Power series

AbstractAlgebra.jl allows the creation of capped relative and absolute power series over any computable commutative ring $R$.

Capped relative power series are power series of the form $a_jx^j + a_{j+1}x^{j+1} + \cdots + a_{k-1}x^{k-1} + O(x^k)$ where $a_j \in R$ and the relative precision $k - j$ is at most equal to some specified precision $n$.

Capped absolute power series are power series of the form $a_jx^j + a_{j+1}x^{j+1} + \cdots + a_{n-1}x^{n-1} + O(x^n)$ where $j \geq 0$, $a_j \in R$ and the precision $n$ is fixed.

There are two implementations of relative series: relative power series, implemented in src/RelSeries.jl for which $j > 0$ in the above description, and Laurent series where $j$ can be negative, implemented in src/Laurent.jl. Note that there are two implementations for Laurent series, one over rings and one over fields, though in practice most of the implementation uses the same code in both cases.

There is a single implementation of absolute series: absolute power series, implemented in src/AbsSeries.jl.

Generic power series types

AbstractAlgebra.jl provides generic series types implemented in src/generic/AbsSeries.jl, src/generic/RelSeries.jl and src/generic/LaurentSeries.jl which implement the Series interface.

These generic series have types Generic.RelSeries{T}, Generic.AbsSeries{T}, Generic.LaurentSeriesRingElem{T} and Generic.LaurentSeriesFieldElem{T}. See the file src/generic/GenericTypes.jl for details.

The parent objects have types Generic.AbsPowerSeriesRing{T} and Generic.RelPowerSeriesRing{T} and Generic.LaurentSeriesRing{T} respectively.

The default precision, string representation of the variable and base ring $R$ of a generic power series are stored in its parent object.

Abstract types

Relative power series elements belong to the abstract type RelPowerSeriesRingElem.

Laurent series elements belong directly to either RingElem or FieldElem since it is more useful to be able to distinguish whether they belong to a ring or field than it is to distinguish that they are relative series.

Absolute power series elements belong to AbsPowerSeriesRingElem.

The parent types for relative and absolute power series, Generic.RelPowerSeriesRing{T} and Generic.AbsPowerSeriesRing{T} respectively, belong to SeriesRing{T}.

The parent types of Laurent series belong directly to Ring and Field respectively.

Series ring constructors

In order to construct series in AbstractAlgebra.jl, one must first construct the ring itself. This is accomplished with any of the following constructors.

power_series_ring(R::Ring, prec_max::Int, s::VarName; cached::Bool = true, model::Symbol=:capped_relative)

laurent_series_ring(R::Ring, prec_max::Int, s::VarName; cached::Bool = true)

laurent_series_ring(R::Field, prec_max::Int, s::VarName; cached::Bool = true)

Given a base ring R, a maximum precision (relative or absolute, depending on the model) and a string s specifying how the generator (variable) should be printed, return a tuple S, x representing the series ring and its generator.

By default, S will depend only on S, x and the maximum precision and will be cached. Setting the optional argument cached to false will prevent this.

In the case of power series, the optional argument model can be set to either :capped_absolute or :capped_relative, depending on which power series model is required.

It is also possible to construct absolute and relative power series with a default variable. These are lightweight constructors and should be used in generic algorithms wherever possible when creating series rings where the symbol does not matter.

AbsPowerSeriesRing(R::Ring, prec::Int)
RelPowerSeriesRing(R::Ring, prec::Int)

Return the absolute or relative power series ring over the given base ring $R$ and with precision cap given by prec. Note that a tuple is not returned, only the power series ring itself, not a generator.

Here are some examples of constructing various kinds of series rings and coercing various elements into those rings.

Examples

julia> R, x = power_series_ring(ZZ, 10, :x)
(Univariate power series ring over integers, x + O(x^11))

julia> S, y = power_series_ring(ZZ, 10, :y; model=:capped_absolute)
(Univariate power series ring over integers, y + O(y^10))

julia> T, z = laurent_series_ring(ZZ, 10, :z)
(Laurent series ring in z over integers, z + O(z^11))

julia> U, w = laurent_series_field(QQ, 10, :w)
(Laurent series field in w over rationals, w + O(w^11))

julia> f = R()
O(x^10)

julia> g = S(123)
123 + O(y^10)

julia> h = U(BigInt(1234))
1234 + O(w^10)

julia> k = T(z + 1)
1 + z + O(z^10)

julia> V = AbsPowerSeriesRing(ZZ, 10)
Univariate power series ring in x with precision 10
  over integers

Power series constructors

Series can be constructed using arithmetic operators using the generator of the series. Also see the big-oh notation below for specifying the precision.

All of the standard ring constructors can also be used to construct power series.

(R::SeriesRing)() # constructs zero
(R::SeriesRing)(c::Integer)
(R::SeriesRing)(c::elem_type(R))
(R::SeriesRing{T})(a::T) where T <: RingElement

In addition, the following constructors that are specific to power series are provided. They take an array of coefficients, a length, precision and valuation. Coefficients will be coerced into the coefficient ring if they are not already in that ring.

For relative series we have:

(S::SeriesRing{T})(A::Vector{T}, len::Int, prec::Int, val::Int) where T <: RingElem
(S::SeriesRing{T})(A::Vector{U}, len::Int, prec::Int, val::Int) where {T <: RingElem, U <: RingElem}
(S::SeriesRing{T})(A::Vector{U}, len::Int, prec::Int, val::Int) where {T <: RingElem, U <: Integer}

And for absolute series:

(S::SeriesRing{T})(A::Vector{T}, len::Int, prec::Int) where T <: RingElem

It is also possible to create series directly without having to create the corresponding series ring.

abs_series(R::Ring, arr::Vector{T}, len::Int, prec::Int, var::VarName=:x; max_precision::Int=prec, cached::Bool=true) where T
rel_series(R::Ring, arr::Vector{T}, len::Int, prec::Int, val::Int, var::VarName=:x; max_precision::Int=prec, cached::Bool=true) where T
laurent_series(R::Ring, arr::Vector{T}, len::Int, prec::Int, val::Int, scale::Int, var::VarName=:x; max_precision::Int=prec, cached::Bool=true) where T

Examples

julia> S, x = power_series_ring(QQ, 10, :x; model=:capped_absolute)
(Univariate power series ring over rationals, x + O(x^10))

julia> f = S(Rational{BigInt}[0, 2, 3, 1], 4, 6)
2*x + 3*x^2 + x^3 + O(x^6)

julia> f = abs_series(ZZ, [1, 2, 3], 3, 5, :y)
1 + 2*y + 3*y^2 + O(y^5)

julia> g = rel_series(ZZ, [1, 2, 3], 3, 7, 4)
x^4 + 2*x^5 + 3*x^6 + O(x^7)

julia> k = abs_series(ZZ, [1, 2, 3], 1, 6, cached=false)
1 + O(x^6)

julia> p = rel_series(ZZ, BigInt[], 0, 3, 1)
O(x^3)

julia> q = abs_series(ZZ, [], 0, 6)
O(x^6)

julia> s = abs_series(ZZ, [1, 2, 3], 3, 5; max_precision=10)
1 + 2*x + 3*x^2 + O(x^5)

julia> s = laurent_series(ZZ, [1, 2, 3], 3, 5, 0, 2; max_precision=10)
1 + 2*x^2 + 3*x^4 + O(x^5)

Big-oh notation

Series elements can be given a precision using the big-oh notation. This is provided by a function of the following form, (or something equivalent for Laurent series):

O(x::SeriesElem)

Examples

julia> R, x = power_series_ring(ZZ, 10, :x)
(Univariate power series ring over integers, x + O(x^11))

julia> S, y = laurent_series_ring(ZZ, 10, :y)
(Laurent series ring in y over integers, y + O(y^11))

julia> f = 1 + 2x + O(x^5)
1 + 2*x + O(x^5)

julia> g = 2y + 7y^2 + O(y^7)
2*y + 7*y^2 + O(y^7)

What is happening here in practice is that O(x^n) is creating the series 0 + O(x^n) and the rules for addition of series dictate that if this is added to a series of greater precision, then the lower of the two precisions must be used.

Of course it may be that the precision of the series that O(x^n) is added to is already lower than n, in which case adding O(x^n) has no effect. This is the case if the default precision is too low, since x on its own has the default precision.

Power series models

Capped relative power series have their maximum relative precision capped at some value prec_max. This means that if the leading term of a nonzero power series element is $c_ax^a$ and the precision is $b$ then the power series is of the form $c_ax^a + c_{a+1}x^{a+1} + \ldots + O(x^{a + b})$.

The zero power series is simply taken to be $0 + O(x^b)$.

The capped relative model has the advantage that power series are stable multiplicatively. In other words, for nonzero power series $f$ and $g$ we have that divexact(f*g), g) == f.

However, capped relative power series are not additively stable, i.e. we do not always have $(f + g) - g = f$.

Similar comments apply to Laurent series.

