Real intervals with a fixed absolute precision¶
- class sage.rings.real_interval_absolute.Factory[source]¶
- Bases: - UniqueFactory
- 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: from sage.rings.real_interval_absolute import RealIntervalAbsoluteField sage: R = RealIntervalAbsoluteField(100) sage: R(1).sin() 0.841470984807896506652502321631? - >>> from sage.all import * >>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField >>> R = RealIntervalAbsoluteField(Integer(100)) >>> R(Integer(1)).sin() 0.841470984807896506652502321631? 
- class sage.rings.real_interval_absolute.RealIntervalAbsoluteElement[source]¶
- Bases: - 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.? sage: RIF(CIF(NaN)) [.. NaN ..] - >>> 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 ..] - abs()[source]¶
- 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) - >>> 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) 
 - absolute_diameter()[source]¶
- 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 - >>> 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 
 - contains_zero()[source]¶
- Return whether - selfcontains 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 - >>> 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 
 - diameter()[source]¶
- 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 - >>> 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 
 - endpoints()[source]¶
- 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) - >>> 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) 
 - is_negative()[source]¶
- Return whether - selfis 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 - >>> 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 
 - is_positive()[source]¶
- Return whether - selfis 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 - >>> 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 
 - lower()[source]¶
- 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 - >>> from sage.all import * >>> from sage.rings.real_interval_absolute import RealIntervalAbsoluteField >>> R = RealIntervalAbsoluteField(Integer(50)) >>> R(Integer(1)/Integer(4)).lower() 1/4 
 - midpoint()[source]¶
- 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 - >>> 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 
 - mpfi_prec()[source]¶
- 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 - >>> 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 
 - sqrt()[source]¶
- 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) - >>> 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) 
 - upper()[source]¶
- 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 - >>> 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.rings.real_interval_absolute.RealIntervalAbsoluteField(*args, **kwds)[source]¶
- This field is similar to the - RealIntervalFieldexcept 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 - >>> 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 
- class sage.rings.real_interval_absolute.RealIntervalAbsoluteField_class[source]¶
- Bases: - Field- This field is similar to the - RealIntervalFieldexcept 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 - >>> 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 - absprec()[source]¶
- Return 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 - >>> 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.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: 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 - >>> 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.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: from sage.rings.real_interval_absolute import shift_floor sage: shift_floor(15, 2) 3 sage: shift_floor(-15, 2) -4 - >>> 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