Real intervals with a fixed absolute precision¶

class sage.rings.real_interval_absolute.Factory[source]¶

Bases: UniqueFactory

create_key(prec)[source]¶: The only piece of data is the precision.

create_object(version, prec)[source]¶: Ensures uniqueness.

class sage.rings.real_interval_absolute.MpfrOp[source]¶

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

sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R(1).sin()
0.841470984807896506652502321631?

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R(Integer(1)).sin()
0.841470984807896506652502321631?

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R(1).sin()

class sage.rings.real_interval_absolute.RealIntervalAbsoluteElement[source]¶

Bases: FieldElement

Create a RealIntervalAbsoluteElement.

EXAMPLES:

Sage

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.?
sage: RIF(CIF(NaN))
[.. NaN ..]

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(50))
>>> R(Integer(1))
1
>>> R(Integer(1)/Integer(3))
0.333333333333334?
>>> R(RealNumber('1.3'))
1.300000000000000?
>>> R(pi)
3.141592653589794?
>>> R((Integer(11), Integer(12)))
12.?
>>> R((Integer(11), RealNumber('11.00001')))
11.00001?

>>> R100 = RealIntervalAbsoluteField(Integer(100))
>>> R(R100((Integer(5),Integer(6))))
6.?
>>> R100(R((Integer(5),Integer(6))))
6.?
>>> RIF(CIF(NaN))
[.. NaN ..]

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(50)
R(1)
R(1/3)
R(1.3)
R(pi)
R((11, 12))
R((11, 11.00001))
R100 = RealIntervalAbsoluteField(100)
R(R100((5,6)))
R100(R((5,6)))
RIF(CIF(NaN))

abs()[source]¶

Return the absolute value of self.

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R(Integer(1)/Integer(3)).abs()
0.333333333333333333333333333334?
>>> R(-Integer(1)/Integer(3)).abs()
0.333333333333333333333333333334?
>>> R((-Integer(1)/Integer(3), Integer(1)/Integer(2))).abs()
1.?
>>> R((-Integer(1)/Integer(3), Integer(1)/Integer(2))).abs().endpoints()
(0, 1/2)
>>> R((-Integer(3)/Integer(2), Integer(1)/Integer(2))).abs().endpoints()
(0, 3/2)

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R(1/3).abs()
R(-1/3).abs()
R((-1/3, 1/2)).abs()
R((-1/3, 1/2)).abs().endpoints()
R((-3/2, 1/2)).abs().endpoints()

absolute_diameter()[source]¶

Return the diameter self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(1)/Integer(4)).absolute_diameter()
0
>>> a = R(pi)
>>> a.absolute_diameter()
1/1024
>>> a.upper() - a.lower()
1/1024

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(1/4).absolute_diameter()
a = R(pi)
a.absolute_diameter()
a.upper() - a.lower()

contains_zero()[source]¶

Return whether self contains zero.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(10)).contains_zero()
False
>>> R((Integer(10),Integer(11))).contains_zero()
False
>>> R((Integer(0),Integer(11))).contains_zero()
True
>>> R((-Integer(10),Integer(11))).contains_zero()
True
>>> R((-Integer(10),-Integer(1))).contains_zero()
False
>>> R((-Integer(10),Integer(0))).contains_zero()
True
>>> R(pi).contains_zero()
False

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(10).contains_zero()
R((10,11)).contains_zero()
R((0,11)).contains_zero()
R((-10,11)).contains_zero()
R((-10,-1)).contains_zero()
R((-10,0)).contains_zero()
R(pi).contains_zero()

diameter()[source]¶

Return the diameter self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(1)/Integer(4)).absolute_diameter()
0
>>> a = R(pi)
>>> a.absolute_diameter()
1/1024
>>> a.upper() - a.lower()
1/1024

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(1/4).absolute_diameter()
a = R(pi)
a.absolute_diameter()
a.upper() - a.lower()

endpoints()[source]¶

Return the left and right endpoints of self, as a tuple.

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(1)/Integer(4)).endpoints()
(1/4, 1/4)
>>> R((Integer(1),Integer(2))).endpoints()
(1, 2)

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(1/4).endpoints()
R((1,2)).endpoints()

is_negative()[source]¶

Return whether self is definitely negative.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R(Integer(10)).is_negative()
False
>>> R((Integer(10),Integer(11))).is_negative()
False
>>> R((Integer(0),Integer(11))).is_negative()
False
>>> R((-Integer(10),Integer(11))).is_negative()
False
>>> R((-Integer(10),-Integer(1))).is_negative()
True
>>> R(pi).is_negative()
False

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R(10).is_negative()
R((10,11)).is_negative()
R((0,11)).is_negative()
R((-10,11)).is_negative()
R((-10,-1)).is_negative()
R(pi).is_negative()

is_positive()[source]¶

Return whether self is definitely positive.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(10)).is_positive()
True
>>> R((Integer(10),Integer(11))).is_positive()
True
>>> R((Integer(0),Integer(11))).is_positive()
False
>>> R((-Integer(10),Integer(11))).is_positive()
False
>>> R((-Integer(10),-Integer(1))).is_positive()
False
>>> R(pi).is_positive()
True

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(10).is_positive()
R((10,11)).is_positive()
R((0,11)).is_positive()
R((-10,11)).is_positive()
R((-10,-1)).is_positive()
R(pi).is_positive()

lower()[source]¶

Return the lower bound of self.

