Access interval bounds.

constexpr double	operator[] (int) const
	Access interval bounds.

Ia	operator+ () const
	Overloaded.

Ia	operator- () const
	Overloaded.

Ia &	operator+= (const Ia &)
	Destructive addition.

Ia &	operator-= (const Ia &)
	Subtracts two intervals.

Ia	AngleTwoPi () const
	Caculate the minimal 2 Math::Pi periodic interval.

bool	operator> (const double &) const
	Compare interval to a real.

bool	operator< (const double &) const
	Compare interval to a real.

bool	operator< (const Ia &) const
	Compare two intervals.

bool	operator> (const Ia &) const
	Compare two intervals.

Ia	Intersection (const Ia &) const
	Compute the intersection of two intervals.

bool	Intersect (const Ia &) const
	Check if two intervals intersect.

double	Length () const
	Compute the length of the interval.

double	Center () const
	Compute the center of the interval.

double	Extent () const
	Compute the radius r of the interval [-r,r] embedding the interval.

Static Public Member Functions
static Ia	Lerp (const Ia &, const Ia &, const double &)
	Linear interpolation of two intervals.

Static Public Attributes
static const Ia	Empty
	Empty interval.

static const Ia	Null
	Null interval is zero-set.

static const Ia	Unit
	Unit interval [0,1].

static const Ia	Infinity
	Infinity interval.

Friends
Ia	operator+ (const Ia &x, const Ia &y)
	Adds two intervals.

Ia	operator- (const Ia &x, const Ia &y)
	Subtracts two intervals.

Ia	operator- (const Ia &x, const double &a)
	Translates (shift left) an interval by a real value.

Ia	operator+ (const Ia &x, const double &a)
	Translates (shift right) an interval by a real value.

Ia	operator- (const double &a, const Ia &x)
	Overloaded.

Ia	operator+ (const double &a, const Ia &x)
	Overloaded.

Ia	operator* (const Ia &x, const Ia &y)
	Multiplies two intervals.

Ia	operator/ (const Ia &x, const Ia &y)
	Divides two intervals.

Ia	operator* (const Ia &x, const double &a)
	Scales a interval by a double.

Ia	operator* (const double &a, const Ia &x)
	Left multiply, same as scaling.

Ia	operator/ (const Ia &x, const double &a)
	Left divide by a double, same as scaling.

Ia	operator/ (const double &a, const Ia &x)
	Divides a real number by an interval.

Ia	Min (const Ia &x, const Ia &y)
	Computes the minimum of two intervals.

Ia	Max (const Ia &x, const Ia &y)
	Computes the maximum of two intervals.

Ia	sqrt (const Ia &x)
	Computes the square root of an interval.

Ia	sqr (const Ia &x)
	Compute the squared interval.

Ia	abs (const Ia &x)
	Computes the absolute value of an interval.

Ia	sin (const Ia &x)
	Computes the sine of an interval.

Ia	cos (const Ia &x)
	Computes the cosine of an interval.

Ia	pow (const Ia &x, const double &n)
	Computes the power of an interval.

bool	operator== (const Ia &x, const Ia &y)
	Compares two intervals.

bool	operator!= (const Ia &x, const Ia &y)
	Compares two intervals.

std::ostream &	operator<< (std::ostream &s, const Ia &x)
	Overloaded output-stream operator.

Detailed Description

Intervals of two reals.

Most interval arithmetics operators are intensive because of the tests needed to sort the extrema of the resulting interval. Only addition and subtraction need not use tests and, therefore, are efficient.

Several functions have been implemented with the same syntax as for doubles, such as sqrt() and abs().

Constructor & Destructor Documentation

◆ Ia() [1/4]

Ia::Ia ( const double & x )

inlineexplicit

Creates an interval.

The length of the interval is 0.

Parameters

x Real.

◆ Ia() [2/4]

Ia::Ia	(	const double &	x,
		const double &	y )

inlineexplicit

Creates an interval.

