DifferentialSparseMultivariatePolynomial(R, S, V)

dpolcat.spad line 407 [edit on github]

• R : Ring
• S : OrderedSet
• V : DifferentialVariableCategory(S)

DifferentialSparseMultivariatePolynomial implements an ordinary differential polynomial ring by combining a domain belonging to the category DifferentialVariableCategory with the domain SparseMultivariatePolynomial.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

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

from RightModule(Fraction(Integer))

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

from RightModule(Integer)

* : (R, %) -> %

from LeftModule(R)

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

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, R) -> % if R has Field

from AbelianMonoidRing(R, IndexedExponents(V))

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : % -> % if R has DifferentialRing

from DifferentialRing

D : (%, V) -> %

from PartialDifferentialRing(V)

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

from PartialDifferentialRing(V)

D : (%, List(V)) -> %

from PartialDifferentialRing(V)

D : (%, List(V), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(V)

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

from DifferentialExtension(R)

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

from DifferentialExtension(R)

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

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

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

from PartialDifferentialRing(Symbol)

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean if R has EntireRing

from EntireRing

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

from NonAssociativeRng

binomThmExpt : (%, %, NonNegativeInteger) -> % if % has CommutativeRing

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

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

from CharacteristicNonZero

coefficient : (%, V, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

coefficient : (%, List(V), List(NonNegativeInteger)) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

coefficient : (%, IndexedExponents(V)) -> R

from AbelianMonoidRing(R, IndexedExponents(V))

coefficients : % -> List(R)

from FreeModuleCategory(R, IndexedExponents(V))

coerce : % -> % if R has CommutativeRing

from Algebra(%)

coerce : R -> %

from Algebra(R)

coerce : S -> %

from CoercibleFrom(S)

coerce : V -> %

from CoercibleFrom(V)

coerce : Fraction(Integer) -> % if R has RetractableTo(Fraction(Integer)) or R has Algebra(Fraction(Integer))

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : SparseMultivariatePolynomial(R, S) -> %

from CoercibleFrom(SparseMultivariatePolynomial(R, S))

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

from PolynomialFactorizationExplicit

construct : List(Record(k : IndexedExponents(V), c : R)) -> %

from IndexedProductCategory(R, IndexedExponents(V))

constructOrdered : List(Record(k : IndexedExponents(V), c : R)) -> %

from IndexedProductCategory(R, IndexedExponents(V))

content : (%, V) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(V), V)

content : % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

convert : % -> InputForm if V has ConvertibleTo(InputForm) and R has ConvertibleTo(InputForm)

from ConvertibleTo(InputForm)

convert : % -> Pattern(Float) if V has ConvertibleTo(Pattern(Float)) and R has ConvertibleTo(Pattern(Float))

from ConvertibleTo(Pattern(Float))

convert : % -> Pattern(Integer) if V has ConvertibleTo(Pattern(Integer)) and R has ConvertibleTo(Pattern(Integer))

from ConvertibleTo(Pattern(Integer))

degree : % -> IndexedExponents(V)

from AbelianMonoidRing(R, IndexedExponents(V))

degree : (%, List(V)) -> List(NonNegativeInteger)

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

degree : (%, S) -> NonNegativeInteger

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

degree : (%, V) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

differentialVariables : % -> List(S)

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

differentiate : % -> % if R has DifferentialRing

from DifferentialRing

differentiate : (%, V) -> %

from PartialDifferentialRing(V)

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

from PartialDifferentialRing(V)

differentiate : (%, List(V)) -> %

from PartialDifferentialRing(V)

differentiate : (%, List(V), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(V)

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

from DifferentialExtension(R)

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

from DifferentialExtension(R)

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

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

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

from PartialDifferentialRing(Symbol)

discriminant : (%, V) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents(V), V)

eval : (%, %, %) -> %

from InnerEvalable(%, %)

eval : (%, S, %) -> % if R has DifferentialRing

from InnerEvalable(S, %)

eval : (%, S, R) -> % if R has DifferentialRing

from InnerEvalable(S, R)

eval : (%, V, %) -> %

from InnerEvalable(V, %)

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

from InnerEvalable(V, R)

eval : (%, Equation(%)) -> %

from Evalable(%)

eval : (%, List(%), List(%)) -> %

from InnerEvalable(%, %)

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

from InnerEvalable(S, %)

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

from InnerEvalable(S, R)

eval : (%, List(V), List(%)) -> %

from InnerEvalable(V, %)

eval : (%, List(V), List(R)) -> %

from InnerEvalable(V, R)

eval : (%, List(Equation(%))) -> %

from Evalable(%)

exquo : (%, %) -> Union(%, "failed") if R has EntireRing

from EntireRing

exquo : (%, R) -> Union(%, "failed") if R has EntireRing

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

factor : % -> Factored(%) if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

