SparseMultivariateTaylorSeries(Coef, Var, SMP)¶
mts.spad line 1 [edit on github]
Coef: Ring
Var: OrderedSet
SMP: PolynomialCategory(Coef, IndexedExponents Var, Var)
This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain SMP
. The n
th element of the stream is a form of degree n
. SMTS is an internal domain.
- 0: %
from AbelianMonoid
- 1: %
from MagmaWithUnit
- *: (%, %) -> %
from Magma
- *: (%, Coef) -> %
from RightModule Coef
- *: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
from RightModule Fraction Integer
- *: (Coef, %) -> %
from LeftModule Coef
- *: (Fraction Integer, %) -> % if Coef has Algebra Fraction Integer
from LeftModule Fraction Integer
- *: (Integer, %) -> %
from AbelianGroup
- *: (NonNegativeInteger, %) -> %
from AbelianMonoid
- *: (PositiveInteger, %) -> %
from AbelianSemiGroup
- *: (SMP, %) -> %
smp*ts
multiplies a TaylorSeriests
by a monomialsmp
.
- +: (%, %) -> %
from AbelianSemiGroup
- -: % -> %
from AbelianGroup
- -: (%, %) -> %
from AbelianGroup
- /: (%, Coef) -> % if Coef has Field
from AbelianMonoidRing(Coef, IndexedExponents Var)
- ^: (%, %) -> % if Coef has Algebra Fraction Integer
- ^: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
from RadicalCategory
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associates?: (%, %) -> Boolean if Coef has IntegralDomain
from EntireRing
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- coefficient: (%, IndexedExponents Var) -> Coef
from AbelianMonoidRing(Coef, IndexedExponents Var)
- coefficient: (%, List Var, List NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Var)
- coefficient: (%, NonNegativeInteger) -> SMP
coefficient(s, n)
gives the terms of total degreen
.- coefficient: (%, Var, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Var)
- coefficients: % -> Stream SMP
coefficients(s)
gives stream of coefficients ofs
, i.e. [coefficient(s
,0), coefficient(s
,1), …]
- coerce: % -> % if Coef has CommutativeRing
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: Coef -> % if Coef has CommutativeRing
from Algebra Coef
- coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
- coerce: Integer -> %
from NonAssociativeRing
- coerce: SMP -> %
coerce(poly)
regroups the terms by total degree and forms a series.
- coerce: Var -> %
coerce(var)
converts a variable to a Taylor series
- commutator: (%, %) -> %
from NonAssociativeRng
- complete: % -> %
from PowerSeriesCategory(Coef, IndexedExponents Var, Var)
- construct: List Record(k: IndexedExponents Var, c: Coef) -> %
from IndexedProductCategory(Coef, IndexedExponents Var)
- constructOrdered: List Record(k: IndexedExponents Var, c: Coef) -> %
from IndexedProductCategory(Coef, IndexedExponents Var)
- D: (%, List Var) -> %
from PartialDifferentialRing Var
- D: (%, List Var, List NonNegativeInteger) -> %
from PartialDifferentialRing Var
- D: (%, Var) -> %
from PartialDifferentialRing Var
- D: (%, Var, NonNegativeInteger) -> %
from PartialDifferentialRing Var
- degree: % -> IndexedExponents Var
from PowerSeriesCategory(Coef, IndexedExponents Var, Var)
- differentiate: (%, List Var) -> %
from PartialDifferentialRing Var
- differentiate: (%, List Var, List NonNegativeInteger) -> %
from PartialDifferentialRing Var
- differentiate: (%, Var) -> %
from PartialDifferentialRing Var
- differentiate: (%, Var, NonNegativeInteger) -> %
from PartialDifferentialRing Var
- eval: (%, %, %) -> %
from InnerEvalable(%, %)
- eval: (%, Equation %) -> %
from Evalable %
- eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
- eval: (%, List Equation %) -> %
from Evalable %
- eval: (%, List Var, List %) -> %
from InnerEvalable(Var, %)
- eval: (%, Var, %) -> %
from InnerEvalable(Var, %)
- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
from EntireRing
- extend: (%, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Var)
- fintegrate: (() -> %, Var, Coef) -> % if Coef has Algebra Fraction Integer
fintegrate(f, v, c)
is the integral off()
with respect tov
and havingc
as the constant of integration. The evaluation off()
is delayed.
- integrate: (%, Var) -> % if Coef has Algebra Fraction Integer
from MultivariateTaylorSeriesCategory(Coef, Var)
- integrate: (%, Var, Coef) -> % if Coef has Algebra Fraction Integer
integrate(s, v, c)
is the integral ofs
with respect tov
and havingc
as the constant of integration.
