Definition
An instance of the data type interval represents a real interval I = [a, b]. The basic interval operations + , - ,*,/, are available. Type interval can be used to approximate exact real arithmetic operations by inexact interval operations, as follows. Each input number x_{i} is converted into the interval {x_{i}} and all real operations are replaced by interval operations. If x is the result of the exact real calculation and I the interval computed by type interval, it is guaranteed that I contains x. I can be seen as a more or less accurate approximation of x. In many cases the computed interval I is small enough to provide a useful approximation of x and the exact sign of x. There are four different implementations of intervals (consult the implementation section below for details):
#include < LEDA/numbers/interval.h >
interval | x | creates an instance x of type interval and initializes it with the interval {0} |
interval | x(VOLATILE_I double a) | creates an instance x of type interval and initializes it with {a} |
interval | x(int a) | creates an instance x of type interval and initializes it with {a} |
interval | x(long a) | creates an instance x of type interval and initializes it with {a} |
interval | x(const integer& a) | creates an instance x of type interval and initializes it with the smallest possible interval containing a |
interval | x(const bigfloat& a) | creates an instance x of type interval and initializes it with the smallest possible interval containing a |
interval | x(const real& a) | creates an instance x of type interval and initializes it with the smallest possible interval containing a |
interval | x(const rational& a) | creates an instance x of type interval and initializes it with the smallest possible interval containing a |
Operations
The arithmetic operations +,-,*,/,sqrt,+=,-=,*=,/= and the stream operators are all available. Important: If the advanced implementation interval_round_outside is used, the user has to guarantee that for each interval operation the IEEE754 rounding mode ''towards + '' is active. This can be achieved by calling the function fpu::round_up(). To avoid side effects with library functions that require the default IEEE754 rounding mode to_nearest, the function fpu::round_nearest() can be used to reset the rounding mode.
double | x.to_double() | returns the midpoint of the interval x as an approximation for the exact real number represented by x. |
double | x.get_double_error() | returns the diameter of the interval x which is the maximal error of the approximation x.to_double() of the exact real number represented by x. |
bool | x.is_a_point() | returns true if and only if the interval x consists of a single point. |
bool | x.is_finite() | returns true if and only if the interval x is a finite interval. |
bool | x.contains(double x) | returns true if and only if the interval x contains the number x |
double | x.upper_bound() | returns the upper bound of the interval x. |
double | x.lower_bound() | returns the lower bound of the interval x. |
void | x.set_range(VOLATILE_I double x, VOLATILE_I double y) | |
sets the current interval to [x, y]. | ||
void | x.set_midpoint(VOLATILE_I double num, VOLATILE_I double error) | |
sets the current interval to a superset of [num - error, num + error], i.e., to an interval with midpoint num and radius error. | ||
bool | x.sign_is_known() | returns true if and only if all numbers in the interval x have the same sign |
int | x.sign() | returns the sign of all numbers in the interval x if this sign is unique; aborts with an error message if x.sign_is_known() gives false |
Implementation
The types interval_round_inside and interval_round_outside represent intervals directly by (the negative of) its lower bound and its upper bound as doubles. Here all arithmetic operations require that the IEEE754 rounding mode ''towards + '' is active. For type interval_round_inside this is done inside each operation, and for type interval_round_outside the user has to do this manually ''from outside the operations'' by an explicit call of fpu::round_up().
The types interval_bound_absolute and interval_bound_relative represent intervals by their double midpoint NUM and diameter ERROR. The interpretation is that NUM is the numerical approximation of a real number and ERROR is a bound for the absolute, respectively relative error of NUM.