InnerSparseUnivariatePowerSeries Coef

sups.spad line 1 [edit on github]

• Coef: Ring

InnerSparseUnivariatePowerSeries is an internal domain used for creating sparse Taylor and Laurent series.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, 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

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from AbelianMonoidRing(Coef, Integer)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, Integer) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

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

from EntireRing

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

from NonAssociativeRng

cAcos: % -> % if Coef has Algebra Fraction Integer

cAcos(f) computes the arccosine of the power series f. For use when the coefficient ring is commutative.

cAcosh: % -> % if Coef has Algebra Fraction Integer

cAcosh(f) computes the inverse hyperbolic cosine of the power series f. For use when the coefficient ring is commutative.

cAcot: % -> % if Coef has Algebra Fraction Integer

cAcot(f) computes the arccotangent of the power series f. For use when the coefficient ring is commutative.

cAcoth: % -> % if Coef has Algebra Fraction Integer

cAcoth(f) computes the inverse hyperbolic cotangent of the power series f. For use when the coefficient ring is commutative.

cAcsc: % -> % if Coef has Algebra Fraction Integer

cAcsc(f) computes the arccosecant of the power series f. For use when the coefficient ring is commutative.

cAcsch: % -> % if Coef has Algebra Fraction Integer

cAcsch(f) computes the inverse hyperbolic cosecant of the power series f. For use when the coefficient ring is commutative.

cAsec: % -> % if Coef has Algebra Fraction Integer

cAsec(f) computes the arcsecant of the power series f. For use when the coefficient ring is commutative.

cAsech: % -> % if Coef has Algebra Fraction Integer

cAsech(f) computes the inverse hyperbolic secant of the power series f. For use when the coefficient ring is commutative.

cAsin: % -> % if Coef has Algebra Fraction Integer

cAsin(f) computes the arcsine of the power series f. For use when the coefficient ring is commutative.

cAsinh: % -> % if Coef has Algebra Fraction Integer

cAsinh(f) computes the inverse hyperbolic sine of the power series f. For use when the coefficient ring is commutative.

cAtan: % -> % if Coef has Algebra Fraction Integer

cAtan(f) computes the arctangent of the power series f. For use when the coefficient ring is commutative.

cAtanh: % -> % if Coef has Algebra Fraction Integer

cAtanh(f) computes the inverse hyperbolic tangent of the power series f. For use when the coefficient ring is commutative.

cCos: % -> % if Coef has Algebra Fraction Integer

cCos(f) computes the cosine of the power series f. For use when the coefficient ring is commutative.

cCosh: % -> % if Coef has Algebra Fraction Integer

cCosh(f) computes the hyperbolic cosine of the power series f. For use when the coefficient ring is commutative.

cCot: % -> % if Coef has Algebra Fraction Integer

cCot(f) computes the cotangent of the power series f. For use when the coefficient ring is commutative.

cCoth: % -> % if Coef has Algebra Fraction Integer

cCoth(f) computes the hyperbolic cotangent of the power series f. For use when the coefficient ring is commutative.

cCsc: % -> % if Coef has Algebra Fraction Integer

cCsc(f) computes the cosecant of the power series f. For use when the coefficient ring is commutative.

cCsch: % -> % if Coef has Algebra Fraction Integer

cCsch(f) computes the hyperbolic cosecant of the power series f. For use when the coefficient ring is commutative.

center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

cExp: % -> % if Coef has Algebra Fraction Integer

cExp(f) computes the exponential of the power series f. For use when the coefficient ring is commutative.

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

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

from CharacteristicNonZero

cLog: % -> % if Coef has Algebra Fraction Integer

cLog(f) computes the logarithm of the power series f. For use when the coefficient ring is commutative.

coefficient: (%, Integer) -> Coef

from AbelianMonoidRing(Coef, Integer)

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

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

construct: List Record(k: Integer, c: Coef) -> %

from IndexedProductCategory(Coef, Integer)

constructOrdered: List Record(k: Integer, c: Coef) -> %

from IndexedProductCategory(Coef, Integer)

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

cPower(f, r) computes f^r, where f has constant coefficient 1. For use when the coefficient ring is commutative.

