From 646de0ed93c1c4fb74a51a3bd638a982c950a64d Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 6 Dec 2023 03:29:01 +0000 Subject: [PATCH] build based on e9eb01b --- previews/PR333/.documenter-siteinfo.json | 1 + previews/PR333/api/index.html | 54 +- previews/PR333/assets/documenter.js | 752 +++++++++++++++--- previews/PR333/assets/search.js | 267 ------- .../PR333/assets/themes/documenter-dark.css | 4 +- .../PR333/assets/themes/documenter-light.css | 4 +- previews/PR333/assets/themeswap.js | 82 +- previews/PR333/assets/warner.js | 87 +- previews/PR333/background/index.html | 4 +- previews/PR333/examples/index.html | 6 +- previews/PR333/index.html | 2 +- previews/PR333/search/index.html | 2 - previews/PR333/search_index.js | 2 +- previews/PR333/userguide/index.html | 4 +- 14 files changed, 791 insertions(+), 480 deletions(-) create mode 100644 previews/PR333/.documenter-siteinfo.json delete mode 100644 previews/PR333/assets/search.js delete mode 100644 previews/PR333/search/index.html diff --git a/previews/PR333/.documenter-siteinfo.json b/previews/PR333/.documenter-siteinfo.json new file mode 100644 index 00000000..4bd22bed --- /dev/null +++ b/previews/PR333/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-06T03:28:55","documenter_version":"1.2.1"}} \ No newline at end of file diff --git a/previews/PR333/api/index.html b/previews/PR333/api/index.html index 199a9405..2095cc99 100644 --- a/previews/PR333/api/index.html +++ b/previews/PR333/api/index.html @@ -1,17 +1,17 @@ -API · TaylorSeries.jl
Edit on GitHub

Library

Module

Types

TaylorSeries.Taylor1 — Type
Taylor1{T<:Number} <: AbstractSeries{T}

DataType for polynomial expansions in one independent variable.

Fields:

  • coeffs :: Array{T,1} Expansion coefficients; the $i$-th component is the coefficient of degree $i-1$ of the expansion.
  • order :: Int Maximum order (degree) of the polynomial.

Note that Taylor1 variables are callable. For more information, see evaluate.

source
TaylorSeries.HomogeneousPolynomial — Type
HomogeneousPolynomial{T<:Number} <: AbstractSeries{T}

DataType for homogeneous polynomials in many (>1) independent variables.

Fields:

  • coeffs :: Array{T,1} Expansion coefficients of the homogeneous

polynomial; the $i$-th component is related to a monomial, where the degrees of the independent variables are specified by coeff_table[order+1][i].

  • order :: Int order (degree) of the homogeneous polynomial.

Note that HomogeneousPolynomial variables are callable. For more information, see evaluate.

source
TaylorSeries.TaylorN — Type
TaylorN{T<:Number} <: AbstractSeries{T}

DataType for polynomial expansions in many (>1) independent variables.

Fields:

  • coeffs :: Array{HomogeneousPolynomial{T},1} Vector containing the

HomogeneousPolynomial entries. The $i$-th component corresponds to the homogeneous polynomial of degree $i-1$.

  • order :: Int maximum order of the polynomial expansion.

Note that TaylorN variables are callable. For more information, see evaluate.

source

Functions and methods

TaylorSeries.Taylor1 — Method
Taylor1([T::Type=Float64], order::Int)

Shortcut to define the independent variable of a Taylor1{T} polynomial of given order. The default type for T is Float64.

julia> Taylor1(16)
+API · TaylorSeries.jl
GitHub
Library
Module
Types
TaylorSeries.Taylor1 — Type
Taylor1{T<:Number} <: AbstractSeries{T}
DataType for polynomial expansions in one independent variable.
Fields:
  • coeffs :: Array{T,1} Expansion coefficients; the $i$-th   component is the coefficient of degree $i-1$ of the expansion.
  • order  :: Int Maximum order (degree) of the polynomial.
Note that Taylor1 variables are callable. For more information, see evaluate.
source
TaylorSeries.HomogeneousPolynomial — Type
HomogeneousPolynomial{T<:Number} <: AbstractSeries{T}
DataType for homogeneous polynomials in many (>1) independent variables.
Fields:
  • coeffs  :: Array{T,1} Expansion coefficients of the homogeneous
polynomial; the $i$-th component is related to a monomial, where the degrees of the independent variables are specified by coeff_table[order+1][i].
  • order   :: Int order (degree) of the homogeneous polynomial.
Note that HomogeneousPolynomial variables are callable. For more information, see evaluate.
source
TaylorSeries.TaylorN — Type
TaylorN{T<:Number} <: AbstractSeries{T}
DataType for polynomial expansions in many (>1) independent variables.
Fields:
  • coeffs  :: Array{HomogeneousPolynomial{T},1} Vector containing the
HomogeneousPolynomial entries. The $i$-th component corresponds to the homogeneous polynomial of degree $i-1$.
  • order   :: Int  maximum order of the polynomial expansion.
Note that TaylorN variables are callable. For more information, see evaluate.
source
Functions and methods
TaylorSeries.Taylor1 — Method
Taylor1([T::Type=Float64], order::Int)
Shortcut to define the independent variable of a Taylor1{T} polynomial of given order. The default type for T is Float64.
julia> Taylor1(16)
  1.0 t + 𝒪(t¹⁷)
 
 julia> Taylor1(Rational{Int}, 4)
- 1//1 t + 𝒪(t⁵)
source
TaylorSeries.HomogeneousPolynomial — Method
HomogeneousPolynomial([T::Type=Float64], nv::Int])
Shortcut to define the nv-th independent HomogeneousPolynomial{T}. The default type for T is Float64.
julia> HomogeneousPolynomial(1)
+ 1//1 t + 𝒪(t⁵)
source
TaylorSeries.HomogeneousPolynomial — Method
HomogeneousPolynomial([T::Type=Float64], nv::Int])
Shortcut to define the nv-th independent HomogeneousPolynomial{T}. The default type for T is Float64.
julia> HomogeneousPolynomial(1)
  1.0 x₁
 
 julia> HomogeneousPolynomial(Rational{Int}, 2)
- 1//1 x₂
source
TaylorSeries.TaylorN — Method
TaylorN([T::Type=Float64], nv::Int; [order::Int=get_order()])
Shortcut to define the nv-th independent TaylorN{T} variable as a polynomial. The order is defined through the keyword parameter order, whose default corresponds to get_order(). The default of type for T is Float64.
julia> TaylorN(1)
+ 1//1 x₂
source
TaylorSeries.TaylorN — Method
TaylorN([T::Type=Float64], nv::Int; [order::Int=get_order()])
Shortcut to define the nv-th independent TaylorN{T} variable as a polynomial. The order is defined through the keyword parameter order, whose default corresponds to get_order(). The default of type for T is Float64.
julia> TaylorN(1)
  1.0 x₁ + 𝒪(‖x‖⁷)
 
 julia> TaylorN(Rational{Int},2)
- 1//1 x₂ + 𝒪(‖x‖⁷)
source
TaylorSeries.set_variables — Function
set_variables([T::Type], names::String; [order=get_order(), numvars=-1])
Return a TaylorN{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 order is the one defined globally. Changing the order or numvars resets the hash_tables.
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", order=4)
+ 1//1 x₂ + 𝒪(‖x‖⁷)
source
TaylorSeries.set_variables — Function
set_variables([T::Type], names::String; [order=get_order(), numvars=-1])
Return a TaylorN{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 order is the one defined globally. Changing the order or numvars resets the hash_tables.
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", order=4)
 3-element Array{TaylorSeries.TaylorN{Int},1}:
   1 x + 𝒪(‖x‖⁵)
   1 y + 𝒪(‖x‖⁵)
@@ -25,57 +25,57 @@
 julia> set_variables("x", order=6, numvars=2)
 2-element Array{TaylorSeries.TaylorN{Float64},1}:
   1.0 x₁ + 𝒪(‖x‖⁷)
-  1.0 x₂ + 𝒪(‖x‖⁷)
source
TaylorSeries.get_variables — Function
get_variables(T::Type, [order::Int=get_order()])
Return a TaylorN{T} vector with each entry representing an independent variable. It takes the default _params_TaylorN_ values if set_variables hasn't been changed with the exception that order can be explicitly established by the user without changing internal values for num_vars or variable_names. Omitting T defaults to Float64.
source
TaylorSeries.getcoeff — Function
getcoeff(a, n)
Return the coefficient of order n::Int of a a::Taylor1 polynomial.
source
getcoeff(a, v)
Return the coefficient of a::HomogeneousPolynomial, specified by v, which is a tuple (or vector) with the indices of the specific monomial.
source
getcoeff(a, v)
Return the coefficient of a::TaylorN, specified by v, which is a tuple (or vector) with the indices of the specific monomial.
source
TaylorSeries.evaluate — Function
evaluate(a, [dx])
Evaluate a Taylor1 polynomial using Horner's rule (hand coded). If dx is omitted, its value is considered as zero. Note that the syntax a(dx) is equivalent to evaluate(a,dx), and a() is equivalent to evaluate(a).
source
evaluate(x, δt)
Evaluates each element of x::AbstractArray{Taylor1{T}}, representing the dependent variables of an ODE, at time δt. Note that the syntax x(δt) is equivalent to evaluate(x, δt), and x() is equivalent to evaluate(x).
source
evaluate(a, x)
Substitute x::Taylor1 as independent variable in a a::Taylor1 polynomial. Note that the syntax a(x) is equivalent to evaluate(a, x).
source
evaluate(a, [vals])
Evaluate a HomogeneousPolynomial polynomial at vals. If vals is omitted, it's evaluated at zero. Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a).
source
evaluate(a, [vals]; sorting::Bool=true)
Evaluate the TaylorN polynomial a at vals. If vals is omitted, it's evaluated at zero. The keyword parameter sorting can be used to avoid sorting (in increasing order by abs2) the terms that are added.
Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a). No extension exists that incorporates sorting.
source
TaylorSeries.evaluate! — Function
evaluate!(x, δt, x0)
Evaluates each element of x::AbstractArray{Taylor1{T}}, representing the Taylor expansion for the dependent variables of an ODE at time δt. It updates the vector x0 with the computed values.
source
TaylorSeries.taylor_expand — Function
taylor_expand(f, x0; order)
Computes the Taylor expansion of the function f around the point x0.
If x0 is a scalar, a Taylor1 expansion will be returned. If x0 is a vector, a TaylorN expansion will be computed. If the dimension of x0 (length(x0)) is different from the variables set for TaylorN (get_numvars()), an AssertionError will be thrown.
source
TaylorSeries.differentiate — Function
differentiate(a)
Return the Taylor1 polynomial of the differential of a::Taylor1. The order of the result is a.order-1.
The function derivative is an exact synonym of differentiate.
source
differentiate(a, n)
Compute recursively the Taylor1 polynomial of the n-th derivative of a::Taylor1. The order of the result is a.order-n.
source
differentiate(n, a)
Return the value of the n-th differentiate of the polynomial a.
source
differentiate(a, r)
Partial differentiation of a::HomogeneousPolynomial series with respect to the r-th variable.
source
differentiate(a, r)
Partial differentiation of a::TaylorN series with respect to the r-th variable. The r-th variable may be also specified through its symbol.
source
differentiate(a::TaylorN{T}, ntup::NTuple{N,Int})
Return a TaylorN with the partial derivative of a defined by ntup::NTuple{N,Int}, where the first entry is the number of derivatives with respect to the first variable, the second is the number of derivatives with respect to the second, and so on.
source
differentiate(ntup::NTuple{N,Int}, a::TaylorN{T})
Returns the value of the coefficient of a specified by ntup::NTuple{N,Int}, multiplied by the corresponding factorials.
source
TaylorSeries.integrate — Function
integrate(a, [x])
Return the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if omitted.
source
integrate(a, r)
Integrate the a::HomogeneousPolynomial with respect to the r-th variable. The returned HomogeneousPolynomial has no added constant of integration. If the order of a corresponds to get_order(), a zero HomogeneousPolynomial of 0-th order is returned.
source
integrate(a, r, [x0])
Integrate the a::TaylorN series with respect to the r-th variable, where x0 the integration constant and must be independent of the r-th variable; if x0 is omitted, it is taken as zero.
source
TaylorSeries.jacobian — Function
    jacobian(vf)
-    jacobian(vf, [vals])
Compute the jacobian matrix of vf, a vector of TaylorN polynomials, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.
source
TaylorSeries.jacobian! — Function
    jacobian!(jac, vf)
-    jacobian!(jac, vf, [vals])
Compute the jacobian matrix of vf, a vector of TaylorN polynomials evaluated at the vector vals, and write results to jac. If vals is omitted, it is evaluated at zero.
source
TaylorSeries.hessian — Function
    hessian(f)
-    hessian(f, [vals])
Return the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.
source
TaylorSeries.hessian! — Function
    hessian!(hes, f)
-    hessian!(hes, f, [vals])
Return the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals, and write results to hes. If vals is omitted, it is evaluated at zero.
source
TaylorSeries.constant_term — Function
constant_term(a)
Return the constant value (zero order coefficient) for Taylor1 and TaylorN. The fallback behavior is to return a itself if a::Number, or a[1] when a::Vector.
source
TaylorSeries.linear_polynomial — Function
linear_polynomial(a)
Returns the linear part of a as a polynomial (Taylor1 or TaylorN), without the constant term. The fallback behavior is to return a itself.
source
TaylorSeries.inverse — Function
inverse(f)
Return the Taylor expansion of $f^{-1}(t)$, of order N = f.order, for f::Taylor1 polynomial if the first coefficient of f is zero. Otherwise, a DomainError is thrown.
The algorithm implements Lagrange inversion at $t=0$ if $f(0)=0$:
\[\begin{equation*}
+  1.0 x₂ + 𝒪(‖x‖⁷)
source
TaylorSeries.get_variables — Function
get_variables(T::Type, [order::Int=get_order()])

Return a TaylorN{T} vector with each entry representing an independent variable. It takes the default _params_TaylorN_ values if set_variables hasn't been changed with the exception that order can be explicitly established by the user without changing internal values for num_vars or variable_names. Omitting T defaults to Float64.

source
TaylorSeries.getcoeff — Function
getcoeff(a, n)

Return the coefficient of order n::Int of a a::Taylor1 polynomial.

source
getcoeff(a, v)

Return the coefficient of a::HomogeneousPolynomial, specified by v, which is a tuple (or vector) with the indices of the specific monomial.

source
getcoeff(a, v)

Return the coefficient of a::TaylorN, specified by v, which is a tuple (or vector) with the indices of the specific monomial.

source
TaylorSeries.evaluate — Function
evaluate(a, [dx])

Evaluate a Taylor1 polynomial using Horner's rule (hand coded). If dx is omitted, its value is considered as zero. Note that the syntax a(dx) is equivalent to evaluate(a,dx), and a() is equivalent to evaluate(a).

source
evaluate(x, δt)

Evaluates each element of x::AbstractArray{Taylor1{T}}, representing the dependent variables of an ODE, at time δt. Note that the syntax x(δt) is equivalent to evaluate(x, δt), and x() is equivalent to evaluate(x).

source
evaluate(a, x)

Substitute x::Taylor1 as independent variable in a a::Taylor1 polynomial. Note that the syntax a(x) is equivalent to evaluate(a, x).

source
evaluate(a, [vals])

Evaluate a HomogeneousPolynomial polynomial at vals. If vals is omitted, it's evaluated at zero. Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a).

source
evaluate(a, [vals]; sorting::Bool=true)

Evaluate the TaylorN polynomial a at vals. If vals is omitted, it's evaluated at zero. The keyword parameter sorting can be used to avoid sorting (in increasing order by abs2) the terms that are added.

Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a). No extension exists that incorporates sorting.

source
TaylorSeries.evaluate! — Function
evaluate!(x, δt, x0)

Evaluates each element of x::AbstractArray{Taylor1{T}}, representing the Taylor expansion for the dependent variables of an ODE at time δt. It updates the vector x0 with the computed values.

source
TaylorSeries.taylor_expand — Function
taylor_expand(f, x0; order)

Computes the Taylor expansion of the function f around the point x0.

If x0 is a scalar, a Taylor1 expansion will be returned. If x0 is a vector, a TaylorN expansion will be computed. If the dimension of x0 (length(x0)) is different from the variables set for TaylorN (get_numvars()), an AssertionError will be thrown.

source
TaylorSeries.differentiate — Function
differentiate(a)

Return the Taylor1 polynomial of the differential of a::Taylor1. The order of the result is a.order-1.

