SparseMultivariateTaylorSeries(Coef, Var, SMP)¶

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 LeftModule %

*: (%, Coef) -> %

from RightModule Coef

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

from LeftModule Coef

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (SMP, %) -> %

`smp*ts` multiplies a TaylorSeries by a monomial `SMP`.

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from AbelianMonoidRing(Coef, IndexedExponents Var)

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> % if Coef has Algebra Fraction Integer
^: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> % if Coef has Algebra Fraction Integer
acosh: % -> % if Coef has Algebra Fraction Integer
acot: % -> % if Coef has Algebra Fraction Integer
acoth: % -> % if Coef has Algebra Fraction Integer
acsc: % -> % if Coef has Algebra Fraction Integer
acsch: % -> % if Coef has Algebra Fraction Integer
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
asec: % -> % if Coef has Algebra Fraction Integer
asech: % -> % if Coef has Algebra Fraction Integer
asin: % -> % if Coef has Algebra Fraction Integer
asinh: % -> % if Coef has Algebra Fraction Integer
associates?: (%, %) -> Boolean if Coef has IntegralDomain

from EntireRing

associator: (%, %, %) -> %
atan: % -> % if Coef has Algebra Fraction Integer
atanh: % -> % if Coef has Algebra Fraction Integer
characteristic: () -> NonNegativeInteger
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 degree `n`.

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

from MultivariateTaylorSeriesCategory(Coef, Var)

coefficients: % -> Stream SMP

`coefficients(s)` gives stream of coefficients of `s`, i.e. [coefficient(`s`,0), coefficient(`s`,1), …]

coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
coerce: Integer -> %
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: (%, %) -> %
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)

cos: % -> % if Coef has Algebra Fraction Integer
cosh: % -> % if Coef has Algebra Fraction Integer
cot: % -> % if Coef has Algebra Fraction Integer
coth: % -> % if Coef has Algebra Fraction Integer
csc: % -> % if Coef has Algebra Fraction Integer
csch: % -> % if Coef has Algebra Fraction Integer
csubst: (List Var, List Stream SMP) -> SMP -> Stream SMP

`csubst(a, b)` is for internal use only

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, %)

exp: % -> % if Coef has Algebra Fraction Integer
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 of `f()` with respect to `v` and having `c` as the constant of integration. The evaluation of `f()` 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 of `s` with respect to `v` and having `c` as the constant of integration.

latex: % -> String

from SetCategory

from PowerSeriesCategory(Coef, IndexedExponents Var, Var)

from PowerSeriesCategory(Coef, IndexedExponents Var, 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

log: % -> % if Coef has Algebra Fraction Integer
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)

nthRoot: (%, Integer) -> % if Coef has Algebra Fraction Integer

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: (%, Var) -> NonNegativeInteger

from MultivariateTaylorSeriesCategory(Coef, Var)

order: (%, Var, NonNegativeInteger) -> NonNegativeInteger

from MultivariateTaylorSeriesCategory(Coef, Var)

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

sec: % -> % if Coef has Algebra Fraction Integer
sech: % -> % if Coef has Algebra Fraction Integer
series: Stream SMP -> %

`series(st)` creates a series from a stream of coefficients.

sin: % -> % if Coef has Algebra Fraction Integer
sinh: % -> % if Coef has Algebra Fraction Integer
sqrt: % -> % if Coef has Algebra Fraction Integer

subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
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

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, IndexedExponents Var)

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Coef, Coef)

BiModule(Fraction Integer, Fraction Integer) if Coef has Algebra Fraction Integer

CancellationAbelianMonoid

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

EntireRing if Coef has IntegralDomain

IndexedProductCategory(Coef, IndexedExponents Var)

InnerEvalable(%, %)

InnerEvalable(Var, %)

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Monoid

MultivariateTaylorSeriesCategory(Coef, Var)

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PowerSeriesCategory(Coef, IndexedExponents Var, Var)

RadicalCategory if Coef has Algebra Fraction Integer

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients