#include <CGAL/Gmpfi.h>

Definition

An object of the class Gmpfi is a closed interval, with endpoints represented as Gmpfr floating-point numbers.

An interval can have finite or infinite endpoints and its meaning is straightforward. It can also have one (or both) NaN endpoint(s): this indicates that an invalid operation has been performed and that the resulting interval has no mathematical meaning.

All the operations of Gmpfi were designed in such a way that the mathematical correct result is always contained in the resulting interval.

This type is ImplicitInteroperable with Gmpfr, Gmpz, Gmpq, long, unsigned long, int, double and long double.

Is model of: FieldWithKthRoot
: RealEmbeddable

Implementation

All interval operations are performed by the MPFI library. The class Gmpfi is not reference counted, but its members are.

The default precision of Gmpfi is local to each thread and independent of the default precision of Gmpfr (in contrast to the behaviour of the MPFI and MPFR libraries, which share a default precision).

See also: CGAL::Gmpfr; CGAL::Interval_nt; CGAL::Uncertain; RealEmbeddable; FieldWithKthRoot

Related Functions
(Note that these are not member functions.)
std::istream &	operator>> (std::istream &is, Gmpfi i)
	Reads `i` from `is`.

std::ostream &	operator<< (std::ostream &os, const Gmpfi &i)
	Writes `i` to `os`, in the form `[i.inf(),i.sup()]`.

Types
typedef unspecified_type	Precision_type
	Type representing the precision (number of bits used to represent the mantissa) of a number.

Creation
All constructors accept as an optional last argument a precision, which can be used to specify the precision of the `Gmpfr` endpoints. If none is specified, the default precision will be used. As the endpoints are represented with a fixed number of bits, they may need to be rounded. In this case, the number from which the `Gmpfi` was constructed is guaranteed to be included in the constructed interval.
	Gmpfi (Precision_type p=get_default_precision())
	creates an uninitialized `Gmpfi` interval `i`.

template<class T >
	Gmpfi (const T &t, Precision_type p=get_default_precision())
	creates a `Gmpfi` initialized with the value of `t`.

	Gmpfi (const Gmpfr &left, const Gmpfr &right, Precision_type p=get_default_precision())
	creates a `Gmpfi` initialized with endpoints `left` and `right`.

template<class L , class R >
	Gmpfi (const std::pair< L, R > &endpoints, Precision_type p=get_default_precision())
	creates a `Gmpfi` initialized with endpoints `endpoints.first` and `endpoints.second`.

Operations
Gmpfr	inf () const
	Returns the smallest (or inferior) `Gmpfr` endpoint of `i`.

Gmpfr	sup () const
	Returns the largest (or superior) `Gmpfr` endpoint of `i`.

Precision_type	get_precision () const
	Returns the precision of `i`.

Gmpfi	round (Precision_type p) const
	Returns the value of the number, rounded with precision `p`.

static Precision_type	get_default_precision ()
	Returns the default precision.

static Precision_type	set_default_precision (Precision_type prec)
	Sets the default precision to `prec` and returns the old value.

Arithmetic Operations
Arithmetic operators `+`, `-`, and `/` are overloaded, but special care must be taken when applying them. The precision of an operation between two `Gmpfi`s is defined as the maximum of the operands precision and the default precision. The second operand of the former operations can be a `Gmpfi`, `Gmpfr`, `int`, `long`, `unsigned`, `unsigned long`, `Gmpz` or `Gmpq`. The precision of an operation between a `Gmpfi` and a number of another type is defined as the `Gmpfi`'s precision (even when operating with a `Gmpfr`). To specify the rounding mode and/or the precision to perform an operation, this class provides the four static functions `add`, `sub`, `mul` and `div`. Only one of them is shown here, since their interfaces are similar: When the precision is not specified in this family of functions, it is defined as in the overloaded operators.
Gmpfi	abs (Precision_type p) const
	Returns the absolute value of `i`, with precision `p`.

Gmpfi	sqrt (Precision_type p) const
	Returns the square root of `i`, with precision `p`.

Gmpfi	kthroot (int k, Precision_type p) const
	Returns the k-th root of `i`, with precision `p`.

Gmpfi	square (Precision_type p) const
	Returns the square of `i`, with precision `p`.

std::pair< double, double >	to_interval () const
	Returns an interval of doubles which contains `i`.

std::pair< double, long >	to_double_exp () const
	Returns \( (m,e)\) such that \( m \times2^e\) is the center of `i`, rounded to nearest.

std::pair< std::pair< double, double >, long >	to_interval_exp () const
	Returns \( ((m_1,m_2),e)\), such that \( [m_1 \times2^e,m_2 \times2^e]\) contains `i`.

static Gmpfi	add (const Gmpfi &a, const Gmpfi &b, Precision_type p=0)

Comparisons
The semantics of the comparison operators is the same than on `Interval_nt<Protected>`. The result of the comparison is always an `Uncertain<bool>` (this type is convertible to `bool`, but may throw an exception). If compared intervals have no common points, the result is `true` or `false`; otherwise, `Uncertain::indeterminate()` will be returned. In the same way, we can explain the semantics of `Uncertain<Comparison_result>` and `Uncertain<Sign>`. With the semantics described above, this class provides comparisons between `Gmpfi` and `Gmpfi`, `Gmpfr`, `long`, `unsigned long`, `int`, `double`, `Gmpz` and `Gmpq`. Comparison operators `==`, `!=`, `>`, `<`, `>=` and `<=` are overloaded. The class provides also functions to test efficiently some special kinds of comparisons:
bool	is_same (const Gmpfi &j) const
	Returns `true` iff left endpoints of `i` and `j` are equal and right endpoints of them are also equal.

bool	do_overlap (const Gmpfi &j) const
	Returns `true` iff `i` and `j` overlap, i.e., iff they have points in common.

Uncertain< Comparison_result >	compare (const Gmpfi &j) const
	If `i` and `j` do not overlap, this function returns the result of the comparison.

Query Functions
bool	is_point () const
	Returns `true` iff both endpoints are equal.

bool	is_nan () const
	Returns `true` iff at least one of the endpoints is `NaN`.

bool	is_inf () const
	Returns `true` iff at least one of the endpoints is plus or minus infinity.

bool	is_number () const
	Returns `true` iff `i` is a bounded interval, i.e., its endpoints are neither invalid nor infinite.

Uncertain< bool >	is_zero () const
	Returns `true` if both endpoints are zero, `false` if the interval does not contain zero and `indeterminate` otherwise.

Uncertain< bool >	is_one () const
	Returns `true` if both endpoints are one, `false` if the interval does not contain one and `indeterminate` otherwise.

Uncertain< Sign >	sign () const
	If all numbers contained in the interval have the same sign, this function returns it.

Uncertain< bool >	is_positive () const
	Returns `true` if all numbers contained in the interval are positive, false if all of them are negative or zero and `indeterminate` otherwise.

Uncertain< bool >	is_negative () const
	Returns `true` if all numbers contained in the interval are negative, false if all of them are positive or zero and `indeterminate` otherwise.

Constructor & Destructor Documentation

◆ Gmpfi() [1/3]

template<class T >

CGAL::Gmpfi::Gmpfi	(	const T &	t,
		Precision_type	p = `get_default_precision()`
	)

creates a Gmpfi initialized with the value of t.

T is Gmpfr, Gmpq, or any type from which Gmpfr can be constructed from. The rounding of the endpoints will guarantee that t is included in i.

◆ Gmpfi() [2/3]

CGAL::Gmpfi::Gmpfi	(	const Gmpfr &	left,
		const Gmpfr &	right,
		Precision_type	p = `get_default_precision()`
	)

creates a Gmpfi initialized with endpoints left and right.

The rounding of the endpoints will guarantee that ([left,right]) is included in i.