Parameters

x,y	Low and high real values.

◆ Ia() [3/4]

Ia::Ia	(	const double	v[],
		int	n )

explicit

Create an interval by computing the minimum and maximum values.

Parameters

v	Array.
n	Size.

◆ Ia() [4/4]

Ia::Ia	(	const Ia &	x,
		const Ia &	y )

inlineexplicit

Creates an embedding interval.

It is the tightest embedding interval.

Parameters

x,y	Intervals.

Member Function Documentation

◆ Center()

double Ia::Center ( ) const

inline

Compute the center of the interval.

Same as:

Ia i;

double l=i.Center(); // double l=0.5*(i[1]+i[0]);

Ia::Ia

Ia()

Empty.

Definition ia.h:12

Ia::Center

double Center() const

Compute the center of the interval.

Definition ia.h:254

◆ Extent()

double Ia::Extent ( ) const

inline

Compute the radius r of the interval [-r,r] embedding the interval.

Same as:

Ia i;

double l=i.Extent(); // double l=Math::Max(fabs(i[0]),fabs(i[1]));

Ia::Extent

double Extent() const

Compute the radius r of the interval [-r,r] embedding the interval.

Definition ia.h:268

◆ Intersect()

bool Ia::Intersect ( const Ia & x ) const

Check if two intervals intersect.

Parameters

x	Other interval.

◆ Intersection()

Ia Ia::Intersection ( const Ia & x ) const

Compute the intersection of two intervals.

Parameters

x	Other interval.

◆ Length()

double Ia::Length ( ) const

inline

Compute the length of the interval.

Same as:

Ia i;

double l=i.Length(); // double l=i[1]-i[0];

Ia::Length

double Length() const

Compute the length of the interval.

Definition ia.h:240

◆ Lerp()

Ia Ia::Lerp	(	const Ia &	x,
		const Ia &	y,
		const double &	t )

static

Linear interpolation of two intervals.

Parameters

x,y	Intervals.
t	Real.

This code is equivalent but more efficient than an interpolation with overloaded operators, as coefficients are known to be within unit interval.

const double t=0.25;

Ia z=(1.0-t)*Ia(2.0,3.0)+t*Ia(-1.0,2.0); // Linear interpolation with overloaded operators

◆ operator<() [1/2]

bool Ia::operator< ( const double & x ) const

Compare interval to a real.

Check if the entire interval is superior to the real. This is equivalent to:

Ia i;
double x;
if (i[1]<x) {} // if (i<x) {}

Parameters

x Real.

◆ operator<() [2/2]

bool Ia::operator< ( const Ia & y ) const

Compare two intervals.

Parameters

y Interval.

◆ operator>() [1/2]

bool Ia::operator> ( const double & x ) const

Compare interval to a real.

Check if the entire interval is superior to the real. This is equivalent to:

Ia i;
double x;
if (i[0]>x) {} // if (i>x) {}

Parameters

x Real.

◆ operator>() [2/2]

bool Ia::operator> ( const Ia & y ) const

Compare two intervals.

Parameters

y Interval.

Friends And Related Symbol Documentation

◆ abs

Ia abs ( const Ia & x )

friend

Computes the absolute value of an interval.

Parameters

x Interval.

◆ cos

Ia cos ( const Ia & x )

friend

Computes the cosine of an interval.

Parameters

x Interval.

◆ Max

Ia Max	(	const Ia &	x,
		const Ia &	y )

friend

Computes the maximum of two intervals.

Parameters

x,y	Intervals.

◆ Min

Ia Min	(	const Ia &	x,
		const Ia &	y )

friend

Computes the minimum of two intervals.

Parameters

x,y	Intervals.

◆ operator!=

bool operator!=	(	const Ia &	x,
		const Ia &	y )

friend

Compares two intervals.

Parameters

x,y	Intervals.

◆ operator* [1/3]

Ia operator*	(	const double &	a,
		const Ia &	x )

friend

Left multiply, same as scaling.

