Taylor expansions in independent variables (support AbstractGraph)

TaylorSeries

A Julia package for Taylor expansions in one or more independent variables.

The basic constructors is TaylorSeries.

ParamsTaylor

DataType holding the current parameters for `TaylorSeries`. 
This part of code is adopted from TaylorSeries.jl (https://github.com/JuliaDiff/TaylorSeries.jl)

Fields:

• orders :: Int Orders (degree) of the polynomials
• num_vars :: Int Number of variables
• variable_names :: Vector{String} Names of the variables
• variable_symbols :: Vector{Symbol} Symbols of the variables

These parameters can be changed using set_variables

mutable struct TaylorSeries{T}

A representation of a taylor series.

Members:

• name::Symbol name of the diagram
• coeffs::Dict{Vector{Int},T} The taylor expansion coefficients. The integer array define the order of corresponding coefficient.

function TaylorSeries(::Type{T}, nv::Int) where {T}

Create a taylor series equal to variable with index nv. For example, if global variables are "x y", in put nv=2 generate series t=y.

Arguments:

• ::Type{T} DataType of coefficients in taylor series.
• nv::Int Index of variable.

function Base.:*(g1::TaylorSeries{T}, g2::TaylorSeries{T}) where {T}

Returns a taylor series `g1 * g2` representing the product of `g2` with `g1`.

Arguments:

• g1 First taylor series
• g2 Second taylor series

function Base.:*(g1::TaylorSeries{T}, c2::Number) where {T}

Returns a TaylorSeries representing the scalar multiplication `g1*c2`.

Arguments:

• g1 TaylorSeries
• c2 scalar multiple

function Base.:*(c1::Number, g2::TaylorSeries{T}) where {T}

Returns a TaylorSeries representing the scalar multiplication `g2*c1`.

Arguments:

• g2 TaylorSeries
• c1 scalar multiple

function Base.:+(g1::TaylorSeries{T}, c::S) where {T,S<:Number}

Returns a taylor series `g1 + c` representing the addition of constant `c` with `g1`.

Arguments:

• g1 Taylor series
• c Constant

function Base.:+(g1::TaylorSeries{T}, g2::TaylorSeries{T}) where {T}

Returns a taylor series `g1 + g2` representing the addition of `g2` with `g1`.

Arguments:

• g1 First taylor series
• g2 Second taylor series

function Base.:+(c::S, g1::TaylorSeries{T}) where {S<:Number,T}

Returns a taylor series `g1 + c` representing the addition of constant `c` with `g1`.

Arguments:

• g1 Taylor series
• c Constant

function Base.:-(g1::TaylorSeries{T,V}, g2::TaylorSeries{T,V}) where {T,V}

Returns a taylor series `g1 - g2` representing the difference of `g2` with `g1`.

Arguments:

• g1 First taylor series
• g2 Second taylor series

function Base.:^(x::TaylorSeries, p::Integer)

Return the power of taylor series x^p, where p is an integer.

Arguments:

• x Taylor series
• 'p' Power index

function  Base.one(g::TaylorSeries{T}) where {T}

Return a constant one for a given taylor series.

Arguments:

• g Taylor series

displayBigO(d::Bool) --> nothing

Set/unset displaying of the big 𝒪 notation in the output of Taylor1 and TaylorN polynomials. The initial value is true.

function getcoeff(g::TaylorSeries, order::Vector{Int})

Return the taylor coefficients with given order in taylor series g.

Arguments:

• g Taylor series
• order Order of target coefficients

function getderivative(g::TaylorSeries, order::Vector{Int})

Return the derivative with given order in taylor series g.

Arguments:

• g Taylor series
• order Order of derivative

set_variables([T::Type], names::String; [orders=get_orders(), numvars=-1])

Return a TaylorSeries{T} vector with each entry representing an independent variable. names defines the output for each variable (separated by a space). The default type T is Float64, and the default for orders is the one defined globally.

If numvars is not specified, it is inferred from names. If only one variable name is defined and numvars>1, it uses this name with subscripts for the different variables.

julia> set_variables(Int, "x y z", orders=[4,4,4])
3-element Array{TaylorSeries.Taylor{Int},1}:
  1 x + 𝒪(x⁵y⁵z⁵)
  1 y + 𝒪(x⁵y⁵z⁵)
  1 z + 𝒪(x⁵y⁵z⁵)

function taylor_binomial(o1::Vector{Int}, o2::Vector{Int})

Return the taylor binomial prefactor when product two high-order derivatives with order o1 and o2.

# Arguments:
- `o1`  Order of first derivative
- `o2`  Order of second derivative

function taylor_factorial(o::Vector{Int})

Return the taylor factorial prefactor with order o.

# Arguments:
- `o`  Order of the taylor coefficient

use_Base_show(d::Bool) --> nothing

Use Base.show_default method (default show method in Base), or a custom display. The initial value is false, so customized display is used.