The function derivative is an exact synonym of differentiate.

source
differentiate(a, n)

Compute recursively the Taylor1 polynomial of the n-th derivative of a::Taylor1. The order of the result is a.order-n.

source
differentiate(n, a)

Return the value of the n-th differentiate of the polynomial a.

source
differentiate(a, r)

Partial differentiation of a::HomogeneousPolynomial series with respect to the r-th variable.

source
differentiate(a, r)

Partial differentiation of a::TaylorN series with respect to the r-th variable. The r-th variable may be also specified through its symbol.

source
differentiate(a::TaylorN{T}, ntup::NTuple{N,Int})

Return a TaylorN with the partial derivative of a defined by ntup::NTuple{N,Int}, where the first entry is the number of derivatives with respect to the first variable, the second is the number of derivatives with respect to the second, and so on.

source
differentiate(ntup::NTuple{N,Int}, a::TaylorN{T})

Returns the value of the coefficient of a specified by ntup::NTuple{N,Int}, multiplied by the corresponding factorials.

source
TaylorSeries.integrate — Function
integrate(a, [x])

Return the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if omitted.

source
integrate(a, r)

Integrate the a::HomogeneousPolynomial with respect to the r-th variable. The returned HomogeneousPolynomial has no added constant of integration. If the order of a corresponds to get_order(), a zero HomogeneousPolynomial of 0-th order is returned.

source
integrate(a, r, [x0])

Integrate the a::TaylorN series with respect to the r-th variable, where x0 the integration constant and must be independent of the r-th variable; if x0 is omitted, it is taken as zero.

source
TaylorSeries.jacobian — Function
    jacobian(vf)
+    jacobian(vf, [vals])

Compute the jacobian matrix of vf, a vector of TaylorN polynomials, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.

source
TaylorSeries.jacobian! — Function
    jacobian!(jac, vf)
+    jacobian!(jac, vf, [vals])

Compute the jacobian matrix of vf, a vector of TaylorN polynomials evaluated at the vector vals, and write results to jac. If vals is omitted, it is evaluated at zero.

source
TaylorSeries.hessian — Function
    hessian(f)
+    hessian(f, [vals])

Return the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.

source
TaylorSeries.hessian! — Function
    hessian!(hes, f)
+    hessian!(hes, f, [vals])

Return the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals, and write results to hes. If vals is omitted, it is evaluated at zero.

source
TaylorSeries.constant_term — Function
constant_term(a)

Return the constant value (zero order coefficient) for Taylor1 and TaylorN. The fallback behavior is to return a itself if a::Number, or a[1] when a::Vector.

source
TaylorSeries.linear_polynomial — Function
linear_polynomial(a)

Returns the linear part of a as a polynomial (Taylor1 or TaylorN), without the constant term. The fallback behavior is to return a itself.

source
TaylorSeries.inverse — Function
inverse(f)

Return the Taylor expansion of $f^{-1}(t)$, of order N = f.order, for f::Taylor1 polynomial if the first coefficient of f is zero. Otherwise, a DomainError is thrown.

The algorithm implements Lagrange inversion at $t=0$ if $f(0)=0$:

\[\begin{equation*} f^{-1}(t) = \sum_{n=1}^{N} \frac{t^n}{n!} \left. \frac{{\rm d}^{n-1}}{{\rm d} z^{n-1}}\left(\frac{z}{f(z)}\right)^n \right\vert_{z=0}. -\end{equation*}\]

source
Base.abs — Function
abs(a)

For a Real type returns a if constant_term(a) > 0 and -a if constant_term(a) < 0 for a <:Union{Taylor1,TaylorN}. For a Complex type, such as Taylor1{ComplexF64}, returns sqrt(real(a)^2 + imag(a)^2).

Notice that typeof(abs(a)) <: AbstractSeries and that for a Complex argument a Real type is returned (e.g. typeof(abs(a::Taylor1{ComplexF64})) == Taylor1{Float64}).

source
LinearAlgebra.norm — Function
norm(x::AbstractSeries, p::Real)

Returns the p-norm of an x::AbstractSeries, defined by

\[\begin{equation*} +\end{equation*}\]

source
Base.abs — Function
abs(a)

For a Real type returns a if constant_term(a) > 0 and -a if constant_term(a) < 0 for a <:Union{Taylor1,TaylorN}. For a Complex type, such as Taylor1{ComplexF64}, returns sqrt(real(a)^2 + imag(a)^2).

Notice that typeof(abs(a)) <: AbstractSeries and that for a Complex argument a Real type is returned (e.g. typeof(abs(a::Taylor1{ComplexF64})) == Taylor1{Float64}).

source
LinearAlgebra.norm — Function
norm(x::AbstractSeries, p::Real)

Returns the p-norm of an x::AbstractSeries, defined by

\[\begin{equation*} \left\Vert x \right\Vert_p = \left( \sum_k | x_k |^p \right)^{\frac{1}{p}}, -\end{equation*}\]

which returns a non-negative number.

source
Base.isapprox — Function
isapprox(x::AbstractSeries, y::AbstractSeries; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false)

Inexact equality comparison between polynomials: returns true if norm(x-y,1) <= atol + rtol*max(norm(x,1), norm(y,1)), where x and y are polynomials. For more details, see Base.isapprox.

source
Base.isless — Function
isless(a::Taylor1{<:Real}, b::Real)
+\end{equation*}\]
which returns a non-negative number.
source
Base.isapprox — Function
isapprox(x::AbstractSeries, y::AbstractSeries; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false)

Inexact equality comparison between polynomials: returns true if norm(x-y,1) <= atol + rtol*max(norm(x,1), norm(y,1)), where x and y are polynomials. For more details, see Base.isapprox.

source
Base.isless — Function
isless(a::Taylor1{<:Real}, b::Real)
 isless(a::TaylorN{<:Real}, b::Real)

Compute isless by comparing the constant_term(a) and b. If they are equal, returns a[nz] < 0, with nz the first non-zero coefficient after the constant term. This defines a total order.

For many variables, the ordering includes a lexicographical convention in order to be total. We have opted for the simplest one, where the larger variable appears before when the TaylorN variables are defined (e.g., through set_variables).

Refs:

  • M. Berz, AIP Conference Proceedings 177, 275 (1988); https://doi.org/10.1063/1.37800
  • M. Berz, "Automatic Differentiation as Nonarchimedean Analysis", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450.
isless(a::Taylor1{<:Real}, b::Taylor1{<:Real})
-isless(a::TaylorN{<:Real}, b::Taylor1{<:Real})

Returns isless(a - b, zero(b)).

source
Base.isfinite — Function
isfinite(x::AbstractSeries) -> Bool

Test whether the coefficients of the polynomial x are finite.

source
TaylorSeries.displayBigO — Function
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.

source
TaylorSeries.use_show_default — Function
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.

source

Internals

TaylorSeries.ParamsTaylorN — Type
ParamsTaylorN

DataType holding the current parameters for TaylorN and HomogeneousPolynomial.

Fields:

  • order :: Int Order (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

source
TaylorSeries._InternalMutFuncs — Type

_InternalMutFuncs

Contains parameters and expressions that allow a simple programmatic construction for calling the internal mutating functions.

source
TaylorSeries.generate_tables — Function
generate_tables(num_vars, order)

Return the hash tables coeff_table, index_table, size_table and pos_table. Internally, these are treated as const.

Hash tables

coeff_table :: Array{Array{Array{Int,1},1},1}

The $i+1$-th component contains a vector with the vectors of all the possible combinations of monomials of a HomogeneousPolynomial of order $i$.

index_table :: Array{Array{Int,1},1}

The $i+1$-th component contains a vector of (hashed) indices that represent the distinct monomials of a HomogeneousPolynomial of order (degree) $i$.

size_table :: Array{Int,1}

The $i+1$-th component contains the number of distinct monomials of the HomogeneousPolynomial of order $i$, equivalent to length(coeff_table[i]).

pos_table :: Array{Dict{Int,Int},1}

The $i+1$-th component maps the hash index to the (lexicographic) position of the corresponding monomial in coeffs_table.

source
TaylorSeries.in_base — Function
in_base(order, v)

Convert vector v of non-negative integers to base oorder, where oorder is the next odd integer of order.

source
TaylorSeries.make_inverse_dict — Function
make_inverse_dict(v)

Return a Dict with the enumeration of v: the elements of v point to the corresponding index.

It is used to construct pos_table from index_table.

source
TaylorSeries.resize_coeffs1! — Function
resize_coeffs1!{T<Number}(coeffs::Array{T,1}, order::Int)

If the length of coeffs is smaller than order+1, it resizes coeffs appropriately filling it with zeros.

source
TaylorSeries.resize_coeffsHP! — Function
resize_coeffsHP!{T<Number}(coeffs::Array{T,1}, order::Int)

If the length of coeffs is smaller than the number of coefficients correspondinf to order (given by size_table[order+1]), it resizes coeffs appropriately filling it with zeros.

source
LinearAlgebra.mul! — Function
mul!(c, a, b, k::Int) --> nothing

Update the k-th expansion coefficient c[k] of c = a * b, where all c, a, and b are either Taylor1 or TaylorN. Note that for TaylorN the result of a * b is accumulated in c[k].

The coefficients are given by

\[c_k = \sum_{j=0}^k a_j b_{k-j}.\]

source
LinearAlgebra.mul! — Method
mul!(c, a, b) --> nothing

Accumulates in c[k] the result of a*b with minimum allocation; all arguments are HomogeneousPolynomial.

source
TaylorSeries.div! — Function
div!(c, a, b, k::Int)

Compute the k-th expansion coefficient c[k] of c = a / b, where all c, a and b are either Taylor1 or TaylorN.

The coefficients are given by

\[c_k = \frac{1}{b_0} \big(a_k - \sum_{j=0}^{k-1} c_j b_{k-j}\big).\]

For Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.

source
TaylorSeries.pow! — Function
pow!(c, a, r::Real, k::Int)

Update the k-th expansion coefficient c[k] of c = a^r, for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[c_k = \frac{1}{k a_0} \sum_{j=0}^{k-1} \big(r(k-j) -j\big)a_{k-j} c_j.\]

For Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.

source
TaylorSeries.sqr! — Function
sqr!(c, a, k::Int) --> nothing

Update the k-th expansion coefficient c[k] of c = a^2, for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} +isless(a::TaylorN{<:Real}, b::Taylor1{<:Real})

Returns isless(a - b, zero(b)).

source
Base.isfinite — Function
isfinite(x::AbstractSeries) -> Bool

Test whether the coefficients of the polynomial x are finite.

source
TaylorSeries.displayBigO — Function
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.

source
TaylorSeries.use_show_default — Function
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.

source

Internals

TaylorSeries.ParamsTaylorN — Type
ParamsTaylorN

DataType holding the current parameters for TaylorN and HomogeneousPolynomial.

Fields:

  • order :: Int Order (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

source
TaylorSeries._InternalMutFuncs — Type

_InternalMutFuncs

Contains parameters and expressions that allow a simple programmatic construction for calling the internal mutating functions.

source
TaylorSeries.generate_tables — Function
generate_tables(num_vars, order)

Return the hash tables coeff_table, index_table, size_table and pos_table. Internally, these are treated as const.

Hash tables

coeff_table :: Array{Array{Array{Int,1},1},1}

The $i+1$-th component contains a vector with the vectors of all the possible combinations of monomials of a HomogeneousPolynomial of order $i$.

index_table :: Array{Array{Int,1},1}

The $i+1$-th component contains a vector of (hashed) indices that represent the distinct monomials of a HomogeneousPolynomial of order (degree) $i$.

size_table :: Array{Int,1}

The $i+1$-th component contains the number of distinct monomials of the HomogeneousPolynomial of order $i$, equivalent to length(coeff_table[i]).

pos_table :: Array{Dict{Int,Int},1}

The $i+1$-th component maps the hash index to the (lexicographic) position of the corresponding monomial in coeffs_table.

source
TaylorSeries.in_base — Function
in_base(order, v)

Convert vector v of non-negative integers to base oorder, where oorder is the next odd integer of order.

source
TaylorSeries.make_inverse_dict — Function
make_inverse_dict(v)

Return a Dict with the enumeration of v: the elements of v point to the corresponding index.

It is used to construct pos_table from index_table.

source
TaylorSeries.resize_coeffs1! — Function
resize_coeffs1!{T<Number}(coeffs::Array{T,1}, order::Int)

If the length of coeffs is smaller than order+1, it resizes coeffs appropriately filling it with zeros.

source
TaylorSeries.resize_coeffsHP! — Function
resize_coeffsHP!{T<Number}(coeffs::Array{T,1}, order::Int)

If the length of coeffs is smaller than the number of coefficients correspondinf to order (given by size_table[order+1]), it resizes coeffs appropriately filling it with zeros.

source
LinearAlgebra.mul! — Function
mul!(c, a, b, k::Int) --> nothing

Update the k-th expansion coefficient c[k] of c = a * b, where all c, a, and b are either Taylor1 or TaylorN. Note that for TaylorN the result of a * b is accumulated in c[k].

The coefficients are given by

\[c_k = \sum_{j=0}^k a_j b_{k-j}.\]

source
LinearAlgebra.mul! — Method
mul!(c, a, b) --> nothing

Accumulates in c[k] the result of a*b with minimum allocation; all arguments are HomogeneousPolynomial.

source
TaylorSeries.div! — Function
div!(c, a, b, k::Int)

Compute the k-th expansion coefficient c[k] of c = a / b, where all c, a and b are either Taylor1 or TaylorN.

The coefficients are given by

\[c_k = \frac{1}{b_0} \big(a_k - \sum_{j=0}^{k-1} c_j b_{k-j}\big).\]

For Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.

source
TaylorSeries.pow! — Function
pow!(c, a, r::Real, k::Int)

Update the k-th expansion coefficient c[k] of c = a^r, for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[c_k = \frac{1}{k a_0} \sum_{j=0}^{k-1} \big(r(k-j) -j\big)a_{k-j} c_j.\]

For Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.

source
TaylorSeries.sqr! — Function
sqr!(c, a, k::Int) --> nothing

Update the k-th expansion coefficient c[k] of c = a^2, for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} c_k &= 2 \sum_{j=0}^{(k-1)/2} a_{k-j} a_j, \text{ if $k$ is odd,} \\ c_k &= 2 \sum_{j=0}^{(k-2)/2} a_{k-j} a_j + (a_{k/2})^2, \text{ if $k$ is even.} -\end{aligned}\]

source
Missing docstring.

Missing docstring for sqr!(::HomogeneousPolynomial{T}, ::HomogeneousPolynomial{T}) where {T<:NumberNotSeriesN}. Check Documenter's build log for details.

TaylorSeries.sqrt! — Function
sqrt!(c, a, k::Int, k0::Int=0)

Compute the k-th expansion coefficient c[k] of c = sqrt(a) for bothc and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} +\end{aligned}\]

source
TaylorSeries.accsqr! — Function
accsqr!(c, a)

Returns c += a*a with no allocation; all parameters are HomogeneousPolynomial.

source
TaylorSeries.sqrt! — Function
sqrt!(c, a, k::Int, k0::Int=0)

Compute the k-th expansion coefficient c[k] of c = sqrt(a) for bothc and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} c_k &= \frac{1}{2 c_0} \big( a_k - 2 \sum_{j=1}^{(k-1)/2} c_{k-j}c_j\big), \text{ if $k$ is odd,} \\ c_k &= \frac{1}{2 c_0} \big( a_k - 2 \sum_{j=1}^{(k-2)/2} c_{k-j}c_j - (c_{k/2})^2\big), \text{ if $k$ is even.} -\end{aligned}\]

For Taylor1 polynomials, k0 is the order of the first non-zero coefficient, which must be even.

source
TaylorSeries.exp! — Function
exp!(c, a, k) --> nothing

Update the k-th expansion coefficient c[k+1] of c = exp(a) for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{equation*} +\end{aligned}\]

For Taylor1 polynomials, k0 is the order of the first non-zero coefficient, which must be even.

source
TaylorSeries.exp! — Function
exp!(c, a, k) --> nothing

Update the k-th expansion coefficient c[k+1] of c = exp(a) for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{equation*} c_k = \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} c_j. -\end{equation*}\]

source
TaylorSeries.log! — Function
log!(c, a, k) --> nothing

Update the k-th expansion coefficient c[k+1] of c = log(a) for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.log! — Function
log!(c, a, k) --> nothing

Update the k-th expansion coefficient c[k+1] of c = log(a) for both c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{equation*} c_k = \frac{1}{a_0} \big(a_k - \frac{1}{k} \sum_{j=0}^{k-1} j a_{k-j} c_j \big). -\end{equation*}\]

source
TaylorSeries.sincos! — Function
sincos!(s, c, a, k) --> nothing

Update the k-th expansion coefficients s[k+1] and c[k+1] of s = sin(a) and c = cos(a) simultaneously, for s, c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} +\end{equation*}\]

source
TaylorSeries.sincos! — Function
sincos!(s, c, a, k) --> nothing

Update the k-th expansion coefficients s[k+1] and c[k+1] of s = sin(a) and c = cos(a) simultaneously, for s, c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} s_k &= \frac{1}{k}\sum_{j=0}^{k-1} (k-j) a_{k-j} c_j ,\\ c_k &= -\frac{1}{k}\sum_{j=0}^{k-1} (k-j) a_{k-j} s_j. -\end{aligned}\]

source
TaylorSeries.tan! — Function
tan!(c, a, p, k::Int) --> nothing

Update the k-th expansion coefficients c[k+1] of c = tan(a), for c and a either Taylor1 or TaylorN; p = c^2 and is passed as an argument for efficiency.