- latex: % -> String
from SetCategory
- leadingCoefficient: % -> Coef
from PowerSeriesCategory(Coef, IndexedExponents Var, Var)
- leadingMonomial: % -> %
from PowerSeriesCategory(Coef, IndexedExponents Var, Var)
- leadingSupport: % -> IndexedExponents Var
from IndexedProductCategory(Coef, IndexedExponents Var)
- leadingTerm: % -> Record(k: IndexedExponents Var, c: Coef)
from IndexedProductCategory(Coef, IndexedExponents Var)
- leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
from Magma
- leftRecip: % -> Union(%, failed)
from MagmaWithUnit
- map: (Coef -> Coef, %) -> %
from IndexedProductCategory(Coef, IndexedExponents Var)
- monomial?: % -> Boolean
from IndexedProductCategory(Coef, IndexedExponents Var)
- monomial: (%, List Var, List NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Var)
- monomial: (%, Var, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Var)
- monomial: (Coef, IndexedExponents Var) -> %
from IndexedProductCategory(Coef, IndexedExponents Var)
- one?: % -> Boolean
from MagmaWithUnit
- opposite?: (%, %) -> Boolean
from AbelianMonoid
- order: (%, Var) -> NonNegativeInteger
from MultivariateTaylorSeriesCategory(Coef, Var)
- order: (%, Var, NonNegativeInteger) -> NonNegativeInteger
from MultivariateTaylorSeriesCategory(Coef, Var)
- plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra Fraction Integer
from NonAssociativeAlgebra Coef
- pole?: % -> Boolean
from PowerSeriesCategory(Coef, IndexedExponents Var, Var)
- polynomial: (%, NonNegativeInteger) -> Polynomial Coef
from MultivariateTaylorSeriesCategory(Coef, Var)
- polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial Coef
from MultivariateTaylorSeriesCategory(Coef, Var)
- recip: % -> Union(%, failed)
from MagmaWithUnit
- reductum: % -> %
from IndexedProductCategory(Coef, IndexedExponents Var)
- rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
from Magma
- rightRecip: % -> Union(%, failed)
from MagmaWithUnit
- sample: %
from AbelianMonoid
- series: Stream SMP -> %
series(st)
creates a series from a stream of coefficients.
- sqrt: % -> % if Coef has Algebra Fraction Integer
from RadicalCategory
- subtractIfCan: (%, %) -> Union(%, failed)
- unit?: % -> Boolean if Coef has IntegralDomain
from EntireRing
- unitCanonical: % -> % if Coef has IntegralDomain
from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %) if Coef has IntegralDomain
from EntireRing
- zero?: % -> Boolean
from AbelianMonoid
AbelianMonoidRing(Coef, IndexedExponents Var)
Algebra % if Coef has CommutativeRing
Algebra Coef if Coef has CommutativeRing
Algebra Fraction Integer if Coef has Algebra Fraction Integer
ArcHyperbolicFunctionCategory if Coef has Algebra Fraction Integer
ArcTrigonometricFunctionCategory if Coef has Algebra Fraction Integer
BiModule(%, %)
BiModule(Coef, Coef)
BiModule(Fraction Integer, Fraction Integer) if Coef has Algebra Fraction Integer
CharacteristicNonZero if Coef has CharacteristicNonZero
CharacteristicZero if Coef has CharacteristicZero
CommutativeRing if Coef has CommutativeRing
CommutativeStar if Coef has CommutativeRing
ElementaryFunctionCategory if Coef has Algebra Fraction Integer
EntireRing if Coef has IntegralDomain
Evalable %
HyperbolicFunctionCategory if Coef has Algebra Fraction Integer
IndexedProductCategory(Coef, IndexedExponents Var)
InnerEvalable(%, %)
InnerEvalable(Var, %)
IntegralDomain if Coef has IntegralDomain
LeftModule Coef
LeftModule Fraction Integer if Coef has Algebra Fraction Integer
Module % if Coef has CommutativeRing
Module Coef if Coef has CommutativeRing
Module Fraction Integer if Coef has Algebra Fraction Integer
MultivariateTaylorSeriesCategory(Coef, Var)
NonAssociativeAlgebra % if Coef has CommutativeRing
NonAssociativeAlgebra Coef if Coef has CommutativeRing
NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer
noZeroDivisors if Coef has IntegralDomain
PowerSeriesCategory(Coef, IndexedExponents Var, Var)
RadicalCategory if Coef has Algebra Fraction Integer
RightModule Coef
RightModule Fraction Integer if Coef has Algebra Fraction Integer
TranscendentalFunctionCategory if Coef has Algebra Fraction Integer
TrigonometricFunctionCategory if Coef has Algebra Fraction Integer
TwoSidedRecip if Coef has CommutativeRing