CGAL 5.6  Number Types

#include <CGAL/Gmpfi.h>
An object of the class Gmpfi
is a closed interval, with endpoints represented as Gmpfr
floatingpoint 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
.
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).
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 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 (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 The precision of an operation between two The second operand of the former operations can be a To specify the rounding mode and/or the precision to perform an operation, this class provides the four static functions  
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 kth 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 The result of the comparison is always an In the same way, we can explain the semantics of 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_result >  compare (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< Sign >  sign () 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.  
CGAL::Gmpfi::Gmpfi  (  const T &  t, 
Precision_type  p = get_default_precision() 

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

) 
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.
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
.
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
.
Gmpfi CGAL::Gmpfi::kthroot  (  int  k, 
Precision_type  p  
)  const 
Returns the kth root of i
, with precision p
.
If p
is not specified, the precision used is the maximum between i
's precision and the default.
If all numbers contained in the interval have the same sign, this function returns it.
Otherwise it returns indeterminate
.
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.
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.
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.
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.
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.

related 
Writes i
to os
, in the form [i.inf(),i.sup()]
.
The endpoints are written according to the Gmpfr
formatting.