BalancedPAdicRational(p)
padic.spad line 559
[edit on github]
Stream-based implementation of Qp: numbers are represented as sum(i = k.., a[i] * p^i), where the a[i] lie in -(p - 1)/2, ..., (p - 1)/2.
- * : (%, %) -> %
- from Magma
- * : (%, BalancedPAdicInteger(p)) -> %
- from RightModule(BalancedPAdicInteger(p))
- * : (%, Fraction(Integer)) -> %
- from RightModule(Fraction(Integer))
- * : (%, Integer) -> % if BalancedPAdicInteger(p) has LinearlyExplicitOver(Integer)
- from RightModule(Integer)
- * : (BalancedPAdicInteger(p), %) -> %
- from LeftModule(BalancedPAdicInteger(p))
- * : (Fraction(Integer), %) -> %
- from LeftModule(Fraction(Integer))
- * : (Integer, %) -> %
- from AbelianGroup
- * : (NonNegativeInteger, %) -> %
- from AbelianMonoid
- * : (PositiveInteger, %) -> %
- from AbelianSemiGroup
- + : (%, %) -> %
- from AbelianSemiGroup
- - : % -> %
- from AbelianGroup
- - : (%, %) -> %
- from AbelianGroup
- / : (%, %) -> %
- from Field
- / : (BalancedPAdicInteger(p), BalancedPAdicInteger(p)) -> %
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- 0 : () -> %
- from AbelianMonoid
- 1 : () -> %
- from MagmaWithUnit
- < : (%, %) -> Boolean if BalancedPAdicInteger(p) has OrderedSet
- from PartialOrder
- <= : (%, %) -> Boolean if BalancedPAdicInteger(p) has OrderedSet
- from PartialOrder
- = : (%, %) -> Boolean
- from BasicType
- > : (%, %) -> Boolean if BalancedPAdicInteger(p) has OrderedSet
- from PartialOrder
- >= : (%, %) -> Boolean if BalancedPAdicInteger(p) has OrderedSet
- from PartialOrder
- D : % -> % if BalancedPAdicInteger(p) has DifferentialRing
- from DifferentialRing
- D : (%, List(Symbol)) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- D : (%, List(Symbol), List(NonNegativeInteger)) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- D : (%, Mapping(BalancedPAdicInteger(p), BalancedPAdicInteger(p))) -> %
- from DifferentialExtension(BalancedPAdicInteger(p))
- D : (%, Mapping(BalancedPAdicInteger(p), BalancedPAdicInteger(p)), NonNegativeInteger) -> %
- from DifferentialExtension(BalancedPAdicInteger(p))
- D : (%, NonNegativeInteger) -> % if BalancedPAdicInteger(p) has DifferentialRing
- from DifferentialRing
- D : (%, Symbol) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- D : (%, Symbol, NonNegativeInteger) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- ^ : (%, Integer) -> %
- from DivisionRing
- ^ : (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^ : (%, PositiveInteger) -> %
- from Magma
- abs : % -> % if BalancedPAdicInteger(p) has OrderedIntegralDomain
- from OrderedRing
- annihilate? : (%, %) -> Boolean
- from Rng
- antiCommutator : (%, %) -> %
- from NonAssociativeSemiRng
- approximate : (%, Integer) -> Fraction(Integer)
- associates? : (%, %) -> Boolean
- from EntireRing
- associator : (%, %, %) -> %
- from NonAssociativeRng
- ceiling : % -> BalancedPAdicInteger(p) if BalancedPAdicInteger(p) has IntegerNumberSystem
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- characteristic : () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot : % -> Union(%, "failed") if BalancedPAdicInteger(p) has CharacteristicNonZero or % has CharacteristicNonZero and BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from CharacteristicNonZero
- coerce : % -> %
- from Algebra(%)
- coerce : BalancedPAdicInteger(p) -> %
- from CoercibleFrom(BalancedPAdicInteger(p))
- coerce : Fraction(Integer) -> %
- from CoercibleFrom(Fraction(Integer))
- coerce : Integer -> %
- from NonAssociativeRing
- coerce : Symbol -> % if BalancedPAdicInteger(p) has RetractableTo(Symbol)
- from CoercibleFrom(Symbol)
- coerce : % -> OutputForm
- from CoercibleTo(OutputForm)
- commutator : (%, %) -> %
- from NonAssociativeRng
