InnerSparseUnivariatePowerSeries(Coef)

sups.spad line 1 [edit on github]

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

* : (%, %) -> %
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)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
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)
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
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.

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.

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.

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.

center : % -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if Coef has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, Integer) -> Coef
from AbelianMonoidRing(Coef, Integer)
coerce : % -> % if Coef has CommutativeRing
from Algebra(%)
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
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
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)
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 : (Mapping(Coef, Coef), %) -> %
from IndexedProductCategory(Coef, Integer)
monomial : (Coef, Integer) -> %
from IndexedProductCategory(Coef, Integer)
monomial? : % -> Boolean

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

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

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(Coef)
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
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

AbelianMonoidRing(Coef, Integer)

BiModule(%, %)

Rng

TwoSidedRecip

RightModule(Coef)

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

unitsKnown

IndexedProductCategory(Coef, Integer)

MagmaWithUnit

AbelianProductCategory(Coef)

Magma

SemiGroup

IntegralDomain

LeftModule(%)

UnivariatePowerSeriesCategory(Coef, Integer)

NonAssociativeRing

PartialDifferentialRing(Symbol)

CharacteristicZero

Algebra(%)

DifferentialRing

RightModule(Fraction(Integer))

CommutativeRing

Eltable(%, %)

NonAssociativeSemiRng

EntireRing

CancellationAbelianMonoid

VariablesCommuteWithCoefficients

SemiRing

CommutativeStar

AbelianMonoid

RightModule(%)

BiModule(Coef, Coef)

LeftModule(Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

SetCategory

noZeroDivisors

NonAssociativeRng

BiModule(Fraction(Integer), Fraction(Integer))

AbelianGroup