The coefficients are given by

\[\begin{equation*} +\end{aligned}\]

source
TaylorSeries.tan! — Function
tan!(c, a, p, k::Int) --> nothing

Update the k-th expansion coefficients c[k+1] of c = tan(a), for c and a either Taylor1 or TaylorN; p = c^2 and is passed as an argument for efficiency.

The coefficients are given by

\[\begin{equation*} c_k = a_k + \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} p_j. -\end{equation*}\]

source
TaylorSeries.asin! — Function
asin!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = asin(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.asin! — Function
asin!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = asin(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} c_k = \frac{1}{ \sqrt{r_0} } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j \big). -\end{equation*}\]

source
TaylorSeries.acos! — Function
acos!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = acos(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.acos! — Function
acos!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = acos(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} c_k = - \frac{1}{ r_0 } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j \big). -\end{equation*}\]

source
TaylorSeries.atan! — Function
atan!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = atan(a), for c and a either Taylor1 or TaylorN; r = 1+a^2 and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.atan! — Function
atan!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = atan(a), for c and a either Taylor1 or TaylorN; r = 1+a^2 and is passed as an argument for efficiency.

\[\begin{equation*} c_k = \frac{1}{r_0}\big(a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j\big). -\end{equation*}\]

source
TaylorSeries.sinhcosh! — Function
sinhcosh!(s, c, a, k)

Update the k-th expansion coefficients s[k+1] and c[k+1] of s = sinh(a) and c = cosh(a) simultaneously, for s, c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} +\end{equation*}\]

source
TaylorSeries.sinhcosh! — Function
sinhcosh!(s, c, a, k)

Update the k-th expansion coefficients s[k+1] and c[k+1] of s = sinh(a) and c = cosh(a) simultaneously, for s, c and a either Taylor1 or TaylorN.

The coefficients are given by

\[\begin{aligned} s_k = \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} c_j, \\ c_k = \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} s_j. -\end{aligned}\]

source
TaylorSeries.tanh! — Function
tanh!(c, a, p, k)

Update the k-th expansion coefficients c[k+1] of c = tanh(a), for c and a either Taylor1 or TaylorN; p = a^2 and is passed as an argument for efficiency.

\[\begin{equation*} +\end{aligned}\]

source
TaylorSeries.tanh! — Function
tanh!(c, a, p, k)

Update the k-th expansion coefficients c[k+1] of c = tanh(a), for c and a either Taylor1 or TaylorN; p = a^2 and is passed as an argument for efficiency.

\[\begin{equation*} c_k = a_k - \frac{1}{k} \sum_{j=0}^{k-1} (k-j) a_{k-j} p_j. -\end{equation*}\]

source
TaylorSeries.asinh! — Function
asinh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = asinh(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.asinh! — Function
asinh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = asinh(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.

\[\begin{equation*} c_k = \frac{1}{ \sqrt{r_0} } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j \big). -\end{equation*}\]

source
TaylorSeries.acosh! — Function
acosh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = acosh(a), for c and a either Taylor1 or TaylorN; r = sqrt(c^2-1) and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.acosh! — Function
acosh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = acosh(a), for c and a either Taylor1 or TaylorN; r = sqrt(c^2-1) and is passed as an argument for efficiency.

\[\begin{equation*} c_k = \frac{1}{ r_0 } \big( a_k - \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j \big). -\end{equation*}\]

source
TaylorSeries.atanh! — Function
atanh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = atanh(a), for c and a either Taylor1 or TaylorN; r = 1-a^2 and is passed as an argument for efficiency.

\[\begin{equation*} +\end{equation*}\]

source
TaylorSeries.atanh! — Function
atanh!(c, a, r, k)

Update the k-th expansion coefficients c[k+1] of c = atanh(a), for c and a either Taylor1 or TaylorN; r = 1-a^2 and is passed as an argument for efficiency.

\[\begin{equation*} c_k = \frac{1}{r_0}\big(a_k + \frac{1}{k} \sum_{j=1}^{k-1} j r_{k-j} c_j\big). -\end{equation*}\]

source
TaylorSeries.differentiate! — Function
differentiate!(res, a) --> nothing

In-place version of differentiate. Compute the Taylor1 polynomial of the differential of a::Taylor1 and return it as res (order of res remains unchanged).

source
differentiate!(p, a, k) --> nothing

Update in-place the k-th expansion coefficient p[k] of p = differentiate(a) for both p and a Taylor1.

The coefficients are given by

\[p_k = (k+1) a_{k+1}.\]

source
TaylorSeries._dict_unary_ops — Constant

_dict_binary_ops

Dict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to unary operations.

The keys correspond to the function symbols.

The arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1, for the required argument, possibly :_aux when there is an auxiliary expression needed, and :_k for the computed order of :_res. When an auxiliary expression is required, an Expr defining its calling pattern is added as the last entry of the vector.

source
TaylorSeries._dict_binary_calls — Constant

_dict_binary_calls::Dict{Symbol, NTuple{2,Expr}}

Dictionary with the expressions that define the internal binary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.

Evaluating the entries generates symbols that represent the actual calls to the internal mutating functions.

source
TaylorSeries._dict_unary_calls — Constant

_dict_unary_calls::Dict{Symbol, NTuple{2,Expr}}

Dictionary with the expressions that define the internal unary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.

Evaluating the entries generates expressions that represent the actual calls to the internal mutating functions.

source
TaylorSeries._dict_binary_ops — Constant

_dict_binary_ops

Dict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to binary operations.

The keys correspond to the function symbols.

The arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1 and _arg2 for the required arguments, and :_k for the computed order of :_res.

source
TaylorSeries.@isonethread — Macro
@isonethread (expr)

Internal macro used to check the number of threads in use, to prevent a data race that modifies coeff_table when using differentiate or integrate; see https://github.com/JuliaDiff/TaylorSeries.jl/issues/318.

This macro is inspired by the macro @threaded; see https://github.com/trixi-framework/Trixi.jl/blob/main/src/auxiliary/auxiliary.jl; and https://github.com/trixi-framework/Trixi.jl/pull/426/files.

source

Index

+\end{equation*}\]

source
TaylorSeries.differentiate! — Function
differentiate!(res, a) --> nothing

In-place version of differentiate. Compute the Taylor1 polynomial of the differential of a::Taylor1 and return it as res (order of res remains unchanged).

source
differentiate!(p, a, k) --> nothing

Update in-place the k-th expansion coefficient p[k] of p = differentiate(a) for both p and a Taylor1.

The coefficients are given by

\[p_k = (k+1) a_{k+1}.\]

source
TaylorSeries._internalmutfunc_call — Function
_internalmutfunc_call( fn :: _InternalMutFuncs )

Creates the appropriate call to the internal mutating function defined by the _InternalMutFuncs object. This is used to construct _dict_unary_calls and _dict_binary_calls. The call contains the prefix TaylorSeries..

source
TaylorSeries._dict_unary_ops — Constant

_dict_binary_ops

Dict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to unary operations.

The keys correspond to the function symbols.

The arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1, for the required argument, possibly :_aux when there is an auxiliary expression needed, and :_k for the computed order of :_res. When an auxiliary expression is required, an Expr defining its calling pattern is added as the last entry of the vector.

source
TaylorSeries._dict_binary_calls — Constant

_dict_binary_calls::Dict{Symbol, NTuple{2,Expr}}

Dictionary with the expressions that define the internal binary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.

Evaluating the entries generates symbols that represent the actual calls to the internal mutating functions.

source
TaylorSeries._dict_unary_calls — Constant

_dict_unary_calls::Dict{Symbol, NTuple{2,Expr}}

Dictionary with the expressions that define the internal unary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.

Evaluating the entries generates expressions that represent the actual calls to the internal mutating functions.

source
TaylorSeries._dict_binary_ops — Constant

_dict_binary_ops

Dict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to binary operations.

The keys correspond to the function symbols.

The arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1 and _arg2 for the required arguments, and :_k for the computed order of :_res.

source
TaylorSeries._populate_dicts! — Function

_populate_dicts!()

Function that populates the internal dictionaries _dict_unary_calls and _dict_binary_calls

source
TaylorSeries.@isonethread — Macro
@isonethread (expr)

Internal macro used to check the number of threads in use, to prevent a data race that modifies coeff_table when using differentiate or integrate; see https://github.com/JuliaDiff/TaylorSeries.jl/issues/318.

This macro is inspired by the macro @threaded; see https://github.com/trixi-framework/Trixi.jl/blob/main/src/auxiliary/auxiliary.jl; and https://github.com/trixi-framework/Trixi.jl/pull/426/files.

source

Index

diff --git a/previews/PR333/assets/documenter.js b/previews/PR333/assets/documenter.js index 6adfbbbf..f5311607 100644 --- a/previews/PR333/assets/documenter.js +++ b/previews/PR333/assets/documenter.js @@ -1,15 +1,16 @@ // Generated by Documenter.jl requirejs.config({ paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', }, shim: { "highlight-julia": { @@ -70,13 +71,90 @@ $(document).ready(function() { hljs.highlightAll(); }) +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + }) //////////////////////////////////////////////////////////////////////////////// require([], function() { function addCopyButtonCallbacks() { for (const el of document.getElementsByTagName("pre")) { const button = document.createElement("button"); - button.classList.add("copy-button", "fas", "fa-copy"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + el.appendChild(button); const success = function () { @@ -85,7 +163,7 @@ function addCopyButtonCallbacks() { }; const failure = function () { - button.classList.add("error", "fa-times"); + button.classList.add("error", "fa-xmark"); button.classList.remove("fa-copy"); }; @@ -94,7 +172,7 @@ function addCopyButtonCallbacks() { setTimeout(function () { button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-times"); + button.classList.remove("success", "fa-check", "fa-xmark"); }, 5000); }); } @@ -138,29 +216,418 @@ require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { // Manages the top navigation bar (hides it when the user starts scrolling down on the // mobile). window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function() { - $('#documenter .docs-navbar').headroom({ - "tolerance": {"up": 10, "down": 10}, +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, }); +}); + }) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // Modal settings dialog -$(document).ready(function() { - var settings = $('#documenter-settings'); - $('#documenter-settings-button').click(function(){ - settings.toggleClass('is-active'); +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); }); // Close the dialog if X is clicked - $('#documenter-settings button.delete').click(function(){ - settings.removeClass('is-active'); + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); }); // Close dialog if ESC is pressed - $(document).keyup(function(e) { - if (e.keyCode == 27) settings.removeClass('is-active'); + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); }); }); @@ -168,151 +635,242 @@ $(document).ready(function() { //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` + +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + // Manages the showing and hiding of the sidebar. -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button") - sidebar_button.click(function(ev) { + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { ev.preventDefault(); - sidebar.toggleClass('visible'); - if (sidebar.hasClass('visible')) { + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { // Makes sure that the current menu item is visible in the sidebar. $("#documenter .docs-menu a.is-active").focus(); } }); - $("#documenter > .docs-main").bind('click', function(ev) { + $("#documenter > .docs-main").bind("click", function (ev) { if ($(ev.target).is(sidebar_button)) { return; } - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); } }); -}) +}); // Resizes the package name / sitename in the sidebar if it is too wide. // Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function() { +$(document).ready(function () { e = $("#documenter .docs-autofit"); function resize() { - var L = parseInt(e.css('max-width'), 10); + var L = parseInt(e.css("max-width"), 10); var L0 = e.width(); - if(L0 > L) { - var h0 = parseInt(e.css('font-size'), 10); - e.css('font-size', L * h0 / L0); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); // TODO: make sure it survives resizes? } } // call once and then register events resize(); $(window).resize(resize); - $(window).on('orientationchange', resize); + $(window).on("orientationchange", resize); }); // Scroll the navigation bar to the currently selected menu item -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter .docs-menu").get(0); var active = $("#documenter .docs-menu .is-active").get(0); - if(typeof active !== 'undefined') { + if (typeof active !== "undefined") { sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -function set_theme(theme) { - var active = null; - var disabled = []; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - var themename = ss.ownerNode.getAttribute("data-theme-name"); - if(themename === null) continue; // ignore non-theme stylesheets - // Find the active theme - if(themename === theme) active = ss; - else disabled.push(ss); - } - if(active !== null) { - active.disabled = false; - if(active.ownerNode.getAttribute("data-theme-primary") === null) { - document.getElementsByTagName('html')[0].className = "theme--" + theme; - } else { - document.getElementsByTagName('html')[0].className = ""; - } - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - - // Store the theme in localStorage - if(typeof(window.localStorage) !== "undefined") { - window.localStorage.setItem("documenter-theme", theme); - } else { - console.error("Browser does not support window.localStorage"); - } -} - // Theme picker setup -$(document).ready(function() { +$(document).ready(function () { // onchange callback - $('#documenter-themepicker').change(function themepick_callback(ev){ - var themename = $('#documenter-themepicker option:selected').attr('value'); - set_theme(themename); + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); }); // Make sure that the themepicker displays the correct theme when the theme is retrieved // from localStorage - if(typeof(window.localStorage) !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if(theme !== null) { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = (e.value === theme); - }) - } else { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = $("html").hasClass(`theme--${e.value}`); - }) + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); } } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function() { +$(document).ready(function () { // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the // siteinfo.js file, we just return immediately and not display the version selector. - if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { return; } var version_selector = $("#documenter .docs-version-selector"); var version_selector_select = $("#documenter .docs-version-selector select"); - version_selector_select.change(function(x) { - target_href = version_selector_select.children("option:selected").get(0).value; + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; window.location.href = target_href; }); // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { - var option = $(""); + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); version_selector_select.append(option); } - if (typeof DOC_VERSIONS !== 'undefined') { + if (typeof DOC_VERSIONS !== "undefined") { var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); - DOC_VERSIONS.forEach(function(each) { - var version_url = documenterBaseURL + "/../" + each; + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; var existing_id = $.inArray(each, existing_versions_texts); // if not already in the version selector, add it as a new option, // otherwise update the old option with the URL and enable it if (existing_id == -1) { - var option = $(""); + var option = $( + "" + ); version_selector_select.append(option); } else { var option = existing_versions[existing_id]; @@ -326,6 +884,6 @@ $(document).ready(function() { if (version_selector_select.children("option").length > 0) { version_selector.toggleClass("visible"); } -}) +}); }) diff --git a/previews/PR333/assets/search.js b/previews/PR333/assets/search.js deleted file mode 100644 index c133f741..00000000 --- a/previews/PR333/assets/search.js +++ /dev/null @@ -1,267 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', - 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', - } -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { - -$(document).ready(function() { - // parseUri 1.2.2 - // (c) Steven Levithan - // MIT License - function parseUri (str) { - var o = parseUri.options, - m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), - uri = {}, - i = 14; - - while (i--) uri[o.key[i]] = m[i] || ""; - - uri[o.q.name] = {}; - uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { - if ($1) uri[o.q.name][$1] = $2; - }); - - return uri; - }; - parseUri.options = { - strictMode: false, - key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], - q: { - name: "queryKey", - parser: /(?:^|&)([^&=]*)=?([^&]*)/g - }, - parser: { - strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, - loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ - } - }; - - $("#search-form").submit(function(e) { - e.preventDefault() - }) - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - lunr.stopWordFilter = lunr.generateStopWordFilter([ - 'a', - 'able', - 'about', - 'across', - 'after', - 'almost', - 'also', - 'am', - 'among', - 'an', - 'and', - 'are', - 'as', - 'at', - 'be', - 'because', - 'been', - 'but', - 'by', - 'can', - 'cannot', - 'could', - 'dear', - 'did', - 'does', - 'either', - 'ever', - 'every', - 'from', - 'got', - 'had', - 'has', - 'have', - 'he', - 'her', - 'hers', - 'him', - 'his', - 'how', - 'however', - 'i', - 'if', - 'into', - 'it', - 'its', - 'just', - 'least', - 'like', - 'likely', - 'may', - 'me', - 'might', - 'most', - 'must', - 'my', - 'neither', - 'no', - 'nor', - 'not', - 'of', - 'off', - 'often', - 'on', - 'or', - 'other', - 'our', - 'own', - 'rather', - 'said', - 'say', - 'says', - 'she', - 'should', - 'since', - 'so', - 'some', - 'than', - 'that', - 'the', - 'their', - 'them', - 'then', - 'there', - 'these', - 'they', - 'this', - 'tis', - 'to', - 'too', - 'twas', - 'us', - 'wants', - 'was', - 'we', - 'were', - 'what', - 'when', - 'who', - 'whom', - 'why', - 'will', - 'would', - 'yet', - 'you', - 'your' - ]) - - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" - // would not find anything if searching for "add!", only for the entire qualification - lunr.tokenizer.separator = /[\s\-\.]+/ - - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - lunr.trimmer = function (token) { - return token.update(function (s) { - return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') - }) - } - - lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') - lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') - - var index = lunr(function () { - this.ref('location') - this.field('title',{boost: 100}) - this.field('text') - documenterSearchIndex['docs'].forEach(function(e) { - this.add(e) - }, this) - }) - var store = {} - - documenterSearchIndex['docs'].forEach(function(e) { - store[e.location] = {title: e.title, category: e.category, page: e.page} - }) - - $(function(){ - searchresults = $('#documenter-search-results'); - searchinfo = $('#documenter-search-info'); - searchbox = $('#documenter-search-query'); - searchform = $('.docs-search'); - sidebar = $('.docs-sidebar'); - function update_search(querystring) { - tokens = lunr.tokenizer(querystring) - results = index.query(function (q) { - tokens.forEach(function (t) { - q.term(t.toString(), { - fields: ["title"], - boost: 100, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["title"], - boost: 10, - usePipeline: true, - editDistance: 2, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["text"], - boost: 1, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - }) - }) - searchinfo.text("Number of results: " + results.length) - searchresults.empty() - results.forEach(function(result) { - data = store[result.ref] - link = $(''+data.title+'') - link.attr('href', documenterBaseURL+'/'+result.ref) - if (data.category != "page"){ - cat = $('('+data.category+', '+data.page+')') - } else { - cat = $('('+data.category+')') - } - li = $('
  • ').append(link).append(" ").append(cat) - searchresults.append(li) - }) - } - - function update_search_box() { - querystring = searchbox.val() - update_search(querystring) - } - - searchbox.keyup(_.debounce(update_search_box, 250)) - searchbox.change(update_search_box) - - // Disable enter-key form submission for the searchbox on the search page - // and just re-run search rather than refresh the whole page. - searchform.keypress( - function(event){ - if (event.which == '13') { - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); - } - update_search_box(); - event.preventDefault(); - } - } - ); - - search_query_uri = parseUri(window.location).queryKey["q"] - if(search_query_uri !== undefined) { - search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) - searchbox.val(search_query) - } - update_search_box(); - }) -}) - -}) diff --git a/previews/PR333/assets/themes/documenter-dark.css b/previews/PR333/assets/themes/documenter-dark.css index c94a294d..9f5449f4 100644 --- a/previews/PR333/assets/themes/documenter-dark.css +++ b/previews/PR333/assets/themes/documenter-dark.css @@ -1,7 +1,7 @@ -@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid 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.notification:not(:last-child),html.theme--documenter-dark .content:not(:last-child),html.theme--documenter-dark .box:not(:last-child){margin-bottom:1.5rem}html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-moz-appearance:none;-webkit-appearance:none;background-color:rgba(10,10,10,0.2);border:none;border-radius:290486px;cursor:pointer;pointer-events:auto;display:inline-block;flex-grow:0;flex-shrink:0;font-size:0;height:20px;max-height:20px;max-width:20px;min-height:20px;min-width:20px;outline:none;position:relative;vertical-align:top;width:20px}html.theme--documenter-dark .modal-close::before,html.theme--documenter-dark .delete::before,html.theme--documenter-dark .modal-close::after,html.theme--documenter-dark .delete::after{background-color:#fff;content:"";display:block;left:50%;position:absolute;top:50%;transform:translateX(-50%) translateY(-50%) rotate(45deg);transform-origin:center center}html.theme--documenter-dark 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:left}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 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.button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark 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kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark 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.button.is-warning.is-active{background-color:#946e00;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:#ad8100;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark 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.button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons 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Gets put into the tag.. function set_theme_from_local_storage() { - // Intialize the theme to null, which means default + // Initialize the theme to null, which means default var theme = null; // If the browser supports the localstorage and is not disabled then try to get the // documenter theme - if(window.localStorage != null) { + if (window.localStorage != null) { // Get the user-picked theme from localStorage. May be `null`, which means the default // theme. - theme = window.localStorage.getItem("documenter-theme"); + theme = window.localStorage.getItem("documenter-theme"); } - // Check if the browser supports user color preference - var darkPreference = false; // Check if the users preference is for dark color scheme - if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { - darkPreference = true; - } + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; // Initialize a few variables for the loop: // // - active: will contain the index of the theme that should be active. 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The names in localStorage much match this. var themename = ss.ownerNode.getAttribute("data-theme-name"); // attribute not set => non-theme stylesheet => ignore - if(themename === null) continue; + if (themename === null) continue; // To distinguish the default (primary) theme, it needs to have the data-theme-primary // attribute set. - var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); - // Check if the theme is primary dark theme - var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); - // If ss is for dark theme then set the value of darkTheme to the name of the theme - if(isDarkTheme) darkTheme = themename; + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } // If we find a matching theme (and it's not the default), we'll set active to non-null - if(themename === theme) active = i; + if (themename === theme) active = i; // Store the style sheets of inactive themes so that we could disable them - if(themename !== theme) disabled.push(ss); + if (themename !== theme) disabled.push(ss); } - if(active !== null) { + var activeTheme = null; + if (active !== null) { // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + theme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - ss.disabled = true; - }); + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. 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    Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} -if (document.readyState === 'loading') { - document.addEventListener('DOMContentLoaded', maybeAddWarning); +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); } else { - maybeAddWarning(); -}; + maybeAddWarning(); +} diff --git a/previews/PR333/background/index.html b/previews/PR333/background/index.html index a29c6483..b813eb88 100644 --- a/previews/PR333/background/index.html +++ b/previews/PR333/background/index.html @@ -1,5 +1,5 @@ -Background · TaylorSeries.jl
    Edit on GitHub