EXAMPLES:

Sage

sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(50)
sage: R(1/4).lower()
1/4

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(50))
>>> R(Integer(1)/Integer(4)).lower()
1/4

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(50)
R(1/4).lower()

midpoint()[source]¶

Return the midpoint of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R(Integer(1)/Integer(4)).midpoint()
1/4
>>> R(pi).midpoint()
7964883625991394727376702227905/2535301200456458802993406410752
>>> R(pi).midpoint().n()
3.14159265358979

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R(1/4).midpoint()
R(pi).midpoint()
R(pi).midpoint().n()

mpfi_prec()[source]¶

Return the precision needed to represent this value as an mpfi interval.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10))
>>> R(Integer(10)).mpfi_prec()
14
>>> R(Integer(1000)).mpfi_prec()
20

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10)
R(10).mpfi_prec()
R(1000).mpfi_prec()

sqrt()[source]¶

Return the square root of self.

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R(Integer(2)).sqrt()
1.414213562373095048801688724210?
>>> R((Integer(4),Integer(9))).sqrt().endpoints()
(2, 3)

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R(2).sqrt()
R((4,9)).sqrt().endpoints()

upper()[source]¶

Return the upper bound of self.

EXAMPLES:

Sage

sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(50)
sage: R(1/4).upper()
1/4

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(50))
>>> R(Integer(1)/Integer(4)).upper()
1/4

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(50)
R(1/4).upper()

sage.rings.real_interval_absolute.RealIntervalAbsoluteField(*args, **kwds)[source]¶

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

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10)); R
Real Interval Field with absolute precision 2^-10
>>> R(Integer(3)/Integer(10))
0.300?
>>> R(Integer(1000003)/Integer(10))
100000.300?
>>> R(RealNumber('1e100')) + R(Integer(1)) - R(RealNumber('1e100'))
1

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10); R
R(3/10)
R(1000003/10)
R(1e100) + R(1) - R(1e100)

class sage.rings.real_interval_absolute.RealIntervalAbsoluteField_class[source]¶

Bases: Field

This field is similar to the RealIntervalField except instead of truncating everything to a fixed relative precision, it maintains a fixed absolute precision.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(10)); R
Real Interval Field with absolute precision 2^-10
>>> R(Integer(3)/Integer(10))
0.300?
>>> R(Integer(1000003)/Integer(10))
100000.300?
>>> R(RealNumber('1e100')) + R(Integer(1)) - R(RealNumber('1e100'))
1

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(10); R
R(3/10)
R(1000003/10)
R(1e100) + R(1) - R(1e100)

absprec()[source]¶

Return the absolute precision of self.

EXAMPLES:

Sage

sage: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
sage: R = RealIntervalAbsoluteField(100)
sage: R.absprec()
100
sage: RealIntervalAbsoluteField(5).absprec()
5

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
>>> R = RealIntervalAbsoluteField(Integer(100))
>>> R.absprec()
100
>>> RealIntervalAbsoluteField(Integer(5)).absprec()
5

Sage Live

from sage.rings.real_interval_absolute import RealIntervalAbsoluteField
R = RealIntervalAbsoluteField(100)
R.absprec()
RealIntervalAbsoluteField(5).absprec()

sage.rings.real_interval_absolute.shift_ceil(x, shift)[source]¶

Return \(x / 2^s\) where \(s\) is the value of shift, rounded towards \(+\infty\). For internal use.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import shift_ceil
>>> shift_ceil(Integer(15), Integer(2))
4
>>> shift_ceil(-Integer(15), Integer(2))
-3
>>> shift_ceil(Integer(32), Integer(2))
8
>>> shift_ceil(-Integer(32), Integer(2))
-8

Sage Live

from sage.rings.real_interval_absolute import shift_ceil
shift_ceil(15, 2)
shift_ceil(-15, 2)
shift_ceil(32, 2)
shift_ceil(-32, 2)

sage.rings.real_interval_absolute.shift_floor(x, shift)[source]¶

Return \(x / 2^s\) where \(s\) is the value of shift, rounded towards \(-\infty\). For internal use.

EXAMPLES:

Sage

sage: from sage.rings.real_interval_absolute import shift_floor
sage: shift_floor(15, 2)
3
sage: shift_floor(-15, 2)
-4

Python

>>> from sage.all import *
>>> from sage.rings.real_interval_absolute import shift_floor
>>> shift_floor(Integer(15), Integer(2))
3
>>> shift_floor(-Integer(15), Integer(2))
-4

Sage Live

from sage.rings.real_interval_absolute import shift_floor
shift_floor(15, 2)
shift_floor(-15, 2)