On the other hand, capped absolute power series have their absolute precision capped. This means that if the leading term of a nonzero power series element is $c_ax^a$ and the precision is $b$ then the power series is of the form $c_ax^a + c_{a+1}x^{a+1} + \ldots + O(x^b)$.

Capped absolute series are additively stable, but not necessarily multiplicatively stable.

For all models, the maximum precision is also used as a default precision in the case of coercing coefficients into the ring and for any computation where the result could mathematically be given to infinite precision.

In all models we say that two power series are equal if they agree up to the minimum absolute precision of the two power series.

Thus, for example, $x^5 + O(x^{10}) == 0 + O(x^5)$, since the minimum absolute precision is $5$.

During computations, it is possible for power series to lose relative precision due to cancellation. For example if $f = x^3 + x^5 + O(x^8)$ and $g = x^3 + x^6 + O(x^8)$ then $f - g = x^5 - x^6 + O(x^8)$ which now has relative precision $3$ instead of relative precision $5$.

Amongst other things, this means that equality is not transitive. For example $x^6 + O(x^{11}) == 0 + O(x^5)$ and $x^7 + O(x^{12}) == 0 + O(x^5)$ but $x^6 + O(x^{11}) \neq x^7 + O(x^{12})$.

Sometimes it is necessary to compare power series not just for arithmetic equality, as above, but to see if they have precisely the same precision and terms. For this purpose we introduce the isequal function.

For example, if $f = x^2 + O(x^7)$ and $g = x^2 + O(x^8)$ and $h = 0 + O(x^2)$ then $f == g$, $f == h$ and $g == h$, but isequal(f, g), isequal(f, h) and isequal(g, h) would all return false. However, if $k = x^2 + O(x^7)$ then isequal(f, k) would return true.

There are further difficulties if we construct polynomial over power series. For example, consider the polynomial in $y$ over the power series ring in $x$ over the rationals. Normalisation of such polynomials is problematic. For instance, what is the leading coefficient of $(0 + O(x^{10}))y + (1 + O(x^{10}))$?

If one takes it to be $(0 + O(x^{10}))$ then some functions may not terminate due to the fact that algorithms may require the degree of polynomials to decrease with each iteration. Instead, the degree may remain constant and simply accumulate leading terms which are arithmetically zero but not identically zero.

On the other hand, when constructing power series over other power series, if we simply throw away terms which are arithmetically equal to zero, our computations may have different output depending on the order in which the power series are added!

One should be aware of these difficulties when working with power series. Power series, as represented on a computer, simply don't satisfy the axioms of a ring. They must be used with care in order to approximate operations in a mathematical power series ring.

Simply increasing the precision will not necessarily give a "more correct" answer and some computations may not even terminate due to the presence of arithmetic zeroes!

An absolute power series ring over a ring $R$ with precision $p$ behaves very much like the quotient $R[x]/(x^p)$ of the polynomial ring over $R$. Therefore one can often treat absolute power series rings as though they were rings. However, this depends on all series being given a precision equal to the specified maximum precision and not a lower precision.

Functions for types and parents of series rings

base_ring(R::SeriesRing)
base_ring(a::SeriesElem)

Return the coefficient ring of the given series ring or series.

parent(a::SeriesElem)

Return the parent of the given series.

characteristic(R::SeriesRing)

Return the characteristic of the given series ring. If the characteristic is not known, an exception is raised.

Series functions

Unless otherwise noted, the functions below are available for all series models, including Laurent series. We denote this by using the abstract type RelPowerSeriesRingElem, even though absolute series and Laurent series types do not belong to this abstract type.

Basic functionality

Series implement the Ring Interface

zero(R::SeriesRing)
one(R::SeriesRing)
iszero(a::SeriesElem)
isone(a::SeriesElem)

divexact(a::T, b::T) where T <: SeriesElem
inv(a::SeriesElem)

Series also implement the Series Interface, the most important basic functions being the following.

var(S::SeriesRing)

Return a symbol for the variable of the given series ring.

max_precision(S::SeriesRing)

Return the precision cap of the given series ring.

precision(f::SeriesElem)
valuation(f::SeriesElem)

gen(R::SeriesRing)

The following functions are also provided for all series.

coeff(a::SeriesElem, n::Int)

Return the degree $n$ coefficient of the given power series. Note coefficients are numbered from $n = 0$ for the constant coefficient. If $n$ exceeds the current precision of the power series, the function returns a zero coefficient.

For power series types, $n$ must be non-negative. Laurent series do not have this restriction.

modulus — Method

modulus(a::SeriesElem{T}) where {T <: ResElem}

Return the modulus of the coefficients of the given power series.