    Background

    Introduction

    TaylorSeries.jl is an implementation of high-order automatic differentiation, as presented in the book by W. Tucker [1]. The general idea is the following.

    The Taylor series expansion of an analytical function $f(t)$ with one independent variable $t$ around $t_0$ can be written as

    \[f(t) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + \cdots + f_k (t-t_0)^k + \cdots,\]

    where $f_0=f(t_0)$, and the Taylor coefficients $f_k = f_k(t_0)$ are the $k$-th normalized derivatives at $t_0$:

    \[f_k = \frac{1}{k!} \frac{{\rm d}^k f} {{\rm d} t^k}(t_0).\]

    Thus, computing the high-order derivatives of $f(t)$ is equivalent to computing its Taylor expansion.

    In the case of many independent variables the same statements hold, though things become more subtle. Following Alex Haro's approach [2], the Taylor expansion is an infinite sum of homogeneous polynomials in the $d$ independent variables $x_1, x_2, \dots, x_d$, which takes the form

    \[f_k (\mathbf{x_0}) = \sum_{m_1+\cdots+m_d = k}\, f_{m_1,\dots,m_d} \;\, +Background · TaylorSeries.jl

    GitHub

    Background

    Introduction

    TaylorSeries.jl is an implementation of high-order automatic differentiation, as presented in the book by W. Tucker [1]. The general idea is the following.

    The Taylor series expansion of an analytical function $f(t)$ with one independent variable $t$ around $t_0$ can be written as

    \[f(t) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + \cdots + f_k (t-t_0)^k + \cdots,\]

    where $f_0=f(t_0)$, and the Taylor coefficients $f_k = f_k(t_0)$ are the $k$-th normalized derivatives at $t_0$:

    \[f_k = \frac{1}{k!} \frac{{\rm d}^k f} {{\rm d} t^k}(t_0).\]

    Thus, computing the high-order derivatives of $f(t)$ is equivalent to computing its Taylor expansion.

    In the case of many independent variables the same statements hold, though things become more subtle. Following Alex Haro's approach [2], the Taylor expansion is an infinite sum of homogeneous polynomials in the $d$ independent variables $x_1, x_2, \dots, x_d$, which takes the form

    \[f_k (\mathbf{x_0}) = \sum_{m_1+\cdots+m_d = k}\, f_{m_1,\dots,m_d} \;\, (x_1-x_{0_1})^{m_1} \cdots (x_d-x_{0_d})^{m_d} = \sum_{|\mathbf{m}|=k} f_\mathbf{m}\, (\mathbf{x}-\mathbf{x_0})^\mathbf{m}.\]

    Here, $\mathbf{m}\in \mathbb{N}^d$ is a multi-index of the $k$-th order homogeneous polynomial and $\mathbf{x}=(x_1,x_2,\ldots,x_d)$ are the $d$ independent variables.

    In both cases, a Taylor series expansion can be represented by a vector containing its coefficients. The difference between the cases of one or more independent variables is that the coefficients are real or complex numbers in the former case, but homogeneous polynomials in the latter case. This motivates the construction of the Taylor1 and TaylorN types.

    Arithmetic operations

    Arithmetic operations involving Taylor series can be expressed as operations on the coefficients:

    \[(f(x) \pm g(x))_k = f_k \pm g_k , \\ (f(x) \cdot g(x))_k = \sum_{i=0}^k f_i \, g_{k-i} , \\ @@ -15,4 +15,4 @@ s(t) = \sin(f(t)) , \qquad s_k = \frac{1}{k}\sum_{j=0}^{k-1} (k-j) \, f_{k-j} \, c_j; \\ c(t) = \cos(f(t)) , \qquad - c_k = -\frac{1}{k}\sum_{j=0}^{k-1} (k-j) \, f_{k-j} \, s_j.\]

    The recursion relations for $s(t) = \sin\big(f(t)\big)$ and $c(t) = \cos\big(f(t)\big)$ depend on each other; this reflects the fact that they are solutions of a second-order differential equation.

    All these relations hold for Taylor expansions in one and more independent variables; in the latter case, the Taylor coefficients $f_k$ are homogeneous polynomials of degree $k$; see [2].

    References

    [1] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press (2011).

    [2] A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, preprint.

    + c_k = -\frac{1}{k}\sum_{j=0}^{k-1} (k-j) \, f_{k-j} \, s_j.\]

    The recursion relations for $s(t) = \sin\big(f(t)\big)$ and $c(t) = \cos\big(f(t)\big)$ depend on each other; this reflects the fact that they are solutions of a second-order differential equation.

    All these relations hold for Taylor expansions in one and more independent variables; in the latter case, the Taylor coefficients $f_k$ are homogeneous polynomials of degree $k$; see [2].

    References

    [1] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press (2011).

    [2] A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, preprint.

    diff --git a/previews/PR333/examples/index.html b/previews/PR333/examples/index.html index 97b8b4a8..73dadd71 100644 --- a/previews/PR333/examples/index.html +++ b/previews/PR333/examples/index.html @@ -1,5 +1,5 @@ -Examples · TaylorSeries.jl
    Edit on GitHub

    Examples

    Expanding exp(x) with taylor_expand()

    The taylor_expand function takes the function to expand as it's first argument, and the point to expand about as the second argument. A keyword argument order determines which order to expand to:

    julia> using TaylorSeries
    julia> taylor_expand(exp, 0, order=2) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

    And voìla! It really is that simple to calculate a simple taylor polynomial. The next example is slightly more complicated.

    Expanding exp(x) with a symbolic object

    An alternative way to compute the single-variable taylor expansion for a function is by defining a variable of type Taylor1, and using it in the function you wish to expand. The argument given to the Taylor1 constructor is the order to expand to:

    julia> using TaylorSeries
    julia> x = Taylor1(2) 1.0 t + 𝒪(t³)
    julia> exp(x) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

    Let's also get rid of the printed error for the next few examples, and set the printed independent variable to x:

    julia> displayBigO(false)false
    julia> set_taylor1_varname("x")"x"
    julia> exp(x) 1.0 + 1.0 x + 0.5 x²

    Changing point to expand about

    A variable constructed with Taylor1() automatically expands about the point x=0. But what if you want to use the symbolic object to expand about a point different from zero? Because expanding exp(x) about x=1 is exactly the same as expanding exp(x+1) about x=0, simply replace the x in your expression with x+1 to expand about x=1:

    julia> p = exp(x+1) 2.718281828459045 + 2.718281828459045 x + 1.3591409142295225 x²
    julia> p(0.01)2.7456005608350584
    julia> exp(1.01)2.7456010150169163

    More examples

    You can even use custum functions

    julia> f(a) = 1/(a+1)f (generic function with 1 method)
    julia> f(x) 1.0 - 1.0 x + 1.0 x²

    Functions can be nested

    julia> sin(f(x)) 0.8414709848078965 - 0.5403023058681398 x + 0.11956681346419151 x²

    and complicated further in a modular way

    julia> sin(exp(x+2))/(x+2)+cos(x+2)+f(x+2) 0.364113974242596 + 0.41259243488717107 x - 11.843864409375039 x²

    Four-square identity

    The first example shows that the four-square identity holds:

    \[(a_1^2 + a_2^2 + a_3^2 + a_4^2)\cdot(b_1^2 + b_2^2 + b_3^2 + b_4^2) = \\ +Examples · TaylorSeries.jl

    GitHub

    Examples

    Expanding exp(x) with taylor_expand()

    The taylor_expand function takes the function to expand as it's first argument, and the point to expand about as the second argument. A keyword argument order determines which order to expand to:

    julia> using TaylorSeries
    julia> taylor_expand(exp, 0, order=2) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

    And voìla! It really is that simple to calculate a simple taylor polynomial. The next example is slightly more complicated.

    Expanding exp(x) with a symbolic object

    An alternative way to compute the single-variable taylor expansion for a function is by defining a variable of type Taylor1, and using it in the function you wish to expand. The argument given to the Taylor1 constructor is the order to expand to:

    julia> using TaylorSeries
    julia> x = Taylor1(2) 1.0 t + 𝒪(t³)
    julia> exp(x) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

    Let's also get rid of the printed error for the next few examples, and set the printed independent variable to x:

    julia> displayBigO(false)false
    julia> set_taylor1_varname("x")"x"
    julia> exp(x) 1.0 + 1.0 x + 0.5 x²

    Changing point to expand about

    A variable constructed with Taylor1() automatically expands about the point x=0. But what if you want to use the symbolic object to expand about a point different from zero? Because expanding exp(x) about x=1 is exactly the same as expanding exp(x+1) about x=0, simply replace the x in your expression with x+1 to expand about x=1:

    julia> p = exp(x+1) 2.718281828459045 + 2.718281828459045 x + 1.3591409142295225 x²
    julia> p(0.01)2.7456005608350584
    julia> exp(1.01)2.7456010150169163

    More examples

    You can even use custum functions

    julia> f(a) = 1/(a+1)f (generic function with 1 method)
    julia> f(x) 1.0 - 1.0 x + 1.0 x²

    Functions can be nested

    julia> sin(f(x)) 0.8414709848078965 - 0.5403023058681398 x + 0.11956681346419151 x²

    and complicated further in a modular way

    julia> sin(exp(x+2))/(x+2)+cos(x+2)+f(x+2) 0.364113974242596 + 0.41259243488717107 x - 11.843864409375039 x²

    Four-square identity

    The first example shows that the four-square identity holds:

    \[(a_1^2 + a_2^2 + a_3^2 + a_4^2)\cdot(b_1^2 + b_2^2 + b_3^2 + b_4^2) = \\ \qquad (a_1 b_1 - a_2 b_2 - a_3 b_3 -a_4 b_4)^2 + (a_1 b_2 - a_2 b_1 - a_3 b_4 -a_4 b_3)^2 + \\ \qquad (a_1 b_3 - a_2 b_4 - a_3 b_1 -a_4 b_2)^2 + @@ -36,9 +36,9 @@ s = one(T) + x + y + z + w s = s^degree s * ( s + one(T) ) - endfateman1 (generic function with 1 method)

    (In the following lines, which are run when the documentation is built, by some reason the timing appears before the command executed.)

    julia> @time fateman1(0);  0.065287 seconds (55.52 k allocations: 34.884 MiB, 15.96% gc time, 54.26% compilation time)
    julia> @time f1 = fateman1(20);  2.462668 seconds (1.93 k allocations: 51.910 MiB, 0.34% gc time)

    Another implementation of the same, but exploiting optimizations related to ^2 yields:

    julia> function fateman2(degree::Int)
+       endfateman1 (generic function with 1 method)

    (In the following lines, which are run when the documentation is built, by some reason the timing appears before the command executed.)

    julia> @time fateman1(0);  0.039320 seconds (54.85 k allocations: 34.846 MiB, 15.37% gc time, 56.34% compilation time)
    julia> @time f1 = fateman1(20);  1.687138 seconds (1.93 k allocations: 51.910 MiB, 0.30% gc time)

    Another implementation of the same, but exploiting optimizations related to ^2 yields:

    julia> function fateman2(degree::Int)
            T = Int128
            s = one(T) + x + y + z + w
            s = s^degree
            s^2 + s
-       endfateman2 (generic function with 1 method)
    julia> fateman2(0);
    julia> @time f2 = fateman2(20); # the timing appears above  1.190777 seconds (2.06 k allocations: 56.059 MiB, 0.66% gc time)

    We note that the above functions use expansions in Int128. This is actually required, since some coefficients are larger than typemax(Int):

    julia> getcoeff(f2, (1,6,7,20)) # coefficient of x y^6 z^7 w^{20}128358585324486316800
    julia> ans > typemax(Int)true
    julia> length(f2)41
    julia> sum(TaylorSeries.size_table)135751

    These examples show that fateman2 is nearly twice as fast as fateman1, and that the series has 135751 monomials in 4 variables.

    Benchmarks

    The functions described above have been compared against Mathematica v11.1. The relevant files used for benchmarking can be found here. Running on a MacPro with Intel-Xeon processors 2.7GHz, we obtain that Mathematica requires on average (5 runs) 3.075957 seconds for the computation, while for fateman1 and fateman2 above we obtain 2.15408 and 1.08337, respectively.

    Then, with the current version of TaylorSeries.jl and using Julia v0.7.0, our implementation of fateman1 is about 30%-40% faster. (The original test by Fateman corresponds to fateman1 above, which avoids some optimizations related to squaring; the implementation in Mathematica is done such that this optimization does not occur.)

    + endfateman2 (generic function with 1 method)
    julia> fateman2(0);
    julia> @time f2 = fateman2(20); # the timing appears above 0.830596 seconds (2.06 k allocations: 56.059 MiB, 0.55% gc time)

    We note that the above functions use expansions in Int128. This is actually required, since some coefficients are larger than typemax(Int):

    julia> getcoeff(f2, (1,6,7,20)) # coefficient of x y^6 z^7 w^{20}128358585324486316800
    julia> ans > typemax(Int)true
    julia> length(f2)41
    julia> sum(TaylorSeries.size_table)135751

    These examples show that fateman2 is nearly twice as fast as fateman1, and that the series has 135751 monomials in 4 variables.

    Benchmarks

    The functions described above have been compared against Mathematica v11.1. The relevant files used for benchmarking can be found here. Running on a MacPro with Intel-Xeon processors 2.7GHz, we obtain that Mathematica requires on average (5 runs) 3.075957 seconds for the computation, while for fateman1 and fateman2 above we obtain 2.15408 and 1.08337, respectively.

    Then, with the current version of TaylorSeries.jl and using Julia v0.7.0, our implementation of fateman1 is about 30%-40% faster. (The original test by Fateman corresponds to fateman1 above, which avoids some optimizations related to squaring; the implementation in Mathematica is done such that this optimization does not occur.)

    diff --git a/previews/PR333/index.html b/previews/PR333/index.html index c3ce2fcd..1d169d58 100644 --- a/previews/PR333/index.html +++ b/previews/PR333/index.html @@ -1,2 +1,2 @@ -Home · TaylorSeries.jl
    Edit on GitHub

    TaylorSeries.jl

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

    Authors

    • Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM).
    • David P. Sanders, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM).

    Citing

    If you find useful this package, please cite the paper:

    Benet, L., & Sanders, D. P. (2019). TaylorSeries.jl: Taylor expansions in one and several variables in Julia. Journal of Open Source Software, 4(36), 1–4. https://doi.org/10.5281/zenodo.2601941

    License

    TaylorSeries is licensed under the MIT "Expat" license; see LICENSE for the full license text.

    Installation

    TaylorSeries.jl is a registered package, and is simply installed by running

    pkg> add("TaylorSeries")

    Acknowledgments

    This project began (using Python) during a Masters' course in the postgraduate programs in Physics and in Mathematics at UNAM, during the second half of 2013. We thank the participants of the course for putting up with the half-baked material and contributing energy and ideas.

    We acknowledge financial support from DGAPA-UNAM PAPIME grants PE-105911 and PE-107114, and PAPIIT grants IG-101113 and IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).

    +Home · TaylorSeries.jl
    GitHub

    TaylorSeries.jl

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

    Authors

    • Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM).
    • David P. Sanders, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM).

    Citing

    If you find useful this package, please cite the paper:

    Benet, L., & Sanders, D. P. (2019). TaylorSeries.jl: Taylor expansions in one and several variables in Julia. Journal of Open Source Software, 4(36), 1–4. https://doi.org/10.5281/zenodo.2601941

    License

    TaylorSeries is licensed under the MIT "Expat" license; see LICENSE for the full license text.

    Installation

    TaylorSeries.jl is a registered package, and is simply installed by running

    pkg> add("TaylorSeries")

    Acknowledgments

    This project began (using Python) during a Masters' course in the postgraduate programs in Physics and in Mathematics at UNAM, during the second half of 2013. We thank the participants of the course for putting up with the half-baked material and contributing energy and ideas.

    We acknowledge financial support from DGAPA-UNAM PAPIME grants PE-105911 and PE-107114, and PAPIIT grants IG-101113 and IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).

    diff --git a/previews/PR333/search/index.html b/previews/PR333/search/index.html deleted file mode 100644 index d4f62302..00000000 --- a/previews/PR333/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · TaylorSeries.jl

    Loading search...

