NMFraction R

jnemo.spad line 1570 [edit on github]

Domain for JL AbstractAlgebra fractions over a NM integral domain Author: G. Vanuxem Date Created: Description:

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> % if R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, NMInteger) -> %

undocumented

*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NMInteger, %) -> %

undocumented

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

/: (R, %) -> %

/ is the division operator.

/: (R, R) -> %

from QuotientFieldCategory R

<=: (%, %) -> Boolean if R has OrderedSet

from PartialOrder

<: (%, %) -> Boolean if R has OrderedSet

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean if R has OrderedSet

from PartialOrder

>: (%, %) -> Boolean if R has OrderedSet

from PartialOrder

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> % if R has OrderedIntegralDomain

from OrderedRing

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

ceiling: % -> R if R has IntegerNumberSystem

from QuotientFieldCategory R

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

from PolynomialFactorizationExplicit

coerce: % -> %

from Algebra %

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: R -> %

from Algebra R

coerce: Symbol -> % if R has RetractableTo Symbol

from CoercibleFrom Symbol

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

convert: % -> DoubleFloat if R has RealConstant

from ConvertibleTo DoubleFloat

convert: % -> Float if R has RealConstant

from ConvertibleTo Float

convert: % -> InputForm if R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: % -> String

from ConvertibleTo String

D: % -> % if R has DifferentialRing

from DifferentialRing

D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

D: (%, R -> R) -> %

from DifferentialExtension R

D: (%, R -> R, NonNegativeInteger) -> %

from DifferentialExtension R

D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

denom: % -> R

from QuotientFieldCategory R

denominator: % -> %

from QuotientFieldCategory R

differentiate: % -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NMInteger -> NMInteger) -> % if R has IntegerNumberSystem

undocumented

differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, R -> R) -> %

from DifferentialExtension R

differentiate: (%, R -> R, NonNegativeInteger) -> %

from DifferentialExtension R

differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

elt: (%, R) -> % if R has Eltable(R, R)

from Eltable(R, %)

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, Equation R) -> % if R has Evalable R

from Evalable R

eval: (%, List Equation R) -> % if R has Evalable R

from Evalable R

eval: (%, List R, List R) -> % if R has Evalable R

from InnerEvalable(R, R)

eval: (%, List Symbol, List R) -> % if R has InnerEvalable(Symbol, R)

from InnerEvalable(Symbol, R)

eval: (%, R, R) -> % if R has Evalable R

from InnerEvalable(R, R)

eval: (%, Symbol, R) -> % if R has InnerEvalable(Symbol, R)

from InnerEvalable(Symbol, R)

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

factorFraction: % -> Fraction Factored R

factorFraction(p) factors the numerator and the denominator of the fraction p.

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

floor: % -> R if R has IntegerNumberSystem

from QuotientFieldCategory R

fractionPart: % -> % if R has EuclideanDomain

from QuotientFieldCategory R

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

init: % if R has StepThrough

from StepThrough

inv: % -> %

from DivisionRing

jlAbout: % -> Void

from JLObjectType

jlApply: (String, %) -> %

from JLObjectType

jlApply: (String, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %, %) -> %

from JLObjectType

jlDisplay: % -> Void

from JLObjectType

jlId: % -> JLInt64

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from NMRing

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlType: % -> String

from JLObjectType

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: (R -> R, %) -> %

from FullyEvalableOver R

max: (%, %) -> % if R has OrderedSet

from OrderedSet

min: (%, %) -> % if R has OrderedSet

from OrderedSet

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

mutable?: % -> Boolean

from JLObjectType

negative?: % -> Boolean if R has OrderedIntegralDomain

from OrderedRing

nextItem: % -> Union(%, failed) if R has StepThrough

from StepThrough

nothing?: % -> Boolean

from JLObjectType

numer: % -> R

from QuotientFieldCategory R

numerator: % -> %

from QuotientFieldCategory R

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

positive?: % -> Boolean if R has OrderedIntegralDomain

from OrderedRing

prime?: % -> Boolean

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

random: (Segment Integer, Segment Integer) -> %

random(x, seg1, seg2) returns a random fraction depending on the base ring. For example: example{FRPRing:=NFRAC(NUP(NINT,’x))} example{random(1..5,-10..10)$FRPRing}

random: PositiveInteger -> % if R has IntegerNumberSystem

random(p) returns a random rational with numerator and denominator having p bits before before canonicalisation.

random: Segment Integer -> %

random(x, seg) returns a random fraction depending on the base ring.

recip: % -> Union(%, failed)

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix R

from LinearlyExplicitOver R

rem: (%, %) -> %

from EuclideanDomain

retract: % -> Fraction Integer if R has RetractableTo Integer

from RetractableTo Fraction Integer

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> R

from RetractableTo R

retract: % -> Symbol if R has RetractableTo Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(R, failed)

from RetractableTo R

retractIfCan: % -> Union(Symbol, failed) if R 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 R has OrderedIntegralDomain

from OrderedRing

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean if R has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

string: % -> String

from JLType

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

wholePart: % -> R if R has EuclideanDomain

from QuotientFieldCategory R

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra R

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(R, R)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

CoercibleFrom Symbol if R has RetractableTo Symbol

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if R has Comparable

ConvertibleTo DoubleFloat if R has RealConstant

ConvertibleTo Float if R has RealConstant

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

ConvertibleTo String

DifferentialExtension R

DifferentialRing if R has DifferentialRing

DivisionRing

Eltable(R, %) if R has Eltable(R, R)

EntireRing

EuclideanDomain

Evalable R if R has Evalable R

Field

FullyEvalableOver R

FullyLinearlyExplicitOver R

FullyPatternMatchable R

GcdDomain

InnerEvalable(R, R) if R has Evalable R

InnerEvalable(Symbol, R) if R has InnerEvalable(Symbol, R)

IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

LeftModule %

LeftModule Fraction Integer

LeftModule R

LeftOreRing

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module R

Monoid

NMCommutativeRing

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra R

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if R has OrderedIntegralDomain

OrderedAbelianMonoid if R has OrderedIntegralDomain

OrderedAbelianSemiGroup if R has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if R has OrderedIntegralDomain

OrderedIntegralDomain if R has OrderedIntegralDomain

OrderedRing if R has OrderedIntegralDomain

OrderedSet if R has OrderedSet

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

PartialOrder if R has OrderedSet

Patternable R

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

PrincipalIdealDomain

QuotientFieldCategory R

RealConstant if R has RealConstant

RetractableTo Fraction Integer if R has RetractableTo Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RetractableTo Symbol if R has RetractableTo Symbol

RightModule %

RightModule Fraction Integer

RightModule Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if R has StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown