Bases: sage.structure.factory.UniqueFactory
INPUT:
EXAMPLES:
sage: from sage.structure.factory import UniqueFactory
sage: fake_factory = UniqueFactory('ZZ')
sage: loads(dumps(fake_factory))
Integer Ring
sage: fake_factory = UniqueFactory('sage.rings.all.QQ')
sage: loads(dumps(fake_factory))
Rational Field
The only piece of data is the precision.
TESTS:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: RealIntervalAbsoluteField.create_key(1000)
1000
Ensures uniqueness.
TESTS:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: RealIntervalAbsoluteField(23) is RealIntervalAbsoluteField(23) # indirect doctest
True
Bases: object
This class is used to endow absolute real interval field elements with all the methods of (relative) real interval field elements.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(1).sin()
0.841470984807896506652502321631?
Bases: sage.structure.element.FieldElement
Create a RealIntervalAbsoluteElement.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(50)
sage: R(1)
1
sage: R(1/3)
0.333333333333334?
sage: R(1.3)
1.300000000000000?
sage: R(pi)
3.141592653589794?
sage: R((11, 12))
12.?
sage: R((11, 11.00001))
11.00001?
sage: R100 = RealIntervalAbsoluteField(100)
sage: R(R100((5,6)))
6.?
sage: R100(R((5,6)))
6.?
Return the absolute value of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(1/3).abs()
0.333333333333333333333333333334?
sage: R(-1/3).abs()
0.333333333333333333333333333334?
sage: R((-1/3, 1/2)).abs()
1.?
sage: R((-1/3, 1/2)).abs().endpoints()
(0, 1/2)
sage: R((-3/2, 1/2)).abs().endpoints()
(0, 3/2)
Return the diameter self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(1/4).absolute_diameter()
0
sage: a = R(pi)
sage: a.absolute_diameter()
1/1024
sage: a.upper() - a.lower()
1/1024
Return whether self contains zero.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(10).contains_zero()
False
sage: R((10,11)).contains_zero()
False
sage: R((0,11)).contains_zero()
True
sage: R((-10,11)).contains_zero()
True
sage: R((-10,-1)).contains_zero()
False
sage: R((-10,0)).contains_zero()
True
sage: R(pi).contains_zero()
False
Return the diameter self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(1/4).absolute_diameter()
0
sage: a = R(pi)
sage: a.absolute_diameter()
1/1024
sage: a.upper() - a.lower()
1/1024
Return the left and right endpoints of self, as a tuple.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(1/4).endpoints()
(1/4, 1/4)
sage: R((1,2)).endpoints()
(1, 2)
Return whether self is definitely negative.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(10).is_negative()
False
sage: R((10,11)).is_negative()
False
sage: R((0,11)).is_negative()
False
sage: R((-10,11)).is_negative()
False
sage: R((-10,-1)).is_negative()
True
sage: R(pi).is_negative()
False
Return whether self is definitely positive.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(10).is_positive()
True
sage: R((10,11)).is_positive()
True
sage: R((0,11)).is_positive()
False
sage: R((-10,11)).is_positive()
False
sage: R((-10,-1)).is_positive()
False
sage: R(pi).is_positive()
True
Return the lower bound of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(50)
sage: R(1/4).lower()
1/4
Return the midpoint of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(1/4).midpoint()
1/4
sage: R(pi).midpoint()
7964883625991394727376702227905/2535301200456458802993406410752
sage: R(pi).midpoint().n()
3.14159265358979
Return the precision needed to represent this value as an mpfi interval.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10)
sage: R(10).mpfi_prec()
14
sage: R(1000).mpfi_prec()
20
Return the square root of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(2).sqrt()
1.414213562373095048801688724210?
sage: R((4,9)).sqrt().endpoints()
(2, 3)
Return the upper bound of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(50)
sage: R(1/4).upper()
1/4
This field is similar to the RealIntervalField except instead of truncating everything to a fixed relative precision, it maintains a fixed absolute precision.
Note that unlike the standard real interval field, elements in this field can have different size and experience coefficient blowup. On the other hand, it avoids precision loss on addition and subtraction. This is useful for, e.g., series computations for special functions.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10); R
Real Interval Field with absolute precision 2^-10
sage: R(3/10)
0.300?
sage: R(1000003/10)
100000.300?
sage: R(1e100) + R(1) - R(1e100)
1
Bases: sage.rings.ring.Field
This field is similar to the RealIntervalField except instead of truncating everything to a fixed relative precision, it maintains a fixed absolute precision.
Note that unlike the standard real interval field, elements in this field can have different size and experience coefficient blowup. On the other hand, it avoids precision loss on addition and subtraction. This is useful for, e.g., series computations for special functions.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(10); R
Real Interval Field with absolute precision 2^-10
sage: R(3/10)
0.300?
sage: R(1000003/10)
100000.300?
sage: R(1e100) + R(1) - R(1e100)
1
Returns the absolute precision of self.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R.absprec()
100
sage: RealIntervalAbsoluteField(5).absprec()
5
Return where
is the value of shift, rounded towards
. For internal use.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import shift_ceil
sage: shift_ceil(15, 2)
4
sage: shift_ceil(-15, 2)
-3
sage: shift_ceil(32, 2)
8
sage: shift_ceil(-32, 2)
-8
Return where
is the value of shift, rounded towards
. For internal use.
EXAMPLES:
sage: from sage.rings.real_interval_absolute import shift_floor
sage: shift_floor(15, 2)
3
sage: shift_floor(-15, 2)
-4