      diff --git a/previews/PR333/search_index.js b/previews/PR333/search_index.js index ea44606f..ad7916a2 100644 --- a/previews/PR333/search_index.js +++ b/previews/PR333/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"api/#Library","page":"API","title":"Library","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"api/#Module","page":"API","title":"Module","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"TaylorSeries","category":"page"},{"location":"api/#TaylorSeries.TaylorSeries","page":"API","title":"TaylorSeries.TaylorSeries","text":"TaylorSeries\n\nA Julia package for Taylor expansions in one or more independent variables.\n\nThe basic constructors are Taylor1 and TaylorN; see also HomogeneousPolynomial.\n\n\n\n\n\n","category":"module"},{"location":"api/#Types","page":"API","title":"Types","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Taylor1\nHomogeneousPolynomial\nTaylorN\nAbstractSeries","category":"page"},{"location":"api/#TaylorSeries.Taylor1","page":"API","title":"TaylorSeries.Taylor1","text":"Taylor1{T<:Number} <: AbstractSeries{T}\n\nDataType for polynomial expansions in one independent variable.\n\nFields:\n\ncoeffs :: Array{T,1} Expansion coefficients; the i-th component is the coefficient of degree i-1 of the expansion.\norder :: Int Maximum order (degree) of the polynomial.\n\nNote that Taylor1 variables are callable. For more information, see evaluate.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.HomogeneousPolynomial","page":"API","title":"TaylorSeries.HomogeneousPolynomial","text":"HomogeneousPolynomial{T<:Number} <: AbstractSeries{T}\n\nDataType for homogeneous polynomials in many (>1) independent variables.\n\nFields:\n\ncoeffs :: Array{T,1} Expansion coefficients of the homogeneous\n\npolynomial; the i-th component is related to a monomial, where the degrees of the independent variables are specified by coeff_table[order+1][i].\n\norder :: Int order (degree) of the homogeneous polynomial.\n\nNote that HomogeneousPolynomial variables are callable. 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For more information, see evaluate.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.AbstractSeries","page":"API","title":"TaylorSeries.AbstractSeries","text":"AbstractSeries{T<:Number} <: Number\n\nParameterized abstract type for Taylor1, HomogeneousPolynomial and TaylorN.\n\n\n\n\n\n","category":"type"},{"location":"api/#Functions-and-methods","page":"API","title":"Functions and methods","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Taylor1(::Type{T}, ::Int) where {T<:Number}\nHomogeneousPolynomial(::Type{T}, ::Int) where {T<:Number}\nTaylorN(::Type{T}, ::Int; ::Int=get_order()) where {T<:Number}\nset_variables\nget_variables\nshow_params_TaylorN\nshow_monomials\ngetcoeff\nevaluate\nevaluate!\ntaylor_expand\nupdate!\ndifferentiate\nderivative\nintegrate\ngradient\njacobian\njacobian!\nhessian\nhessian!\nconstant_term\nlinear_polynomial\nnonlinear_polynomial\ninverse\nabs\nnorm\nisapprox\nisless\nisfinite\ndisplayBigO\nuse_show_default\nset_taylor1_varname","category":"page"},{"location":"api/#TaylorSeries.Taylor1-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.Taylor1","text":"Taylor1([T::Type=Float64], order::Int)\n\nShortcut to define the independent variable of a Taylor1{T} polynomial of given order. The default type for T is Float64.\n\njulia> Taylor1(16)\n 1.0 t + 𝒪(t¹⁷)\n\njulia> Taylor1(Rational{Int}, 4)\n 1//1 t + 𝒪(t⁵)\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.HomogeneousPolynomial-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.HomogeneousPolynomial","text":"HomogeneousPolynomial([T::Type=Float64], nv::Int])\n\nShortcut to define the nv-th independent HomogeneousPolynomial{T}. The default type for T is Float64.\n\njulia> HomogeneousPolynomial(1)\n 1.0 x₁\n\njulia> HomogeneousPolynomial(Rational{Int}, 2)\n 1//1 x₂\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.TaylorN-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.TaylorN","text":"TaylorN([T::Type=Float64], nv::Int; [order::Int=get_order()])\n\nShortcut to define the nv-th independent TaylorN{T} variable as a polynomial. The order is defined through the keyword parameter order, whose default corresponds to get_order(). The default of type for T is Float64.\n\njulia> TaylorN(1)\n 1.0 x₁ + 𝒪(‖x‖⁷)\n\njulia> TaylorN(Rational{Int},2)\n 1//1 x₂ + 𝒪(‖x‖⁷)\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.set_variables","page":"API","title":"TaylorSeries.set_variables","text":"set_variables([T::Type], names::String; [order=get_order(), numvars=-1])\n\nReturn a TaylorN{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 order is the one defined globally. Changing the order or numvars resets the hash_tables.\n\nIf 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.\n\njulia> set_variables(Int, \"x y z\", order=4)\n3-element Array{TaylorSeries.TaylorN{Int},1}:\n 1 x + 𝒪(‖x‖⁵)\n 1 y + 𝒪(‖x‖⁵)\n 1 z + 𝒪(‖x‖⁵)\n\njulia> set_variables(\"α\", numvars=2)\n2-element Array{TaylorSeries.TaylorN{Float64},1}:\n 1.0 α₁ + 𝒪(‖x‖⁵)\n 1.0 α₂ + 𝒪(‖x‖⁵)\n\njulia> set_variables(\"x\", order=6, numvars=2)\n2-element Array{TaylorSeries.TaylorN{Float64},1}:\n 1.0 x₁ + 𝒪(‖x‖⁷)\n 1.0 x₂ + 𝒪(‖x‖⁷)\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.get_variables","page":"API","title":"TaylorSeries.get_variables","text":"get_variables(T::Type, [order::Int=get_order()])\n\nReturn a TaylorN{T} vector with each entry representing an independent variable. 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Omitting T defaults to Float64.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.show_params_TaylorN","page":"API","title":"TaylorSeries.show_params_TaylorN","text":"show_params_TaylorN()\n\nDisplay the current parameters for TaylorN and HomogeneousPolynomial types.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.show_monomials","page":"API","title":"TaylorSeries.show_monomials","text":"show_monomials(ord::Int) --> nothing\n\nList the indices and corresponding of a HomogeneousPolynomial of degree ord.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.getcoeff","page":"API","title":"TaylorSeries.getcoeff","text":"getcoeff(a, n)\n\nReturn the coefficient of order n::Int of a a::Taylor1 polynomial.\n\n\n\n\n\ngetcoeff(a, v)\n\nReturn the coefficient of a::HomogeneousPolynomial, specified by v, which is a tuple (or vector) with the indices of the specific monomial.\n\n\n\n\n\ngetcoeff(a, v)\n\nReturn the coefficient of a::TaylorN, specified by v, which is a tuple (or vector) with the indices of the specific monomial.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.evaluate","page":"API","title":"TaylorSeries.evaluate","text":"evaluate(a, [dx])\n\nEvaluate a Taylor1 polynomial using Horner's rule (hand coded). If dx is omitted, its value is considered as zero. Note that the syntax a(dx) is equivalent to evaluate(a,dx), and a() is equivalent to evaluate(a).\n\n\n\n\n\nevaluate(x, δt)\n\nEvaluates each element of x::AbstractArray{Taylor1{T}}, representing the dependent variables of an ODE, at time δt. Note that the syntax x(δt) is equivalent to evaluate(x, δt), and x() is equivalent to evaluate(x).\n\n\n\n\n\nevaluate(a, x)\n\nSubstitute x::Taylor1 as independent variable in a a::Taylor1 polynomial. Note that the syntax a(x) is equivalent to evaluate(a, x).\n\n\n\n\n\nevaluate(a, [vals])\n\nEvaluate a HomogeneousPolynomial polynomial at vals. If vals is omitted, it's evaluated at zero. Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a).\n\n\n\n\n\nevaluate(a, [vals]; sorting::Bool=true)\n\nEvaluate the TaylorN polynomial a at vals. If vals is omitted, it's evaluated at zero. The keyword parameter sorting can be used to avoid sorting (in increasing order by abs2) the terms that are added.\n\nNote that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a). No extension exists that incorporates sorting.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.evaluate!","page":"API","title":"TaylorSeries.evaluate!","text":"evaluate!(x, δt, x0)\n\nEvaluates each element of x::AbstractArray{Taylor1{T}}, representing the Taylor expansion for the dependent variables of an ODE at time δt. It updates the vector x0 with the computed values.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.taylor_expand","page":"API","title":"TaylorSeries.taylor_expand","text":"taylor_expand(f, x0; order)\n\nComputes the Taylor expansion of the function f around the point x0.\n\nIf x0 is a scalar, a Taylor1 expansion will be returned. If x0 is a vector, a TaylorN expansion will be computed. If the dimension of x0 (length(x0)) is different from the variables set for TaylorN (get_numvars()), an AssertionError will be thrown.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.update!","page":"API","title":"TaylorSeries.update!","text":"update!(a, x0)\n\nTakes a <: Union{Taylo1,TaylorN} and expands it around the coordinate x0.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.differentiate","page":"API","title":"TaylorSeries.differentiate","text":"differentiate(a)\n\nReturn the Taylor1 polynomial of the differential of a::Taylor1. The order of the result is a.order-1.\n\nThe function derivative is an exact synonym of differentiate.\n\n\n\n\n\ndifferentiate(a, n)\n\nCompute recursively the Taylor1 polynomial of the n-th derivative of a::Taylor1. The order of the result is a.order-n.\n\n\n\n\n\ndifferentiate(n, a)\n\nReturn the value of the n-th differentiate of the polynomial a.\n\n\n\n\n\ndifferentiate(a, r)\n\nPartial differentiation of a::HomogeneousPolynomial series with respect to the r-th variable.\n\n\n\n\n\ndifferentiate(a, r)\n\nPartial differentiation of a::TaylorN series with respect to the r-th variable. The r-th variable may be also specified through its symbol.\n\n\n\n\n\ndifferentiate(a::TaylorN{T}, ntup::NTuple{N,Int})\n\nReturn a TaylorN with the partial derivative of a defined by ntup::NTuple{N,Int}, where the first entry is the number of derivatives with respect to the first variable, the second is the number of derivatives with respect to the second, and so on.\n\n\n\n\n\ndifferentiate(ntup::NTuple{N,Int}, a::TaylorN{T})\n\nReturns the value of the coefficient of a specified by ntup::NTuple{N,Int}, multiplied by the corresponding factorials.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.derivative","page":"API","title":"TaylorSeries.derivative","text":"derivative\n\nAn exact synonym of differentiate.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.integrate","page":"API","title":"TaylorSeries.integrate","text":"integrate(a, [x])\n\nReturn the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if omitted.\n\n\n\n\n\nintegrate(a, r)\n\nIntegrate the a::HomogeneousPolynomial with respect to the r-th variable. The returned HomogeneousPolynomial has no added constant of integration. If the order of a corresponds to get_order(), a zero HomogeneousPolynomial of 0-th order is returned.\n\n\n\n\n\nintegrate(a, r, [x0])\n\nIntegrate the a::TaylorN series with respect to the r-th variable, where x0 the integration constant and must be independent of the r-th variable; if x0 is omitted, it is taken as zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.gradient","page":"API","title":"TaylorSeries.gradient","text":" gradient(f)\n ∇(f)\n\nCompute the gradient of the polynomial f::TaylorN.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.jacobian","page":"API","title":"TaylorSeries.jacobian","text":" jacobian(vf)\n jacobian(vf, [vals])\n\nCompute the jacobian matrix of vf, a vector of TaylorN polynomials, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.jacobian!","page":"API","title":"TaylorSeries.jacobian!","text":" jacobian!(jac, vf)\n jacobian!(jac, vf, [vals])\n\nCompute the jacobian matrix of vf, a vector of TaylorN polynomials evaluated at the vector vals, and write results to jac. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.hessian","page":"API","title":"TaylorSeries.hessian","text":" hessian(f)\n hessian(f, [vals])\n\nReturn the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.hessian!","page":"API","title":"TaylorSeries.hessian!","text":" hessian!(hes, f)\n hessian!(hes, f, [vals])\n\nReturn the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals, and write results to hes. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.constant_term","page":"API","title":"TaylorSeries.constant_term","text":"constant_term(a)\n\nReturn the constant value (zero order coefficient) for Taylor1 and TaylorN. The fallback behavior is to return a itself if a::Number, or a[1] when a::Vector.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.linear_polynomial","page":"API","title":"TaylorSeries.linear_polynomial","text":"linear_polynomial(a)\n\nReturns the linear part of a as a polynomial (Taylor1 or TaylorN), without the constant term. The fallback behavior is to return a itself.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.nonlinear_polynomial","page":"API","title":"TaylorSeries.nonlinear_polynomial","text":"nonlinear_polynomial(a)\n\nReturns the nonlinear part of a. The fallback behavior is to return zero(a).\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.inverse","page":"API","title":"TaylorSeries.inverse","text":"inverse(f)\n\nReturn the Taylor expansion of f^-1(t), of order N = f.order, for f::Taylor1 polynomial if the first coefficient of f is zero. Otherwise, a DomainError is thrown.\n\nThe algorithm implements Lagrange inversion at t=0 if f(0)=0:\n\nbeginequation*\nf^-1(t) = sum_n=1^N fract^nn left\n fracrm d^n-1rm d z^n-1left(fraczf(z)right)^n\n rightvert_z=0\nendequation*\n\n\n\n","category":"function"},{"location":"api/#Base.abs","page":"API","title":"Base.abs","text":"abs(a)\n\nFor a Real type returns a if constant_term(a) > 0 and -a if constant_term(a) < 0 for a <:Union{Taylor1,TaylorN}. For a Complex type, such as Taylor1{ComplexF64}, returns sqrt(real(a)^2 + imag(a)^2).\n\nNotice that typeof(abs(a)) <: AbstractSeries and that for a Complex argument a Real type is returned (e.g. typeof(abs(a::Taylor1{ComplexF64})) == Taylor1{Float64}).\n\n\n\n","category":"function"},{"location":"api/#LinearAlgebra.norm","page":"API","title":"LinearAlgebra.norm","text":"norm(x::AbstractSeries, p::Real)\n\nReturns the p-norm of an x::AbstractSeries, defined by\n\nbeginequation*\nleftVert x rightVert_p = left( sum_k x_k ^p right)^frac1p\nendequation*\n\nwhich returns a non-negative number.\n\n\n\n","category":"function"},{"location":"api/#Base.isapprox","page":"API","title":"Base.isapprox","text":"isapprox(x::AbstractSeries, y::AbstractSeries; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false)\n\nInexact equality comparison between polynomials: returns true if norm(x-y,1) <= atol + rtol*max(norm(x,1), norm(y,1)), where x and y are polynomials. For more details, see Base.isapprox.\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.isless","page":"API","title":"Base.isless","text":"isless(a::Taylor1{<:Real}, b::Real)\nisless(a::TaylorN{<:Real}, b::Real)\n\nCompute isless by comparing the constant_term(a) and b. If they are equal, returns a[nz] < 0, with nz the first non-zero coefficient after the constant term. This defines a total order.\n\nFor many variables, the ordering includes a lexicographical convention in order to be total. We have opted for the simplest one, where the larger variable appears before when the TaylorN variables are defined (e.g., through set_variables).\n\nRefs:\n\nM. Berz, AIP Conference Proceedings 177, 275 (1988); https://doi.org/10.1063/1.37800\nM. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450.\n\n\n\nisless(a::Taylor1{<:Real}, b::Taylor1{<:Real})\nisless(a::TaylorN{<:Real}, b::Taylor1{<:Real})\n\nReturns isless(a - b, zero(b)).\n\n\n\n","category":"function"},{"location":"api/#Base.isfinite","page":"API","title":"Base.isfinite","text":"isfinite(x::AbstractSeries) -> Bool\n\nTest whether the coefficients of the polynomial x are finite.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.displayBigO","page":"API","title":"TaylorSeries.displayBigO","text":"displayBigO(d::Bool) --> nothing\n\nSet/unset displaying of the big 𝒪 notation in the output of Taylor1 and TaylorN polynomials. The initial value is true.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.use_show_default","page":"API","title":"TaylorSeries.use_show_default","text":"use_Base_show(d::Bool) --> nothing\n\nUse Base.show_default method (default show method in Base), or a custom display. The initial value is false, so customized display is used.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.set_taylor1_varname","page":"API","title":"TaylorSeries.set_taylor1_varname","text":"set_taylor1_varname(var::String)\n\nChange the displayed variable for Taylor1 objects.\n\n\n\n\n\n","category":"function"},{"location":"api/#Internals","page":"API","title":"Internals","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"ParamsTaylor1\nParamsTaylorN\n_InternalMutFuncs\ngenerate_tables\ngenerate_index_vectors\nin_base\nmake_inverse_dict\nresize_coeffs1!\nresize_coeffsHP!\nnumtype\nmul!\nmul!(::HomogeneousPolynomial, ::HomogeneousPolynomial, ::HomogeneousPolynomial)\nmul!(::Vector{Taylor1{T}}, ::Union{Matrix{T},SparseMatrixCSC{T}},::Vector{Taylor1{T}}) where {T<:Number}\ndiv!\npow!\nsquare\nsqr!\nsqr!(::HomogeneousPolynomial{T}, ::HomogeneousPolynomial{T}) where {T<:NumberNotSeriesN}\nsqrt!\nexp!\nlog!\nsincos!\ntan!\nasin!\nacos!\natan!\nsinhcosh!\ntanh!\nasinh!\nacosh!\natanh!\ndifferentiate!\n_internalmutfunc_call\n_dict_unary_ops\n_dict_binary_calls\n_dict_unary_calls\n_dict_binary_ops\n@isonethread","category":"page"},{"location":"api/#TaylorSeries.ParamsTaylor1","page":"API","title":"TaylorSeries.ParamsTaylor1","text":"ParamsTaylor1\n\nDataType holding the current variable name for Taylor1.\n\nField:\n\nvar_name :: String Names of the variables\n\nThese parameters can be changed using set_taylor1_varname\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.ParamsTaylorN","page":"API","title":"TaylorSeries.ParamsTaylorN","text":"ParamsTaylorN\n\nDataType holding the current parameters for TaylorN and HomogeneousPolynomial.\n\nFields:\n\norder :: Int Order (degree) of the polynomials\nnum_vars :: Int Number of variables\nvariable_names :: Vector{String} Names of the variables\nvariable_symbols :: Vector{Symbol} Symbols of the variables\n\nThese parameters can be changed using set_variables\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries._InternalMutFuncs","page":"API","title":"TaylorSeries._InternalMutFuncs","text":"_InternalMutFuncs\n\nContains parameters and expressions that allow a simple programmatic construction for calling the internal mutating functions.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.generate_tables","page":"API","title":"TaylorSeries.generate_tables","text":"generate_tables(num_vars, order)\n\nReturn the hash tables coeff_table, index_table, size_table and pos_table. Internally, these are treated as const.\n\nHash tables\n\ncoeff_table :: Array{Array{Array{Int,1},1},1}\n\nThe i+1-th component contains a vector with the vectors of all the possible combinations of monomials of a HomogeneousPolynomial of order i.\n\nindex_table :: Array{Array{Int,1},1}\n\nThe i+1-th component contains a vector of (hashed) indices that represent the distinct monomials of a HomogeneousPolynomial of order (degree) i.\n\nsize_table :: Array{Int,1}\n\nThe i+1-th component contains the number of distinct monomials of the HomogeneousPolynomial of order i, equivalent to length(coeff_table[i]).\n\npos_table :: Array{Dict{Int,Int},1}\n\nThe i+1-th component maps the hash index to the (lexicographic) position of the corresponding monomial in coeffs_table.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.generate_index_vectors","page":"API","title":"TaylorSeries.generate_index_vectors","text":"generate_index_vectors(num_vars, degree)\n\nReturn a vector of index vectors with num_vars (number of variables) and degree.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.in_base","page":"API","title":"TaylorSeries.in_base","text":"in_base(order, v)\n\nConvert vector v of non-negative integers to base oorder, where oorder is the next odd integer of order.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.make_inverse_dict","page":"API","title":"TaylorSeries.make_inverse_dict","text":"make_inverse_dict(v)\n\nReturn a Dict with the enumeration of v: the elements of v point to the corresponding index.\n\nIt is used to construct pos_table from index_table.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.resize_coeffs1!","page":"API","title":"TaylorSeries.resize_coeffs1!","text":"resize_coeffs1!{T nothing\n\nUpdate the k-th expansion coefficient c[k] of c = a * b, where all c, a, and b are either Taylor1 or TaylorN. Note that for TaylorN the result of a * b is accumulated in c[k].\n\nThe coefficients are given by\n\nc_k = sum_j=0^k a_j b_k-j\n\n\n\n","category":"function"},{"location":"api/#LinearAlgebra.mul!-Tuple{HomogeneousPolynomial, HomogeneousPolynomial, HomogeneousPolynomial}","page":"API","title":"LinearAlgebra.mul!","text":"mul!(c, a, b) --> nothing\n\nAccumulates in c[k] the result of a*b with minimum allocation; all arguments are HomogeneousPolynomial.\n\n\n\n\n\n","category":"method"},{"location":"api/#LinearAlgebra.mul!-Union{Tuple{T}, Tuple{Array{Taylor1{T}, 1}, Union{SparseArrays.SparseMatrixCSC{T}, Matrix{T}}, Array{Taylor1{T}, 1}}} where T<:Number","page":"API","title":"LinearAlgebra.mul!","text":"mul!(Y, A, B)\n\nMultiply A*B and save the result in Y.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.div!","page":"API","title":"TaylorSeries.div!","text":"div!(c, a, b, k::Int)\n\nCompute the k-th expansion coefficient c[k] of c = a / b, where all c, a and b are either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nc_k = frac1b_0 big(a_k - sum_j=0^k-1 c_j b_k-jbig)\n\nFor Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.pow!","page":"API","title":"TaylorSeries.pow!","text":"pow!(c, a, r::Real, k::Int)\n\nUpdate the k-th expansion coefficient c[k] of c = a^r, for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nc_k = frac1k a_0 sum_j=0^k-1 big(r(k-j) -jbig)a_k-j c_j\n\nFor Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.square","page":"API","title":"TaylorSeries.square","text":"square(a::AbstractSeries) --> typeof(a)\n\nReturn a^2; see TaylorSeries.sqr!.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sqr!","page":"API","title":"TaylorSeries.sqr!","text":"sqr!(c, a, k::Int) --> nothing\n\nUpdate the k-th expansion coefficient c[k] of c = a^2, for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\nc_k = 2 sum_j=0^(k-1)2 a_k-j a_j\n text if k is odd \nc_k = 2 sum_j=0^(k-2)2 a_k-j a_j + (a_k2)^2\n text if k is even\nendaligned\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sqrt!","page":"API","title":"TaylorSeries.sqrt!","text":"sqrt!(c, a, k::Int, k0::Int=0)\n\nCompute the k-th expansion coefficient c[k] of c = sqrt(a) for bothc and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\nc_k = frac12 c_0 big( a_k - 2 sum_j=1^(k-1)2 c_k-jc_jbig)\n text if k is odd \nc_k = frac12 c_0 big( a_k - 2 sum_j=1^(k-2)2 c_k-jc_j\n - (c_k2)^2big) text if k is even\nendaligned\n\nFor Taylor1 polynomials, k0 is the order of the first non-zero coefficient, which must be even.\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.exp!","page":"API","title":"TaylorSeries.exp!","text":"exp!(c, a, k) --> nothing\n\nUpdate the k-th expansion coefficient c[k+1] of c = exp(a) for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = frac1k sum_j=0^k-1 (k-j) a_k-j c_j\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.log!","page":"API","title":"TaylorSeries.log!","text":"log!(c, a, k) --> nothing\n\nUpdate the k-th expansion coefficient c[k+1] of c = log(a) for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = frac1a_0 big(a_k - frac1k sum_j=0^k-1 j a_k-j c_j big)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sincos!","page":"API","title":"TaylorSeries.sincos!","text":"sincos!(s, c, a, k) --> nothing\n\nUpdate the k-th expansion coefficients s[k+1] and c[k+1] of s = sin(a) and c = cos(a) simultaneously, for s, c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\ns_k = frac1ksum_j=0^k-1 (k-j) a_k-j c_j \nc_k = -frac1ksum_j=0^k-1 (k-j) a_k-j s_j\nendaligned\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.tan!","page":"API","title":"TaylorSeries.tan!","text":"tan!(c, a, p, k::Int) --> nothing\n\nUpdate the k-th expansion coefficients c[k+1] of c = tan(a), for c and a either Taylor1 or TaylorN; p = c^2 and is passed as an argument for efficiency.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = a_k + frac1k sum_j=0^k-1 (k-j) a_k-j p_j\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.asin!","page":"API","title":"TaylorSeries.asin!","text":"asin!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = asin(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 sqrtr_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.acos!","page":"API","title":"TaylorSeries.acos!","text":"acos!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = acos(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = - frac1 r_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.atan!","page":"API","title":"TaylorSeries.atan!","text":"atan!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = atan(a), for c and a either Taylor1 or TaylorN; r = 1+a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1r_0big(a_k - frac1k sum_j=1^k-1 j r_k-j c_jbig)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sinhcosh!","page":"API","title":"TaylorSeries.sinhcosh!","text":"sinhcosh!(s, c, a, k)\n\nUpdate the k-th expansion coefficients s[k+1] and c[k+1] of s = sinh(a) and c = cosh(a) simultaneously, for s, c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\ns_k = frac1k sum_j=0^k-1 (k-j) a_k-j c_j \nc_k = frac1k sum_j=0^k-1 (k-j) a_k-j s_j\nendaligned\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.tanh!","page":"API","title":"TaylorSeries.tanh!","text":"tanh!(c, a, p, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = tanh(a), for c and a either Taylor1 or TaylorN; p = a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = a_k - frac1k sum_j=0^k-1 (k-j) a_k-j p_j\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.asinh!","page":"API","title":"TaylorSeries.asinh!","text":"asinh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = asinh(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 sqrtr_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.acosh!","page":"API","title":"TaylorSeries.acosh!","text":"acosh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = acosh(a), for c and a either Taylor1 or TaylorN; r = sqrt(c^2-1) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 r_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.atanh!","page":"API","title":"TaylorSeries.atanh!","text":"atanh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = atanh(a), for c and a either Taylor1 or TaylorN; r = 1-a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1r_0big(a_k + frac1k sum_j=1^k-1 j r_k-j c_jbig)\nendequation*\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.differentiate!","page":"API","title":"TaylorSeries.differentiate!","text":"differentiate!(res, a) --> nothing\n\nIn-place version of differentiate. Compute the Taylor1 polynomial of the differential of a::Taylor1 and return it as res (order of res remains unchanged).\n\n\n\n\n\ndifferentiate!(p, a, k) --> nothing\n\nUpdate in-place the k-th expansion coefficient p[k] of p = differentiate(a) for both p and a Taylor1.\n\nThe coefficients are given by\n\np_k = (k+1) a_k+1\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries._internalmutfunc_call","page":"API","title":"TaylorSeries._internalmutfunc_call","text":"_internalmutfunc_call( fn :: _InternalMutFuncs )\n\nCreates the appropriate call to the internal mutating function defined by the _InternalMutFuncs object. This is used to construct _dict_unary_calls and _dict_binary_calls. The call contains the prefix TaylorSeries..\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries._dict_unary_ops","page":"API","title":"TaylorSeries._dict_unary_ops","text":"_dict_binary_ops\n\nDict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to unary operations.\n\nThe keys correspond to the function symbols.\n\nThe arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1, for the required argument, possibly :_aux when there is an auxiliary expression needed, and :_k for the computed order of :_res. When an auxiliary expression is required, an Expr defining its calling pattern is added as the last entry of the vector.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_binary_calls","page":"API","title":"TaylorSeries._dict_binary_calls","text":"_dict_binary_calls::Dict{Symbol, NTuple{2,Expr}}\n\nDictionary with the expressions that define the internal binary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.\n\nEvaluating the entries generates symbols that represent the actual calls to the internal mutating functions.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_unary_calls","page":"API","title":"TaylorSeries._dict_unary_calls","text":"_dict_unary_calls::Dict{Symbol, NTuple{2,Expr}}\n\nDictionary with the expressions that define the internal unary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.\n\nEvaluating the entries generates expressions that represent the actual calls to the internal mutating functions.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_binary_ops","page":"API","title":"TaylorSeries._dict_binary_ops","text":"_dict_binary_ops\n\nDict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to binary operations.\n\nThe keys correspond to the function symbols.\n\nThe arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1 and _arg2 for the required arguments, and :_k for the computed order of :_res.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries.@isonethread","page":"API","title":"TaylorSeries.@isonethread","text":"@isonethread (expr)\n\nInternal macro used to check the number of threads in use, to prevent a data race that modifies coeff_table when using differentiate or integrate; see https://github.com/JuliaDiff/TaylorSeries.jl/issues/318.\n\nThis macro is inspired by the macro @threaded; see https://github.com/trixi-framework/Trixi.jl/blob/main/src/auxiliary/auxiliary.jl; and https://github.com/trixi-framework/Trixi.jl/pull/426/files.\n\n\n\n\n\n","category":"macro"},{"location":"api/#Index","page":"API","title":"Index","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Pages = [\"api.md\"]\nModule = [\"TaylorSeries\"]\nOrder = [:type, :function]","category":"page"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"examples/#Expanding-exp(x)-with-taylor_expand()","page":"Examples","title":"Expanding exp(x) with taylor_expand()","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The taylor_expand function takes the function to expand as it's first argument, and the point to expand about as the second argument. A keyword argument order determines which order to expand to:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n\ntaylor_expand(exp, 0, order=2)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"And voìla! It really is that simple to calculate a simple taylor polynomial. The next example is slightly more complicated.","category":"page"},{"location":"examples/#Expanding-exp(x)-with-a-symbolic-object","page":"Examples","title":"Expanding exp(x) with a symbolic object","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"An alternative way to compute the single-variable taylor expansion for a function is by defining a variable of type Taylor1, and using it in the function you wish to expand. The argument given to the Taylor1 constructor is the order to expand to:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n\nx = Taylor1(2)\n\nexp(x)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Let's also get rid of the printed error for the next few examples, and set the printed independent variable to x:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"displayBigO(false)\n\nset_taylor1_varname(\"x\")\n\nexp(x)","category":"page"},{"location":"examples/#Changing-point-to-expand-about","page":"Examples","title":"Changing point to expand about","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"A variable constructed with Taylor1() automatically expands about the point x=0. But what if you want to use the symbolic object to expand about a point different from zero? Because expanding exp(x) about x=1 is exactly the same as expanding exp(x+1) about x=0, simply replace the x in your expression with x+1 to expand about x=1:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"p = exp(x+1)\n\np(0.01)\n\nexp(1.01)","category":"page"},{"location":"examples/#More-examples","page":"Examples","title":"More examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"You can even use custum functions","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"f(a) = 1/(a+1)\n\nf(x)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Functions can be nested","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sin(f(x))","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"and complicated further in a modular way","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sin(exp(x+2))/(x+2)+cos(x+2)+f(x+2)","category":"page"},{"location":"examples/#Four-square-identity","page":"Examples","title":"Four-square identity","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The first example shows that the four-square identity holds:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"(a_1^2 + a_2^2 + a_3^2 + a_4^2)cdot(b_1^2 + b_2^2 + b_3^2 + b_4^2) = \n qquad (a_1 b_1 - a_2 b_2 - a_3 b_3 -a_4 b_4)^2 +\n (a_1 b_2 - a_2 b_1 - a_3 b_4 -a_4 b_3)^2 + \n qquad (a_1 b_3 - a_2 b_4 - a_3 b_1 -a_4 b_2)^2 +\n (a_1 b_4 - a_2 b_3 - a_3 b_2 -a_4 b_1)^2","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"which was originally proved by Euler. The code can also be found in this test of the package.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"First, we reset the maximum degree of the polynomial to 4, since the RHS of the equation has a priori terms of fourth order, and define the 8 independent variables.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n# Define the variables α₁, ..., α₄, β₁, ..., β₄\nmake_variable(name, index::Int) = string(name, TaylorSeries.subscriptify(index))\nvariable_names = [make_variable(\"α\", i) for i in 1:4]\nappend!(variable_names, [make_variable(\"β\", i) for i in 1:4])\n# Create the TaylorN variables (order=4, numvars=8)\na1, a2, a3, a4, b1, b2, b3, b4 = set_variables(variable_names, order=4)\na1 # variable a1","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Now we compute each term appearing in Eq. (\\ref{eq:Euler})","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"# left-hand side\nlhs1 = a1^2 + a2^2 + a3^2 + a4^2 ;\nlhs2 = b1^2 + b2^2 + b3^2 + b4^2 ;\nlhs = lhs1 * lhs2;\n# right-hand side\nrhs1 = (a1*b1 - a2*b2 - a3*b3 - a4*b4)^2 ;\nrhs2 = (a1*b2 + a2*b1 + a3*b4 - a4*b3)^2 ;\nrhs3 = (a1*b3 - a2*b4 + a3*b1 + a4*b2)^2 ;\nrhs4 = (a1*b4 + a2*b3 - a3*b2 + a4*b1)^2 ;\nrhs = rhs1 + rhs2 + rhs3 + rhs4;","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"We now compare the two sides of the identity,","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"lhs == rhs","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"The identity is satisfied. square","category":"page"},{"location":"examples/#Fateman-test","page":"Examples","title":"Fateman test","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Richard J. Fateman, from Berkeley, proposed as a stringent test of polynomial multiplication the evaluation of scdot(s+1), where s = (1+x+y+z+w)^20. This is implemented in the function fateman1 below. We shall also consider the form s^2+s in fateman2, which involves fewer operations (and makes a fairer comparison to what Mathematica does).","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\nconst order = 20\nconst x, y, z, w = set_variables(Int128, \"x\", numvars=4, order=2order)\nfunction fateman1(degree::Int)\n T = Int128\n s = one(T) + x + y + z + w\n s = s^degree\n s * ( s + one(T) )\nend","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"(In the following lines, which are run when the documentation is built, by some reason the timing appears before the command executed.)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"@time fateman1(0);\n# hide\n@time f1 = fateman1(20);","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Another implementation of the same, but exploiting optimizations related to ^2 yields:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"function fateman2(degree::Int)\n T = Int128\n s = one(T) + x + y + z + w\n s = s^degree\n s^2 + s\nend\nfateman2(0);\n@time f2 = fateman2(20); # the timing appears above","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"We note that the above functions use expansions in Int128. This is actually required, since some coefficients are larger than typemax(Int):","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"getcoeff(f2, (1,6,7,20)) # coefficient of x y^6 z^7 w^{20}\nans > typemax(Int)\nlength(f2)\nsum(TaylorSeries.size_table)\nset_variables(\"x\", numvars=2, order=6) # hide","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"These examples show that fateman2 is nearly twice as fast as fateman1, and that the series has 135751 monomials in 4 variables.","category":"page"},{"location":"examples/#Benchmarks","page":"Examples","title":"Benchmarks","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The functions described above have been compared against Mathematica v11.1. The relevant files used for benchmarking can be found here. Running on a MacPro with Intel-Xeon processors 2.7GHz, we obtain that Mathematica requires on average (5 runs) 3.075957 seconds for the computation, while for fateman1 and fateman2 above we obtain 2.15408 and 1.08337, respectively.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then, with the current version of TaylorSeries.jl and using Julia v0.7.0, our implementation of fateman1 is about 30%-40% faster. (The original test by Fateman corresponds to fateman1 above, which avoids some optimizations related to squaring; the implementation in Mathematica is done such that this optimization does not occur.)","category":"page"},{"location":"userguide/#User-guide","page":"User guide","title":"User guide","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"For some simple examples, head over to the examples section. For a detailed guide, keep reading.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"TaylorSeries.jl is a basic polynomial algebraic manipulator in one or more variables; these two cases are treated separately. Three new types are defined, Taylor1, HomogeneousPolynomial and TaylorN, which correspond to expansions in one independent variable, homogeneous polynomials of various variables, and the polynomial series in many independent variables, respectively. These types are subtypes of AbstractSeries, which in turn is a subtype of Number, and are defined parametrically.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The package is loaded as usual:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"using TaylorSeries","category":"page"},{"location":"userguide/#One-independent-variable","page":"User guide","title":"One independent variable","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor expansions in one variable are represented by the Taylor1 type, which consists of a vector of coefficients (fieldname coeffs) and the maximum order considered for the expansion (fieldname order). The coefficients are arranged in ascending order with respect to the degree of the monomial, so that coeffs[1] is the constant term, coeffs[2] gives the first order term (t^1), etc. Yet, it is possible to have the natural ordering with respect to the degree; see below. This is a dense representation of the polynomial. The order of the polynomial can be omitted in the constructor, which is then fixed by the length of the vector of coefficients. If the length of the vector does not correspond with the order, order is used, which effectively truncates polynomial to degree order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor1([1, 2, 3],4) # Polynomial of order 4 with coefficients 1, 2, 3\nTaylor1([0.0, 1im]) # Also works with complex numbers\nTaylor1(ones(8), 2) # Polynomial truncated to order 2\nshift_taylor(a) = a + Taylor1(typeof(a),5) ## a + taylor-polynomial of order 5\nt = shift_taylor(0.0) # Independent variable `t`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"warning: Warning\nThe information about the maximum order considered is displayed using a big-𝒪 notation. The convention followed when different orders are combined, and when certain functions are used in a way that they reduce the order (like differentiate), is to be consistent with the mathematics and the big-𝒪 notation, i.e., to propagate the lowest order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"In some cases, it is desirable to not display the big-𝒪 notation. The function displayBigO allows to control whether it is displayed or not.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"displayBigO(false) # turn-off displaying big O notation\nt\ndisplayBigO(true) # turn it on\nt","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similarly, it is possible to control some aspects of the format of the displayed series through the function use_show_default; use_show_default(true) uses the Base.show_default, while use_show_default(false) uses the custom display form (default).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"use_show_default(true) # use Base.show method\nt\nuse_show_default(false) # use custom `show`\nt","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The definition of shift_taylor(a) uses the method Taylor1([::Type{Float64}], order::Int), which is a shortcut to define the independent variable of a Taylor expansion, of given type and order (the default is Float64). Defining the independent variable in advance is one of the easiest ways to use the package.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with the Taylor1 type, including promotions that involve Numbers. The operations return a valid Taylor expansion, consistent with the order of the series. This is apparent in the penultimate example below, where the fist non-zero coefficient is beyond the order of the expansion, and hence the result is zero.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t*(3t+2.5)\n1/(1-t)\nt*(t^2-4)/(t+2)\ntI = im*t\n(1-t)^3.2\n(1+t)^t\nTaylor1(3) + Taylor1(5) == 2Taylor1(3) # big-𝒪 convention applies\nt^6 # t is of order 5\nt^2 / t # The result is of order 4, instead of 5","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that the last example returns a Taylor1 series of order 4, instead of order 5; this is be consistent with the number of known coefficients of the returned series, since the result corresponds to factorize t in the numerator to cancel the same factor in the denominator.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor1 is also equiped with a total order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. This is illustrated by:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"1 > t > 2*t^2 > 100*t^3 >= 0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"If no valid Taylor expansion can be computed an error is thrown, for instance, when a derivative is not defined at the expansion point, or it simply diverges.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"1/t\nt^3.2\nabs(t)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Several elementary functions have been implemented; their coefficients are computed recursively. At the moment of this writing, these functions are exp, log, sqrt, the trigonometric functions sin, cos and tan, their inverses, as well as the hyperbolic functions sinh, cosh and tanh and their inverses; more functions will be added in the future. Note that this way of obtaining the Taylor coefficients is not a lazy way, in particular for many independent variables. Yet, the implementation is efficient enough, especially for the integration of ordinary differential equations, which is among the applications we have in mind (see TaylorIntegration.jl).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exp(t)\nlog(1-t)\nsqrt(1 + t)\nimag(exp(tI)')\nreal(exp(Taylor1([0.0,1im],17))) - cos(Taylor1([0.0,1.0],17)) == 0.0\nconvert(Taylor1{Rational{Int}}, exp(t))","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Again, errors are thrown whenever it is necessary.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"sqrt(t)\nlog(t)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To obtain a specific coefficient, getcoeff can be used. Another alternative is to request the specific degree using the vector notation, where the index corresponds to the degree of the term.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"expon = exp(t)\ngetcoeff(expon, 0) == expon[0]\nrationalize(expon[3])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that certain arithmetic operations, or the application of some functions, may change the order of the result, as mentioned above.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t # order 5 independent variable\nt^2/t # returns an order 4 series expansion\nsqrt(t^2) # returns an order 2 series expansion\n(t^4)^(1/4) # order 1 series","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Differentiating and integrating is straightforward for polynomial expansions in one variable, using differentiate and integrate. (The function derivative is a synonym of differentiate.) These functions return the corresponding Taylor1 expansions. The order of the derivative of a Taylor1 is reduced by 1. For the integral, an integration constant may be set by the user (the default is zero); the order of the integrated polynomial for the integral is kept unchanged. The value of the n-th (n ge 0) derivative is obtained using differentiate(n,a), where a is a Taylor series; likewise, the Taylor1 polynomial of the n-th derivative is obtained as differentiate(a,n); the resulting polynomial is of order get_order(a)-n.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate(exp(t)) # exp(t) is of order 5; the derivative is of order 4\nintegrate(exp(t)) # the resulting TaylorSeries is of order 5\nintegrate( exp(t), 1.0)\nintegrate( differentiate( exp(-t)), 1.0 ) == exp(-t)\ndifferentiate(1, exp(shift_taylor(1.0))) == exp(1.0)\ndifferentiate(5, exp(shift_taylor(1.0))) == exp(1.0) # 5-th differentiate of `exp(1+t)`\nderivative(exp(1+t), 3) # Taylor1 polynomial of the 3-rd derivative of `exp(1+t)`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To evaluate a Taylor series at a given point, Horner's rule is used via the function evaluate(a, dt). Here, dt is the increment from the point t_0 around which the Taylor expansion of a is calculated, i.e., the series is evaluated at t = t_0 + dt. Omitting dt corresponds to dt = 0; see evaluate.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate(exp(shift_taylor(1.0))) - ℯ # exp(t) around t0=1 (order 5), evaluated there (dt=0)\nevaluate(exp(t), 1) - ℯ # exp(t) around t0=0 (order 5), evaluated at t=1\nevaluate(exp( Taylor1(17) ), 1) - ℯ # exp(t) around t0=0, order 17\ntBig = Taylor1(BigFloat, 50) # Independent variable with BigFloats, order 50\neBig = evaluate( exp(tBig), one(BigFloat) )\nℯ - eBig","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Another way to evaluate the value of a Taylor1 polynomial p at a given value x, is to call p as if it was a function, i.e., p(x):","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t = Taylor1(15)\np = sin(t)\nevaluate(p, pi/2) # value of p at pi/2 using `evaluate`\np(pi/2) # value of p at pi/2 by evaluating p as a function\np(pi/2) == evaluate(p, pi/2)\np(0.0)\np() == p(0.0) # p() is a shortcut to obtain the 0-th order coefficient of `p`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that the syntax p(x) is equivalent to evaluate(p, x), whereas p() is equivalent to evaluate(p).