Parameters

x	Interval.
a	Real.

◆ operator* [2/3]

Ia operator*	(	const Ia &	x,
		const double &	a )

friend

Scales a interval by a double.

Parameters

x	Interval.
a	Real.

◆ operator* [3/3]

Ia operator*	(	const Ia &	x,
		const Ia &	y )

friend

Multiplies two intervals.

Parameters

x,y	Intervals.

◆ operator+ [1/3]

Ia operator+	(	const double &	a,
		const Ia &	x )

friend

Overloaded.

Parameters

x	Interval.
a	Real.

◆ operator+ [2/3]

Ia operator+	(	const Ia &	x,
		const double &	a )

friend

Translates (shift right) an interval by a real value.

Parameters

x	Interval.
a	Real.

◆ operator+ [3/3]

Ia operator+	(	const Ia &	x,
		const Ia &	y )

friend

Adds two intervals.

Parameters

x,y	Intervals.

◆ operator- [1/3]

Ia operator-	(	const double &	a,
		const Ia &	x )

friend

Overloaded.

Parameters

x	Interval.
a	Real.

◆ operator- [2/3]

Ia operator-	(	const Ia &	x,
		const double &	a )

friend

Translates (shift left) an interval by a real value.

Parameters

x	Interval.
a	Real.

◆ operator- [3/3]

Ia operator-	(	const Ia &	x,
		const Ia &	y )

friend

Subtracts two intervals.

Parameters

x,y	Intervals.

◆ operator/ [1/3]

Ia operator/	(	const double &	a,
		const Ia &	x )

friend

Divides a real number by an interval.

Parameters

a	Real.
x	Interval.

◆ operator/ [2/3]

Ia operator/	(	const Ia &	x,
		const double &	a )

friend

Left divide by a double, same as scaling.

Parameters

x	Interval.
a	Real.

◆ operator/ [3/3]

Ia operator/	(	const Ia &	x,
		const Ia &	y )

friend

Divides two intervals.

Parameters

x,y	Intervals.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	s,
		const Ia &	x )

friend

Overloaded output-stream operator.

Parameters

s	Stream.
x	Interval.

◆ operator==

bool operator==	(	const Ia &	x,
		const Ia &	y )

friend

Compares two intervals.

Parameters

x,y	Argument intervals.

◆ pow

Ia pow	(	const Ia &	x,
		const double &	n )

friend

Computes the power of an interval.

Parameters

x	Interval.
n	Power.

◆ sin

Ia sin ( const Ia & x )

friend

Computes the sine of an interval.

See also: cos(const Ia&)

Parameters

x Interval.

◆ sqr

Ia sqr ( const Ia & x )

friend

Compute the squared interval.

This function is more efficient than multiplication.

Parameters

x Interval.

◆ sqrt

Ia sqrt ( const Ia & x )

friend

Computes the square root of an interval.

Interval should have positive bounds.

Parameters

x Interval.

Public Member Functions

Static Public Member Functions

Static Public Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ Ia() [1/4]

◆ Ia() [2/4]

◆ Ia() [3/4]

◆ Ia() [4/4]

Member Function Documentation

◆ Center()

◆ Extent()

◆ Intersect()

◆ Intersection()

◆ Length()

◆ Lerp()

◆ operator<() [1/2]

◆ operator<() [2/2]

◆ operator>() [1/2]

◆ operator>() [2/2]

Friends And Related Symbol Documentation

◆ abs

◆ cos

◆ Max

◆ Min

◆ operator!=

◆ operator* [1/3]

◆ operator* [2/3]

◆ operator* [3/3]

◆ operator+ [1/3]

◆ operator+ [2/3]

◆ operator+ [3/3]

◆ operator- [1/3]

◆ operator- [2/3]

◆ operator- [3/3]

◆ operator/ [1/3]

◆ operator/ [2/3]

◆ operator/ [3/3]

◆ operator<<

◆ operator==

◆ pow

◆ sin

◆ sqr

◆ sqrt