- conditionP : Matrix(%) -> Union(Vector(%), "failed") if % has CharacteristicNonZero and BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- continuedFraction : % -> ContinuedFraction(Fraction(Integer))
- convert : % -> DoubleFloat if BalancedPAdicInteger(p) has RealConstant
- from ConvertibleTo(DoubleFloat)
- convert : % -> Float if BalancedPAdicInteger(p) has RealConstant
- from ConvertibleTo(Float)
- convert : % -> InputForm if BalancedPAdicInteger(p) has ConvertibleTo(InputForm)
- from ConvertibleTo(InputForm)
- convert : % -> Pattern(Float) if BalancedPAdicInteger(p) has ConvertibleTo(Pattern(Float))
- from ConvertibleTo(Pattern(Float))
- convert : % -> Pattern(Integer) if BalancedPAdicInteger(p) has ConvertibleTo(Pattern(Integer))
- from ConvertibleTo(Pattern(Integer))
- denom : % -> BalancedPAdicInteger(p)
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- denominator : % -> %
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- differentiate : % -> % if BalancedPAdicInteger(p) has DifferentialRing
- from DifferentialRing
- differentiate : (%, List(Symbol)) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- differentiate : (%, Mapping(BalancedPAdicInteger(p), BalancedPAdicInteger(p))) -> %
- from DifferentialExtension(BalancedPAdicInteger(p))
- differentiate : (%, Mapping(BalancedPAdicInteger(p), BalancedPAdicInteger(p)), NonNegativeInteger) -> %
- from DifferentialExtension(BalancedPAdicInteger(p))
- differentiate : (%, NonNegativeInteger) -> % if BalancedPAdicInteger(p) has DifferentialRing
- from DifferentialRing
- differentiate : (%, Symbol) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- differentiate : (%, Symbol, NonNegativeInteger) -> % if BalancedPAdicInteger(p) has PartialDifferentialRing(Symbol)
- from PartialDifferentialRing(Symbol)
- divide : (%, %) -> Record(quotient : %, remainder : %)
- from EuclideanDomain
- elt : (%, BalancedPAdicInteger(p)) -> % if BalancedPAdicInteger(p) has Eltable(BalancedPAdicInteger(p), BalancedPAdicInteger(p))
- from Eltable(BalancedPAdicInteger(p), %)
- euclideanSize : % -> NonNegativeInteger
- from EuclideanDomain
- eval : (%, BalancedPAdicInteger(p), BalancedPAdicInteger(p)) -> % if BalancedPAdicInteger(p) has Evalable(BalancedPAdicInteger(p))
- from InnerEvalable(BalancedPAdicInteger(p), BalancedPAdicInteger(p))
- eval : (%, Equation(BalancedPAdicInteger(p))) -> % if BalancedPAdicInteger(p) has Evalable(BalancedPAdicInteger(p))
- from Evalable(BalancedPAdicInteger(p))
- eval : (%, List(BalancedPAdicInteger(p)), List(BalancedPAdicInteger(p))) -> % if BalancedPAdicInteger(p) has Evalable(BalancedPAdicInteger(p))
- from InnerEvalable(BalancedPAdicInteger(p), BalancedPAdicInteger(p))
- eval : (%, List(Equation(BalancedPAdicInteger(p)))) -> % if BalancedPAdicInteger(p) has Evalable(BalancedPAdicInteger(p))
- from Evalable(BalancedPAdicInteger(p))
- eval : (%, List(Symbol), List(BalancedPAdicInteger(p))) -> % if BalancedPAdicInteger(p) has InnerEvalable(Symbol, BalancedPAdicInteger(p))
- from InnerEvalable(Symbol, BalancedPAdicInteger(p))
- eval : (%, Symbol, BalancedPAdicInteger(p)) -> % if BalancedPAdicInteger(p) has InnerEvalable(Symbol, BalancedPAdicInteger(p))
- from InnerEvalable(Symbol, BalancedPAdicInteger(p))
- expressIdealMember : (List(%), %) -> Union(List(%), "failed")
- from PrincipalIdealDomain
- exquo : (%, %) -> Union(%, "failed")
- from EntireRing
- extendedEuclidean : (%, %) -> Record(coef1 : %, coef2 : %, generator : %)
- from EuclideanDomain
- extendedEuclidean : (%, %, %) -> Union(Record(coef1 : %, coef2 : %), "failed")
- from EuclideanDomain
- factor : % -> Factored(%)
- from UniqueFactorizationDomain
- factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- floor : % -> BalancedPAdicInteger(p) if BalancedPAdicInteger(p) has IntegerNumberSystem
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- fractionPart : % -> %
