InnerSparseUnivariatePowerSeries CoefΒΆ

sups.spad line 1

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

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
+: (%, %) -> %
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)
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 PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
from DifferentialRing
D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
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 PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
from DifferentialRing
differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
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.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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)
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 AbelianMonoidRing(Coef, Integer)
monomial: (%, List SingletonAsOrderedSet, List Integer) -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, Integer) -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
monomial: (Coef, Integer) -> %
from AbelianMonoidRing(Coef, Integer)
monomial?: % -> Boolean
monomial?(f) tests if f is a single monomial.
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)
pole?: % -> Boolean
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> %
from AbelianMonoidRing(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)
variables: % -> List SingletonAsOrderedSet
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Integer)

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

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

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

UnivariatePowerSeriesCategory(Coef, Integer)

VariablesCommuteWithCoefficients