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

cRationalPower(f, r) computes f^r. For use when the coefficient ring is commutative.

cSec: % -> % if Coef has Algebra Fraction Integer

cSec(f) computes the secant of the power series f. For use when the coefficient ring is commutative.

cSech: % -> % if Coef has Algebra Fraction Integer

cSech(f) computes the hyperbolic secant of the power series f. For use when the coefficient ring is commutative.

cSin: % -> % if Coef has Algebra Fraction Integer

cSin(f) computes the sine of the power series f. For use when the coefficient ring is commutative.

cSinh: % -> % if Coef has Algebra Fraction Integer

cSinh(f) computes the hyperbolic sine of the power series f. For use when the coefficient ring is commutative.

cTan: % -> % if Coef has Algebra Fraction Integer

cTan(f) computes the tangent of the power series f. For use when the coefficient ring is commutative.

cTanh: % -> % if Coef has Algebra Fraction Integer

cTanh(f) computes the hyperbolic tangent of the power series f. For use when the coefficient ring is commutative.

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

from DifferentialRing

D: (%, List Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

D: (%, Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

degree: % -> Integer

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

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

from DifferentialRing

differentiate: (%, List Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

differentiate: (%, Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

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

from EntireRing

extend: (%, Integer) -> %

from UnivariatePowerSeriesCategory(Coef, Integer)

getRef: % -> Reference OrderedCompletion Integer

getRef(f) returns a reference containing the order to which the terms of f have been computed.

getStream: % -> Stream Record(k: Integer, c: Coef)

getStream(f) returns the stream of terms representing the series f.

iCompose: (%, %) -> %

iCompose(f, g) returns f(g(x)). This is an internal function which should only be called for Taylor series f(x) and g(x) such that the constant coefficient of g(x) is zero.

iExquo: (%, %, Boolean) -> Union(%, failed)

iExquo(f, g, taylor?) is the quotient of the power series f and g. If taylor? is true, then we must have order(f) >= order(g).

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

integrate(f(x)) returns an anti-derivative of the power series f(x) with constant coefficient 0. Warning: function does not check for a term of degree -1.

latex: % -> String

from SetCategory

leadingCoefficient: % -> Coef

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

leadingMonomial: % -> %

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

leadingSupport: % -> Integer

from IndexedProductCategory(Coef, Integer)

leadingTerm: % -> Record(k: Integer, c: Coef)

from IndexedProductCategory(Coef, Integer)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

makeSeries: (Reference OrderedCompletion Integer, Stream Record(k: Integer, c: Coef)) -> %

makeSeries(refer, str) creates a power series from the reference refer and the stream str.

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

from IndexedProductCategory(Coef, Integer)

monomial?: % -> Boolean

monomial?(f) tests if f is a single monomial.

monomial: (Coef, Integer) -> %

from IndexedProductCategory(Coef, Integer)

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

multiplyCoefficients(fn, f) returns the series sum(fn(n) * an * x^n, n = n0..), where f is the series sum(an * x^n, n = n0..).

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(Coef, Integer)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> Integer

from UnivariatePowerSeriesCategory(Coef, Integer)

order: (%, Integer) -> Integer

from UnivariatePowerSeriesCategory(Coef, Integer)

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

from NonAssociativeAlgebra %

pole?: % -> Boolean

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(Coef, Integer)

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

series: Stream Record(k: Integer, c: Coef) -> %

series(st) creates a series from a stream of non-zero terms, where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.

seriesToOutputForm: (Stream Record(k: Integer, c: Coef), Reference OrderedCompletion Integer, Symbol, Coef, Fraction Integer) -> OutputForm

seriesToOutputForm(st, refer, var, cen, r) prints the series f((var - cen)^r).

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

from CancellationAbelianMonoid

taylorQuoByVar: % -> %

taylorQuoByVar(a0 + a1 x + a2 x^2 + ...) returns a1 + a2 x + a3 x^2 + ...

terms: % -> Stream Record(k: Integer, c: Coef)

from UnivariatePowerSeriesCategory(Coef, Integer)

truncate: (%, Integer) -> %

from UnivariatePowerSeriesCategory(Coef, Integer)

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

from UnivariatePowerSeriesCategory(Coef, 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

variable: % -> Symbol

from UnivariatePowerSeriesCategory(Coef, Integer)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Integer)

AbelianProductCategory Coef

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

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

DifferentialRing if Coef has *: (Integer, Coef) -> Coef

Eltable(%, %)

EntireRing if Coef has IntegralDomain

IndexedProductCategory(Coef, Integer)

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Monoid

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Symbol if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

UnivariatePowerSeriesCategory(Coef, Integer)

VariablesCommuteWithCoefficients