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- gcd : (%, %) -> %
- from GcdDomain
- gcd : List(%) -> %
- from GcdDomain
- gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)
- from PolynomialFactorizationExplicit
- init : () -> % if BalancedPAdicInteger(p) has StepThrough
- from StepThrough
- inv : % -> %
- from DivisionRing
- latex : % -> String
- from SetCategory
- lcm : (%, %) -> %
- from GcdDomain
- lcm : List(%) -> %
- from GcdDomain
- lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)
- from LeftOreRing
- leftPower : (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower : (%, PositiveInteger) -> %
- from Magma
- leftRecip : % -> Union(%, "failed")
- from MagmaWithUnit
- map : (Mapping(BalancedPAdicInteger(p), BalancedPAdicInteger(p)), %) -> %
- from FullyEvalableOver(BalancedPAdicInteger(p))
- max : (%, %) -> % if BalancedPAdicInteger(p) has OrderedSet
- from OrderedSet
- min : (%, %) -> % if BalancedPAdicInteger(p) has OrderedSet
- from OrderedSet
- multiEuclidean : (List(%), %) -> Union(List(%), "failed")
- from EuclideanDomain
- negative? : % -> Boolean if BalancedPAdicInteger(p) has OrderedIntegralDomain
- from OrderedRing
- nextItem : % -> Union(%, "failed") if BalancedPAdicInteger(p) has StepThrough
- from StepThrough
- numer : % -> BalancedPAdicInteger(p)
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- numerator : % -> %
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- one? : % -> Boolean
- from MagmaWithUnit
- opposite? : (%, %) -> Boolean
- from AbelianMonoid
- patternMatch : (%, Pattern(Float), PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if BalancedPAdicInteger(p) has PatternMatchable(Float)
- from PatternMatchable(Float)
- patternMatch : (%, Pattern(Integer), PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if BalancedPAdicInteger(p) has PatternMatchable(Integer)
- from PatternMatchable(Integer)
- plenaryPower : (%, PositiveInteger) -> %
- from NonAssociativeAlgebra(%)
- positive? : % -> Boolean if BalancedPAdicInteger(p) has OrderedIntegralDomain
- from OrderedRing
- prime? : % -> Boolean
- from UniqueFactorizationDomain
- principalIdeal : List(%) -> Record(coef : List(%), generator : %)
- from PrincipalIdealDomain
- quo : (%, %) -> %
- from EuclideanDomain
- recip : % -> Union(%, "failed")
- from MagmaWithUnit
- reducedSystem : Matrix(%) -> Matrix(BalancedPAdicInteger(p))
- from LinearlyExplicitOver(BalancedPAdicInteger(p))
- reducedSystem : Matrix(%) -> Matrix(Integer) if BalancedPAdicInteger(p) has LinearlyExplicitOver(Integer)
- from LinearlyExplicitOver(Integer)
- reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(BalancedPAdicInteger(p)), vec : Vector(BalancedPAdicInteger(p)))
- from LinearlyExplicitOver(BalancedPAdicInteger(p))
- reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if BalancedPAdicInteger(p) has LinearlyExplicitOver(Integer)
- from LinearlyExplicitOver(Integer)
- rem : (%, %) -> %
- from EuclideanDomain
- removeZeroes : % -> %
- removeZeroes : (Integer, %) -> %
- retract : % -> BalancedPAdicInteger(p)
- from RetractableTo(BalancedPAdicInteger(p))
- retract : % -> Fraction(Integer) if BalancedPAdicInteger(p) has RetractableTo(Integer)
- from RetractableTo(Fraction(Integer))
- retract : % -> Integer if BalancedPAdicInteger(p) has RetractableTo(Integer)
- from RetractableTo(Integer)
- retract : % -> Symbol if BalancedPAdicInteger(p) has RetractableTo(Symbol)
- from RetractableTo(Symbol)
- retractIfCan : % -> Union(BalancedPAdicInteger(p), "failed")
- from RetractableTo(BalancedPAdicInteger(p))
- retractIfCan : % -> Union(Fraction(Integer), "failed") if BalancedPAdicInteger(p) has RetractableTo(Integer)
- from RetractableTo(Fraction(Integer))
- retractIfCan : % -> Union(Integer, "failed") if BalancedPAdicInteger(p) has RetractableTo(Integer)
