QEtaTruncatedTaylorSeries C

qetaser.spad line 432 [edit on github]

• C: Ring

QEtaTruncatedTruncatedSeries(C) implements truncated Taylor series. They work like Laurent polynomials, but throw away any coefficients that fall into the O(q^n) category. Series operation are done directly and not in a lazy fashion. The domain distinguishes between exact and approximated series. An exact series can be expanded (filled with zeros) to any precision. Addition and multiplication of exact series works like polynomials. Addition and multiplication of exact and approximated series gives an approximated series.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, C) -> %

from RightModule C

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

from RightModule Fraction Integer

*: (C, %) -> %

from LeftModule C

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

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from AbelianMonoidRing(C, NonNegativeInteger)

=: (%, %) -> Boolean

from BasicType

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

from ElementaryFunctionCategory

^: (%, C) -> % if C has Field

from UnivariateTaylorSeriesCategory C

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

from RadicalCategory

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acosh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

acot: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acoth: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

acsc: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acsch: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, NonNegativeInteger) -> C if C has ^: (C, NonNegativeInteger) -> C and C has coerce: Symbol -> C

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

asec: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

asech: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

asin: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

asinh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

associates?: (%, %) -> Boolean if C has IntegralDomain

from EntireRing

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

from NonAssociativeRng

atan: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

atanh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

center: % -> C

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if C has CharacteristicNonZero

from CharacteristicNonZero

coefficient: (%, NonNegativeInteger) -> C

from AbelianMonoidRing(C, NonNegativeInteger)

coefficients: % -> Stream C

from UnivariateTaylorSeriesCategory C

coerce: % -> % if C has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: % -> SparseUnivariatePolynomial C

coerce(x) turns the known coefficients into a univariate polynomial.

coerce: C -> % if C has CommutativeRing

from Algebra C

coerce: Fraction Integer -> % if C has Algebra Fraction Integer

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: SparseUnivariatePolynomial C -> %

coerce(p) embeds the univariate polynomial p as exact data into this domain.

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

construct: List Record(k: NonNegativeInteger, c: C) -> %

from IndexedProductCategory(C, NonNegativeInteger)

constructOrdered: List Record(k: NonNegativeInteger, c: C) -> %

from IndexedProductCategory(C, NonNegativeInteger)

cos: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

cosh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

cot: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

coth: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

csc: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

csch: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

D: % -> % if C has *: (NonNegativeInteger, C) -> C

from DifferentialRing

D: (%, List Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if C has *: (NonNegativeInteger, C) -> C

from DifferentialRing

D: (%, Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

degree: % -> NonNegativeInteger

from PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

differentiate: % -> % if C has *: (NonNegativeInteger, C) -> C

from DifferentialRing

differentiate: (%, List Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if C has *: (NonNegativeInteger, C) -> C

from DifferentialRing

differentiate: (%, Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

from PartialDifferentialRing Symbol

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, NonNegativeInteger) -> C

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

eval: (%, C) -> Stream C if C has ^: (C, NonNegativeInteger) -> C

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

exact?: % -> Boolean

If exact?(x) returns true then x represents exact data, i.e. no approximation term is involved. 0 and 1 are exact as are inclusions of univariate polynomials into this domain. precision(x) is then an indicator of the stored coefficients of x, [coefficient(x,i) for i in 0..precision(x)-1]. For any i>=precision(x), coefficient(x,i) returns 0. If exact?(x) is false, then x is know only approximately up to precision(x).

exp: % -> % if C has Algebra Fraction Integer

from ElementaryFunctionCategory

exponentGcd: (%, PositiveInteger) -> Integer

from QEtaTaylorSeriesCategory C

exquo: (%, %) -> Union(%, failed) if C has IntegralDomain

from EntireRing

extend: (%, NonNegativeInteger) -> %

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

integrate: % -> % if C has Algebra Fraction Integer

from UnivariateSeriesWithRationalExponents(C, NonNegativeInteger)

integrate: (%, Symbol) -> % if C has variables: C -> List Symbol and C has Algebra Fraction Integer and C has integrate: (C, Symbol) -> C

from UnivariateSeriesWithRationalExponents(C, NonNegativeInteger)

latex: % -> String

from SetCategory

leadingCoefficient: % -> C

from PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

leadingMonomial: % -> %

from PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

leadingSupport: % -> NonNegativeInteger

from IndexedProductCategory(C, NonNegativeInteger)

leadingTerm: % -> Record(k: NonNegativeInteger, c: C)

from IndexedProductCategory(C, NonNegativeInteger)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

log: % -> % if C has Algebra Fraction Integer

from ElementaryFunctionCategory

map: (C -> C, %) -> %

from IndexedProductCategory(C, NonNegativeInteger)

mapn: ((C, Integer) -> C, %, Integer) -> %

from QEtaTaylorSeriesCategory C

monomial?: % -> Boolean

from IndexedProductCategory(C, NonNegativeInteger)

monomial: (C, NonNegativeInteger) -> %

from IndexedProductCategory(C, NonNegativeInteger)

multiplyCoefficients: (Integer -> C, %) -> %

from UnivariateTaylorSeriesCategory C

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

multisect: (Integer, Integer, %) -> %

from QEtaTaylorSeriesCategory C

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

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> NonNegativeInteger

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

order: (%, NonNegativeInteger) -> NonNegativeInteger

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

pi: () -> % if C has Algebra Fraction Integer

from TranscendentalFunctionCategory

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

from NonAssociativeAlgebra %

pole?: % -> Boolean

from PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

polynomial: (%, NonNegativeInteger) -> Polynomial C

from UnivariateTaylorSeriesCategory C

polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial C

from UnivariateTaylorSeriesCategory C

precision: % -> NonNegativeInteger

precision(x) returns the number of valid coefficients, i.e. the valid coefficients are [coefficient(x,i) for i in 0..precision(x)-1].