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

from PolynomialFactorizationExplicit

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

from PolynomialFactorizationExplicit

fmecg : (%, IndexedExponents(V), R, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

gcd : (%, %) -> % if R has GcdDomain

from GcdDomain

gcd : List(%) -> % if R has GcdDomain

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GcdDomain

from PolynomialFactorizationExplicit

ground : % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

ground? : % -> Boolean

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

hash : % -> SingleInteger if R has Hashable and V has Hashable

from Hashable

hashUpdate! : (HashState, %) -> HashState if R has Hashable and V has Hashable

from Hashable

initial : % -> %

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

isExpt : % -> Union(Record(var : V, exponent : NonNegativeInteger), "failed")

from PolynomialCategory(R, IndexedExponents(V), V)

isPlus : % -> Union(List(%), "failed")

from PolynomialCategory(R, IndexedExponents(V), V)

isTimes : % -> Union(List(%), "failed")

from PolynomialCategory(R, IndexedExponents(V), V)

isobaric? : % -> Boolean

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

latex : % -> String

from SetCategory

lcm : (%, %) -> % if R has GcdDomain

from GcdDomain

lcm : List(%) -> % if R has GcdDomain

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %) if R has GcdDomain

from LeftOreRing

leader : % -> V

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

leadingCoefficient : % -> R

from IndexedProductCategory(R, IndexedExponents(V))

leadingMonomial : % -> %

from IndexedProductCategory(R, IndexedExponents(V))

leadingSupport : % -> IndexedExponents(V)

from IndexedProductCategory(R, IndexedExponents(V))

leadingTerm : % -> Record(k : IndexedExponents(V), c : R)

from IndexedProductCategory(R, IndexedExponents(V))

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

linearExtend : (Mapping(R, IndexedExponents(V)), %) -> R if R has CommutativeRing

from FreeModuleCategory(R, IndexedExponents(V))

listOfTerms : % -> List(Record(k : IndexedExponents(V), c : R))

from IndexedDirectProductCategory(R, IndexedExponents(V))

mainVariable : % -> Union(V, "failed")

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

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

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

makeVariable : S -> Mapping(%, NonNegativeInteger)

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

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

from IndexedProductCategory(R, IndexedExponents(V))

mapExponents : (Mapping(IndexedExponents(V), IndexedExponents(V)), %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

minimumDegree : % -> IndexedExponents(V)

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

minimumDegree : (%, List(V)) -> List(NonNegativeInteger)

from PolynomialCategory(R, IndexedExponents(V), V)

minimumDegree : (%, V) -> NonNegativeInteger

from PolynomialCategory(R, IndexedExponents(V), V)

monicDivide : (%, %, V) -> Record(quotient : %, remainder : %)

from PolynomialCategory(R, IndexedExponents(V), V)

monomial : (%, V, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

monomial : (%, List(V), List(NonNegativeInteger)) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

monomial : (R, IndexedExponents(V)) -> %

from IndexedProductCategory(R, IndexedExponents(V))

monomial? : % -> Boolean

from IndexedProductCategory(R, IndexedExponents(V))

monomials : % -> List(%)

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

multivariate : (SparseUnivariatePolynomial(%), V) -> %

from PolynomialCategory(R, IndexedExponents(V), V)

multivariate : (SparseUnivariatePolynomial(R), V) -> %

from PolynomialCategory(R, IndexedExponents(V), V)

numberOfMonomials : % -> NonNegativeInteger

from IndexedDirectProductCategory(R, IndexedExponents(V))

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

order : % -> NonNegativeInteger

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

order : (%, S) -> NonNegativeInteger

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

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

from PatternMatchable(Float)

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

from PatternMatchable(Integer)

plenaryPower : (%, PositiveInteger) -> % if R has CommutativeRing or R has Algebra(Fraction(Integer))

from NonAssociativeAlgebra(%)

pomopo! : (%, R, IndexedExponents(V), %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(V))

prime? : % -> Boolean if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

primitiveMonomials : % -> List(%)

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

primitivePart : % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(V), V)

primitivePart : (%, V) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(V), V)

recip : % -> Union(%, "failed")

from MagmaWithUnit

reducedSystem : Matrix(%) -> Matrix(R)

from LinearlyExplicitOver(R)

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

from LinearlyExplicitOver(Integer)

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

from LinearlyExplicitOver(R)

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

from LinearlyExplicitOver(Integer)

reductum : % -> %

from IndexedProductCategory(R, IndexedExponents(V))

resultant : (%, %, V) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents(V), V)

retract : % -> R

from RetractableTo(R)

retract : % -> S

from RetractableTo(S)

retract : % -> V

from RetractableTo(V)

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

from RetractableTo(Fraction(Integer))

retract : % -> Integer if R has RetractableTo(Integer)

from RetractableTo(Integer)

retract : % -> SparseMultivariatePolynomial(R, S)

from RetractableTo(SparseMultivariatePolynomial(R, S))

retractIfCan : % -> Union(R, "failed")

from RetractableTo(R)

retractIfCan : % -> Union(S, "failed")

from RetractableTo(S)

retractIfCan : % -> Union(V, "failed")

from RetractableTo(V)

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

from RetractableTo(Fraction(Integer))

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

from RetractableTo(Integer)

retractIfCan : % -> Union(SparseMultivariatePolynomial(R, S), "failed")

from RetractableTo(SparseMultivariatePolynomial(R, S))

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

separant : % -> %

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

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

from Comparable

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

from PolynomialFactorizationExplicit

squareFree : % -> Factored(%) if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(V), V)

squareFreePart : % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(V), V)

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

from PolynomialFactorizationExplicit

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

from CancellationAbelianMonoid

support : % -> List(IndexedExponents(V))

from FreeModuleCategory(R, IndexedExponents(V))

totalDegree : % -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

totalDegree : (%, List(V)) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

totalDegreeSorted : (%, List(V)) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

unit? : % -> Boolean if R has EntireRing

from EntireRing

unitCanonical : % -> % if R has EntireRing

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %) if R has EntireRing

from EntireRing

univariate : (%, V) -> SparseUnivariatePolynomial(%)

from PolynomialCategory(R, IndexedExponents(V), V)

univariate : % -> SparseUnivariatePolynomial(R)

from PolynomialCategory(R, IndexedExponents(V), V)

variables : % -> List(V)

from MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

weight : % -> NonNegativeInteger

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

weight : (%, S) -> NonNegativeInteger

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

weights : % -> List(NonNegativeInteger)

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

weights : (%, S) -> List(NonNegativeInteger)

from DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

PatternMatchable(Integer)

CharacteristicNonZero

Module(Fraction(Integer))

NonAssociativeSemiRing

RetractableTo(SparseMultivariatePolynomial(R, S))

LeftModule(R)

CoercibleFrom(V)

BiModule(%, %)

AbelianProductCategory(R)

ConvertibleTo(InputForm)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

FreeModuleCategory(R, IndexedExponents(V))

unitsKnown

FullyLinearlyExplicitOver(R)

PatternMatchable(Float)

PolynomialCategory(R, IndexedExponents(V), V)

DifferentialPolynomialCategory(R, S, V, IndexedExponents(V))

IndexedProductCategory(R, IndexedExponents(V))

AbelianSemiGroup

noZeroDivisors

CoercibleFrom(R)

CoercibleFrom(S)

InnerEvalable(S, %)

SemiGroup

Magma

GcdDomain

IntegralDomain

LeftModule(%)

InnerEvalable(V, R)

UniqueFactorizationDomain

PartialDifferentialRing(Symbol)

CharacteristicZero

Module(R)

CoercibleFrom(Fraction(Integer))

Algebra(%)

InnerEvalable(S, R)

DifferentialRing

BiModule(R, R)

RightModule(Fraction(Integer))

PartialDifferentialRing(V)

InnerEvalable(V, %)

RightModule(R)

NonAssociativeRng

CommutativeRing

NonAssociativeSemiRng

CancellationAbelianMonoid

RetractableTo(Integer)

LinearlyExplicitOver(R)

AbelianMonoidRing(R, IndexedExponents(V))

SetCategory

CommutativeStar

AbelianMonoid

MagmaWithUnit

Comparable

NonAssociativeRing

RightModule(%)

VariablesCommuteWithCoefficients

NonAssociativeAlgebra(%)

Hashable

RetractableTo(S)

CoercibleFrom(SparseMultivariatePolynomial(R, S))

Module(%)

CoercibleTo(OutputForm)

DifferentialExtension(R)

MaybeSkewPolynomialCategory(R, IndexedExponents(V), V)

FiniteAbelianMonoidRing(R, IndexedExponents(V))

ConvertibleTo(Pattern(Float))

SemiRng

LinearlyExplicitOver(Integer)

Monoid

PolynomialFactorizationExplicit

NonAssociativeAlgebra(R)

LeftOreRing

Algebra(R)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

IndexedDirectProductCategory(R, IndexedExponents(V))

InnerEvalable(%, %)

LeftModule(Fraction(Integer))

Evalable(%)

BiModule(Fraction(Integer), Fraction(Integer))

RetractableTo(Fraction(Integer))

RetractableTo(R)

RetractableTo(V)

ConvertibleTo(Pattern(Integer))

AbelianGroup