- from RetractableTo(Integer)
- retractIfCan : % -> Union(Symbol, "failed") if BalancedPAdicInteger(p) has RetractableTo(Symbol)
- from RetractableTo(Symbol)
- rightPower : (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower : (%, PositiveInteger) -> %
- from Magma
- rightRecip : % -> Union(%, "failed")
- from MagmaWithUnit
- sample : () -> %
- from AbelianMonoid
- sign : % -> Integer if BalancedPAdicInteger(p) has OrderedIntegralDomain
- from OrderedRing
- sizeLess? : (%, %) -> Boolean
- from EuclideanDomain
- smaller? : (%, %) -> Boolean if BalancedPAdicInteger(p) has Comparable
- from Comparable
- solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)), SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)), "failed") if BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- squareFree : % -> Factored(%)
- from UniqueFactorizationDomain
- squareFreePart : % -> %
- from UniqueFactorizationDomain
- squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if BalancedPAdicInteger(p) has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- subtractIfCan : (%, %) -> Union(%, "failed")
- from CancellationAbelianMonoid
- unit? : % -> Boolean
- from EntireRing
- unitCanonical : % -> %
- from EntireRing
- unitNormal : % -> Record(unit : %, canonical : %, associate : %)
- from EntireRing
- wholePart : % -> BalancedPAdicInteger(p)
- from QuotientFieldCategory(BalancedPAdicInteger(p))
- zero? : % -> Boolean
- from AbelianMonoid
- ~= : (%, %) -> Boolean
- from BasicType
Algebra(BalancedPAdicInteger(p))
Module(Fraction(Integer))
ConvertibleTo(Float)
PrincipalIdealDomain
OrderedAbelianSemiGroup
NonAssociativeSemiRing
BiModule(%, %)
ConvertibleTo(InputForm)
Field
canonicalUnitNormal
Rng
CoercibleFrom(Integer)
TwoSidedRecip
RightModule(BalancedPAdicInteger(p))
SemiRing
EntireRing
PatternMatchable(Float)
NonAssociativeAlgebra(Fraction(Integer))
CharacteristicNonZero
InnerEvalable(BalancedPAdicInteger(p), BalancedPAdicInteger(p))
unitsKnown
FullyEvalableOver(BalancedPAdicInteger(p))
FullyLinearlyExplicitOver(BalancedPAdicInteger(p))
QuotientFieldCategory(BalancedPAdicInteger(p))
LinearlyExplicitOver(BalancedPAdicInteger(p))
noZeroDivisors
RetractableTo(Fraction(Integer))
LeftModule(BalancedPAdicInteger(p))
RetractableTo(BalancedPAdicInteger(p))
UniqueFactorizationDomain
SemiGroup
RightModule(Fraction(Integer))
Magma
RetractableTo(Symbol)
IntegralDomain
LeftModule(%)
Patternable(BalancedPAdicInteger(p))
GcdDomain
NonAssociativeAlgebra(%)
Eltable(BalancedPAdicInteger(p), %)
PartialDifferentialRing(Symbol)
CharacteristicZero
CoercibleFrom(BalancedPAdicInteger(p))
BiModule(BalancedPAdicInteger(p), BalancedPAdicInteger(p))
OrderedIntegralDomain
Algebra(%)
Ring
CoercibleFrom(Fraction(Integer))
DifferentialRing
FullyPatternMatchable(BalancedPAdicInteger(p))
OrderedAbelianMonoid
DivisionRing
CommutativeRing
OrderedAbelianGroup
PartialOrder
NonAssociativeSemiRng
CancellationAbelianMonoid
EuclideanDomain
canonicalsClosed
RetractableTo(Integer)
OrderedCancellationAbelianMonoid
Module(BalancedPAdicInteger(p))
OrderedRing
CommutativeStar
AbelianMonoid
MagmaWithUnit
Comparable
NonAssociativeRing
RightModule(%)
RealConstant
InnerEvalable(Symbol, BalancedPAdicInteger(p))
ConvertibleTo(DoubleFloat)
DifferentialExtension(BalancedPAdicInteger(p))
Module(%)
LinearlyExplicitOver(Integer)
CoercibleTo(OutputForm)
ConvertibleTo(Pattern(Float))
SemiRng
CoercibleFrom(Symbol)
Monoid
PolynomialFactorizationExplicit
NonAssociativeAlgebra(BalancedPAdicInteger(p))
LeftOreRing
OrderedSet
StepThrough
Algebra(Fraction(Integer))
BasicType
RightModule(Integer)
LeftModule(Fraction(Integer))
AbelianSemiGroup
SetCategory
NonAssociativeRng
PatternMatchable(Integer)
BiModule(Fraction(Integer), Fraction(Integer))
Evalable(BalancedPAdicInteger(p))
ConvertibleTo(Pattern(Integer))
AbelianGroup