quoByVar: % -> %

from UnivariateTaylorSeriesCategory C

recip: % -> Union(%, failed)

from MagmaWithUnit

reduce: (%, C, %) -> %

reduce(x,c,y) returns x+c*y.

reduce: (%, C, Integer, %) -> %

reduce(x,c,e,y) returns x+c*q^e*y where q is the variable of this domain if e>=0. If e<0, then it returns q^(-e)*x+c*y.

reductum: % -> %

from IndexedProductCategory(C, NonNegativeInteger)

removeInitial: (%, Integer) -> %

reductum(x, n) is the identity for n<=0 and otherwise removes n leading coefficients from the datastructure. It is like dividing x by q^n and removing all terms with negative exponent (where q corresponds to the variable of this domain).

revert: % -> %

from QEtaTaylorSeriesCategory C

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

sech: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

series: Stream C -> %

from UnivariateTaylorSeriesCategory C

series: Stream Record(k: NonNegativeInteger, c: C) -> %

from UnivariateTaylorSeriesCategory C

sin: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

sinh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

sqrt: % -> % if C has Algebra Fraction Integer

from RadicalCategory

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

from CancellationAbelianMonoid

tan: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

tanh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

taylor: List C -> %

taylor(cs) returns an (inexact) truncated series whose coefficients are given by the list cs.

terms: % -> Stream Record(k: NonNegativeInteger, c: C)

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

truncate: (%, NonNegativeInteger) -> %

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

truncate: (%, NonNegativeInteger, NonNegativeInteger) -> %

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

truncateAt: (%, NonNegativeInteger) -> %

truncateAt(t, n) yields a truncated Taylor series at order k=min(n,precision(t)), i.e. precision of the result will be k.

truncateAt: (QEtaTaylorSeries C, NonNegativeInteger) -> %

truncateAt(t, n) yields a truncated Taylor series at order n>0, i.e. precision of the result will be n.

unit?: % -> Boolean if C has IntegralDomain

from EntireRing

unitCanonical: % -> % if C has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if C has IntegralDomain

from EntireRing

variable: % -> Symbol

from UnivariatePowerSeriesCategory(C, NonNegativeInteger)

zero?: % -> Boolean

Since we cannot reliably check whether x respresents 0, this function returns true only of exact?(x) holds and all coefficients vanish. Otherwise it returns always false.

AbelianGroup

AbelianMonoid

AbelianMonoidRing(C, NonNegativeInteger)

AbelianProductCategory C

AbelianSemiGroup

Algebra % if C has CommutativeRing

Algebra C if C has CommutativeRing

Algebra Fraction Integer if C has Algebra Fraction Integer

ArcHyperbolicFunctionCategory if C has Algebra Fraction Integer

ArcTrigonometricFunctionCategory if C has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(C, C)

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

CancellationAbelianMonoid

CharacteristicNonZero if C has CharacteristicNonZero

CharacteristicZero if C has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if C has CommutativeRing

CommutativeStar if C has CommutativeRing

DifferentialRing if C has *: (NonNegativeInteger, C) -> C

ElementaryFunctionCategory if C has Algebra Fraction Integer

Eltable(%, %)

EntireRing if C has IntegralDomain

HyperbolicFunctionCategory if C has Algebra Fraction Integer

IndexedProductCategory(C, NonNegativeInteger)

IntegralDomain if C has IntegralDomain

LeftModule %

LeftModule C

LeftModule Fraction Integer if C has Algebra Fraction Integer

Magma

MagmaWithUnit

Module % if C has CommutativeRing

Module C if C has CommutativeRing

Module Fraction Integer if C has Algebra Fraction Integer

Monoid

NonAssociativeAlgebra % if C has CommutativeRing

NonAssociativeAlgebra C if C has CommutativeRing

NonAssociativeAlgebra Fraction Integer if C has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if C has IntegralDomain

PartialDifferentialRing Symbol if C has PartialDifferentialRing Symbol and C has *: (NonNegativeInteger, C) -> C

PowerSeriesCategory(C, NonNegativeInteger, SingletonAsOrderedSet)

QEtaTaylorSeriesCategory C

RadicalCategory if C has Algebra Fraction Integer

RightModule %

RightModule C

RightModule Fraction Integer if C has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory if C has Algebra Fraction Integer

TrigonometricFunctionCategory if C has Algebra Fraction Integer

TwoSidedRecip if C has CommutativeRing

unitsKnown

UnivariatePowerSeriesCategory(C, NonNegativeInteger)

UnivariateSeriesWithRationalExponents(C, NonNegativeInteger)

UnivariateTaylorSeriesCategory C

VariablesCommuteWithCoefficients