CGAL 5.1.3 - Number Types
CGAL::Gmpfi Class Reference

#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).

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. More...

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

## 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. More...

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

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. More...

## 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 Gmpfis 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. More...

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

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

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

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

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

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. More...

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. More...

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

Uncertain< Comparison_resultcompare (const Gmpfi &j) const
If i and j do not overlap, this function returns the result of the comparison. More...

## 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< Signsign () const
If all numbers contained in the interval have the same sign, this function returns it. More...

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.

## ◆ 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.

## ◆ 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 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 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 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 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,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.

## ◆ operator>>()

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

Reads i from is.

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