◆ Gmpfi() [3/3]

template<class L , class R >

CGAL::Gmpfi::Gmpfi	(	const std::pair< L, R > &	endpoints,
		Precision_type	p = `get_default_precision()`
	)

creates a Gmpfi initialized with endpoints endpoints.first and endpoints.second.

L and R are types from which Gmpfr can be constructed from. The rounding of the endpoints will guarantee that ([endpoints.first,endpoints.second]) is included in i.

Member Function Documentation

◆ abs()

Gmpfi CGAL::Gmpfi::abs ( Precision_type p ) const

Returns the absolute value of i, with precision p.

If p is not specified, the precision used is the maximum between i's precision and the default.

◆ compare()

Uncertain< Comparison_result > CGAL::Gmpfi::compare ( const Gmpfi & j ) const

If i and j do not overlap, this function returns the result of the comparison.

Otherwise, it returns indeterminate.

◆ is_same()

bool CGAL::Gmpfi::is_same ( const Gmpfi & j ) const

Returns true iff left endpoints of i and j are equal and right endpoints of them are also equal.

Note that this does not mean equality between i and j.

◆ kthroot()

Gmpfi CGAL::Gmpfi::kthroot	(	int	k,
		Precision_type	p
	)		const

Returns the k-th root of i, with precision p.

If p is not specified, the precision used is the maximum between i's precision and the default.

◆ sign()

Uncertain< Sign > CGAL::Gmpfi::sign ( ) const

If all numbers contained in the interval have the same sign, this function returns it.

Otherwise it returns indeterminate.

◆ sqrt()

Gmpfi CGAL::Gmpfi::sqrt ( Precision_type p ) const

Returns the square root of i, with precision p.

If p is not specified, the precision used is the maximum between i's precision and the default.

◆ square()

Gmpfi CGAL::Gmpfi::square ( Precision_type p ) const

Returns the square of i, with precision p.

If p is not specified, the precision used is the maximum between i's precision and the default.

◆ to_double_exp()

std::pair< double, long > CGAL::Gmpfi::to_double_exp ( ) const

Returns \( (m,e)\) such that \( m \times2^e\) is the center of i, rounded to nearest.

If one of the endpoints of i is NaN or infinity, then the corresponding double is returned, leaving the exponent undefined and setting the appropriate error flag.

◆ to_interval()

std::pair< double, double > CGAL::Gmpfi::to_interval ( ) const

Returns an interval of doubles which contains i.

If a rounded endpoint does not fit in a double, sets its value to plus or minus infinity and the overflow or underflow flag.

◆ to_interval_exp()

std::pair< std::pair< double, double >, long > CGAL::Gmpfi::to_interval_exp ( ) const

Returns \( ((m_1,m_2),e)\), such that \( [m_1 \times2^e,m_2 \times2^e]\) contains i.

If one of the endpoints of i is NaN or infinity, then the corresponding doubles are returned, leaving the exponent undefined and setting the appropriate error flag.

Friends And Related Function Documentation

◆ operator<<()

std::ostream & operator<<	(	std::ostream &	os,
		const Gmpfi &	i
	)

The endpoints are written according to the Gmpfr formatting.

◆ operator>>()

std::istream & operator>>	(	std::istream &	is,
		Gmpfi	i
	)

is must have the form [inf,sup], where inf and sup have valid Gmpfr input formats.

Definition

Related Functions

Types

Creation

Operations

Arithmetic Operations

Comparisons

Query Functions

Constructor & Destructor Documentation

◆ Gmpfi() [1/3]

◆ Gmpfi() [2/3]

◆ Gmpfi() [3/3]

Member Function Documentation

◆ abs()

◆ compare()

◆ is_same()

◆ kthroot()

◆ sign()

◆ sqrt()

◆ square()

◆ to_double_exp()

◆ to_interval()

◆ to_interval_exp()

Friends And Related Function Documentation

◆ operator<<()

◆ operator>>()