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Useful shortcuts are taylor_expand and update!. The former returns the expansion of a function around a given value t0, mimicking the use of shift_taylor above. In turn, update! provides an in-place update of a given Taylor polynomial, that is, it shifts it further by the provided amount.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"p = taylor_expand( x -> sin(x), pi/2, order=16) # 16-th order expansion of sin(t) around pi/2\nupdate!(p, 0.025) # updates the expansion given by p, by shifting it further by 0.025\np","category":"page"},{"location":"userguide/#Many-variables","page":"User guide","title":"Many variables","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"A polynomial in N1 variables can be represented in (at least) two ways: As a vector whose coefficients are homogeneous polynomials of fixed degree, or as a vector whose coefficients are polynomials in N-1 variables. The current implementation of TaylorSeries.jl corresponds to the first option, though some infrastructure has been built that permits to develop the second one. An elegant (lazy) implementation of the second representation was discussed here.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The structure TaylorN is constructed as a vector of parameterized homogeneous polynomials defined by the type HomogeneousPolynomial, which in turn is an ordered vector of coefficients of given order (degree). This implementation imposes the user to specify the (maximum) order considered and the number of independent variables at the beginning, which can be conveniently done using set_variables. A vector of the resulting Taylor variables is returned:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\")\ntypeof(x)\nx.order\nx.coeffs","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As shown, the resulting objects are of TaylorN{Float64} type. There is an optional order keyword argument in set_variables, used to specify the maximum order of the TaylorN polynomials. Note that one can specify the variables using a vector of symbols.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables([:x, :y], order=10)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similarly, subindexed variables are also available by specifying a single variable name and the optional keyword argument numvars:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables(\"α\", numvars=3)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Alternatively to set_variables, get_variables can be used if one does not want to change internal dictionaries. get_variables returns a vector of TaylorN independent variables of a desired order (lesser than get_order so the internals doesn't have to change) with the length and variable names defined by set_variables initially.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"get_variables(2) # vector of independent variables of order 2","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"warning: Warning\nAn OverflowError is thrown when the construction of the internal tables is not fully consistent, avoiding silent errors; see issue #85.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The function show_params_TaylorN displays the current values of the parameters, in an info block.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"show_params_TaylorN()","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Internally, changing the parameters (maximum order and number of variables) redefines the hash-tables that translate the index of the coefficients of a HomogeneousPolynomial of given order into the corresponding multi-variable monomials, or the other way around. Fixing these values from the start is imperative; the initial (default) values are order = 6 and num_vars=2.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The easiest way to construct a TaylorN object is by defining the independent variables. This can be done using set_variables as above, or through the method TaylorN{T<:Number}(::Type{T}, nv::Int) for the nv independent TaylorN{T} variable; the order can be also specified using the optional keyword argument order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", numvars=2, order=6);\nx\nTaylorN(1, order=4) # variable 1 of order 4\nTaylorN(Int, 2) # variable 2, type Int, order=get_order()=6","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Other ways of constructing TaylorN polynomials involve using HomogeneousPolynomial objects directly, which is uncomfortable.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables(:x, numvars=2); # symbols can be used\nHomogeneousPolynomial([1,-1])\nTaylorN([HomogeneousPolynomial([1,0]), HomogeneousPolynomial([1,2,3])],4)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The Taylor expansions are implemented around 0 for all variables; if the expansion is needed around a different value, the trick is a simple translation of the corresponding independent variable, i.e. x to x+a.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As before, the usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with TaylorN objects, including the appropriate promotions to deal with the usual numeric types. Note that some of the arithmetic operations have been extended for HomogeneousPolynomial, whenever the result is a HomogeneousPolynomial; division, for instance, is not extended. The same convention used for Taylor1 objects is used when combining TaylorN polynomials of different order. Both HomogeneousPolynomial and TaylorN are equiped with a total lexicographical order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. Essentially, the lexicographic order works as follows: smaller order monomials are larger than higher order monomials; when the order is the same, larger monomials appear before in the hash-tables; the function show_monomials.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", order=10);\n\nshow_monomials(2)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Then, the following then holds:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"0 < 1e8 * y^2 < x*y < x^2 < y < x/1e8 < 1.0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The elementary functions have also been implemented, again by computing their coefficients recursively:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy = exp(x+y)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The function getcoeff gives the normalized coefficient of the polynomial that corresponds to the monomial specified by the tuple or vector v containing the powers. For instance, for the polynomial exy above, the coefficient of the monomial x^3 y^5 is obtained using getcoeff(exy, (3,5)) or getcoeff(exy, [3,5]).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"getcoeff(exy, (3,5))\nrationalize(ans)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similar to Taylor1, vector notation can be used to request specific coefficients of HomogeneousPolynomial or TaylorN objects. For TaylorN objects, the index refers to the degree of the HomogeneousPolynomial. In the case of HomogeneousPolynomial the index refers to the position of the hash table. The function show_monomials can be used to obtain the coefficient a specific monomial, given the degree of the HomogeneousPolynomial.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy[8] # get the 8th order term\nshow_monomials(8)\nexy[8][6] # get the 6th coeff of the 8th order term","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Partial differentiation is also implemented for TaylorN objects, through the function differentiate, specifying the number of the variable, or its symbol, as the second argument.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"p = x^3 + 2x^2 * y - 7x + 2\nq = y - x^4\ndifferentiate( p, 1 ) # partial derivative with respect to 1st variable\ndifferentiate( q, :y ) # partial derivative with respect to :y","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"If we ask for the partial derivative with respect to a non-defined variable, an error is thrown.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate( q, 3 ) # error, since we are dealing with 2 variables","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To obtain more specific partial derivatives we have two specialized methods that involve a tuple, which represents the number of derivatives with respect to each variable (so the tuple's length has to be the same as the actual number of variables). These methods either return the TaylorN object in question, or the coefficient corresponding to the specified tuple, normalized by the factorials defined by the tuple. The latter is in essence the 0-th order coefficient of the former.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate(p, (2,1)) # two derivatives on :x and one on :y\ndifferentiate((2,1), p) # 0-th order coefficient of the previous expression\ndifferentiate(p, (1,1)) # one derivative on :x and one on :y\ndifferentiate((1,1), p) # 0-th order coefficient of the previous expression","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Integration with respect to the r-th variable for HomogeneousPolynomials and TaylorN objects is obtained using integrate. Note that integrate for TaylorN objects allows to specify a constant of integration, which must be independent from the integrated variable. Again, the integration variable may be specified by its symbol.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"integrate( differentiate( p, 1 ), 1) # integrate with respect to the first variable\nintegrate( differentiate( p, 1 ), :x, 2) # integration with respect to :x, constant of integration is 2\nintegrate( differentiate( q, 2 ), :y, -x^4) == q\nintegrate( differentiate( q, 2 ), 2, y)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate can also be used for TaylorN objects, using it on vectors of numbers (Real or Complex); the length of the vector must coincide with the number of independent variables. evaluate also allows to specify only one variable and a value.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate(exy, [.1,.02]) == exp(0.12)\nevaluate(exy, :x, 0.0) == exp(y) # evaluate `exy` for :x -> 0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Analogously to Taylor1, another way to obtain the value of a TaylorN polynomial p at a given point x, is to call it as if it were a function: the syntax p(x) for p::TaylorN is equivalent to evaluate(p,x), and p() is equivalent to evaluate(p).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy([.1,.02]) == exp(0.12)\nexy(:x, 0.0)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Internally, evaluate for TaylorN considers separately the contributions of all HomogeneousPolynomials by order, which are finally added up after sorting them in place (which is the default) in increasing order by abs2. This is done in order to use as many significant figures as possible of all terms in the final sum, which then should yield a more accurate result. This default can be changed to a non-sorting sum thought, which may be more performant or useful for certain subtypes of Number which, for instance, do not have isless defined. See this issue for a motivating example. This can be done using the keyword sorting in evaluate, which expects a Bool, or using a that boolean as the first argument in the function-like evaluation.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy([.1,.02]) # default is `sorting=true`\nevaluate(exy, [.1,.02]; sorting=false)\nexy(false, [.1,.02])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"In the examples shown above, the first entry corresponds to the default case (sorting=true), which yields the same result as exp(0.12), and the remaining two illustrate turning off sorting the terms. Note that the results are not identical, since floating point addition is not associative, which may introduce rounding errors.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The functions taylor_expand and update! work as well for TaylorN.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"xysq = x^2 + y^2\nupdate!(xysq, [1.0, -2.0]) # expand around (1,-2)\nxysq\nupdate!(xysq, [-1.0, 2.0]) # shift-back\nxysq == x^2 + y^2","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Functions to compute the gradient, Jacobian and Hessian have also been implemented; note that these functions are not exported, so its use require the prefix TaylorSeries. Using the polynomials p = x^3 + 2x^2 y - 7 x + 2 and q = y-x^4 defined above, we may use TaylorSeries.gradient (or ∇); the results are of type Array{TaylorN{T},1}. To compute the Jacobian and Hessian of a vector field evaluated at a point, we use respectively TaylorSeries.jacobian and TaylorSeries.hessian:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"∇(p)\nTaylorSeries.gradient( q )\nr = p-q-2*p*q\nTaylorSeries.hessian(ans)\nTaylorSeries.jacobian([p,q], [2,1])\nTaylorSeries.hessian(r, [1.0,1.0])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Other specific applications are described in the Examples.","category":"page"},{"location":"userguide/#Mixtures","page":"User guide","title":"Mixtures","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As mentioned above, Taylor1{T}, HomogeneousPolynomial{T} and TaylorN{T} are parameterized structures such that T<:AbstractSeries, the latter is a subtype of Number. Then, we may actually define Taylor expansions in N+1 variables, where one of the variables (the Taylor1 variable) is somewhat special.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", order=3)\nt1N = Taylor1([zero(x), one(x)], 5)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The last line defines a Taylor1{TaylorN{Float64}} variable, which is of order 5 in t and order 3 in x and y. Then, we can evaluate functions involving such polynomials:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"cos(2.1+x+t1N)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"This kind of expansions are of interest when studying the dependence of parameters, for instance in the context of bifurcation theory or when considering the dependence of the solution of a differential equation on the initial conditions, around a given solution. In this case, x and y represent small variations around a given value of the parameters, or around some specific initial condition. Such constructions are exploited in the package TaylorIntegration.jl.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"CurrentModule = nothing","category":"page"},{"location":"background/#Background","page":"Background","title":"Background","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"","category":"page"},{"location":"background/#Introduction","page":"Background","title":"Introduction","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"TaylorSeries.jl is an implementation of high-order automatic differentiation, as presented in the book by W. Tucker [1]. The general idea is the following.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The Taylor series expansion of an analytical function f(t) with one independent variable t around t_0 can be written as","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f(t) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + cdots + f_k (t-t_0)^k + cdots","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"where f_0=f(t_0), and the Taylor coefficients f_k = f_k(t_0) are the k-th normalized derivatives at t_0:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f_k = frac1k fracrm d^k f rm d t^k(t_0)","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Thus, computing the high-order derivatives of f(t) is equivalent to computing its Taylor expansion.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"In the case of many independent variables the same statements hold, though things become more subtle. Following Alex Haro's approach [2], the Taylor expansion is an infinite sum of homogeneous polynomials in the d independent variables x_1 x_2 dots x_d, which takes the form","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f_k (mathbfx_0) = sum_m_1+cdots+m_d = k f_m_1dotsm_d \n(x_1-x_0_1)^m_1 cdots (x_d-x_0_d)^m_d =\nsum_mathbfm=k f_mathbfm (mathbfx-mathbfx_0)^mathbfm","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Here, mathbfmin mathbbN^d is a multi-index of the k-th order homogeneous polynomial and mathbfx=(x_1x_2ldotsx_d) are the d independent variables.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"In both cases, a Taylor series expansion can be represented by a vector containing its coefficients. The difference between the cases of one or more independent variables is that the coefficients are real or complex numbers in the former case, but homogeneous polynomials in the latter case. This motivates the construction of the Taylor1 and TaylorN types.","category":"page"},{"location":"background/#Arithmetic-operations","page":"Background","title":"Arithmetic operations","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Arithmetic operations involving Taylor series can be expressed as operations on the coefficients:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"(f(x) pm g(x))_k = f_k pm g_k \n(f(x) cdot g(x))_k = sum_i=0^k f_i g_k-i \nBig( fracf(x)g(x) Big)_k = frac1g_0 Big f_k -\nsum_i=0^k-1 big(fracf(x)g(x)big)_i g_k-i Big ","category":"page"},{"location":"background/#Elementary-functions-of-polynomials","page":"Background","title":"Elementary functions of polynomials","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Consider a function y(t) that satisfies the ordinary differential equation doty = f(y), y(t_0)=y_0, where t is the independent variable. Writing y(t) and f(t) as Taylor polynomials of t, substituting these in the differential equation and equating equal powers of the independent variable leads to the recursion relation","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"y_n+1 = fracf_nn+1","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The last equation and the corresponding initial condition y(t_0)=y_0 define a recurrence relation for the Taylor coefficients of y(t) around t_0.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The following are examples of such recurrence relations for some elementary functions:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"p(t) =(f(t))^alpha qquad\n p_k = frac1k f_0sum_j=0^k-1big( alpha(k-j)-jbig)\n f_k-j p_j \ne(t) = exp(f(t)) qquad\n e_k = frac1ksum_j=0^k-1 (k-j) f_k-j e_j \nl(t) = log(f(t)) qquad\n l_k = frac1f_0big( f_k - frac1ksum_j=1^k-1 j\n f_k-j l_j big) \ns(t) = sin(f(t)) qquad\n s_k = frac1ksum_j=0^k-1 (k-j) f_k-j c_j \nc(t) = cos(f(t)) qquad\n c_k = -frac1ksum_j=0^k-1 (k-j) f_k-j s_j","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The recursion relations for s(t) = sinbig(f(t)big) and c(t) = cosbig(f(t)big) depend on each other; this reflects the fact that they are solutions of a second-order differential equation.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"All these relations hold for Taylor expansions in one and more independent variables; in the latter case, the Taylor coefficients f_k are homogeneous polynomials of degree k; see [2].","category":"page"},{"location":"background/#refs","page":"Background","title":"References","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"[1] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press (2011).","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"[2] A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, preprint.","category":"page"},{"location":"#TaylorSeries.jl","page":"Home","title":"TaylorSeries.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A Julia package for Taylor expansions in one or more independent variables.","category":"page"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"#Authors","page":"Home","title":"Authors","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM).\nDavid P. Sanders, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM).","category":"page"},{"location":"#Citing","page":"Home","title":"Citing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you find useful this package, please cite the paper:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Benet, L., & Sanders, D. P. (2019). TaylorSeries.jl: Taylor expansions in one and several variables in Julia. Journal of Open Source Software, 4(36), 1–4. https://doi.org/10.5281/zenodo.2601941","category":"page"},{"location":"#License","page":"Home","title":"License","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorSeries is licensed under the MIT \"Expat\" license; see LICENSE for the full license text.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorSeries.jl is a registered package, and is simply installed by running","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add(\"TaylorSeries\")","category":"page"},{"location":"#Related-packages","page":"Home","title":"Related packages","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Polynomials.jl: Polynomial manipulations\nPowerSeries.jl: Truncated power series for Julia\nMultivariatePolynomials.jl: Multivariate polynomials interface\nAbstractAlgebra.jl: Generic abstract algebra functionality in pure Julia\nForwardDiff.jl: Forward Mode Automatic Differentiation for Julia\nReverseDiff.jl: Reverse Mode Automatic Differentiation for Julia\nHyperDualNumbers.jl: Julia implementation of HyperDualNumbers","category":"page"},{"location":"#Acknowledgments","page":"Home","title":"Acknowledgments","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This project began (using Python) during a Masters' course in the postgraduate programs in Physics and in Mathematics at UNAM, during the second half of 2013. We thank the participants of the course for putting up with the half-baked material and contributing energy and ideas.","category":"page"},{"location":"","page":"Home","title":"Home","text":"We acknowledge financial support from DGAPA-UNAM PAPIME grants PE-105911 and PE-107114, and PAPIIT grants IG-101113 and IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).","category":"page"}] +[{"location":"api/#Library","page":"API","title":"Library","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"api/#Module","page":"API","title":"Module","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"TaylorSeries","category":"page"},{"location":"api/#TaylorSeries.TaylorSeries","page":"API","title":"TaylorSeries.TaylorSeries","text":"TaylorSeries\n\nA Julia package for Taylor expansions in one or more independent variables.\n\nThe basic constructors are Taylor1 and TaylorN; see also HomogeneousPolynomial.\n\n\n\n\n\n","category":"module"},{"location":"api/#Types","page":"API","title":"Types","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Taylor1\nHomogeneousPolynomial\nTaylorN\nAbstractSeries","category":"page"},{"location":"api/#TaylorSeries.Taylor1","page":"API","title":"TaylorSeries.Taylor1","text":"Taylor1{T<:Number} <: AbstractSeries{T}\n\nDataType for polynomial expansions in one independent variable.\n\nFields:\n\ncoeffs :: Array{T,1} Expansion coefficients; the i-th component is the coefficient of degree i-1 of the expansion.\norder :: Int Maximum order (degree) of the polynomial.\n\nNote that Taylor1 variables are callable. For more information, see evaluate.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.HomogeneousPolynomial","page":"API","title":"TaylorSeries.HomogeneousPolynomial","text":"HomogeneousPolynomial{T<:Number} <: AbstractSeries{T}\n\nDataType for homogeneous polynomials in many (>1) independent variables.\n\nFields:\n\ncoeffs :: Array{T,1} Expansion coefficients of the homogeneous\n\npolynomial; the i-th component is related to a monomial, where the degrees of the independent variables are specified by coeff_table[order+1][i].\n\norder :: Int order (degree) of the homogeneous polynomial.\n\nNote that HomogeneousPolynomial variables are callable. For more information, see evaluate.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.TaylorN","page":"API","title":"TaylorSeries.TaylorN","text":"TaylorN{T<:Number} <: AbstractSeries{T}\n\nDataType for polynomial expansions in many (>1) independent variables.\n\nFields:\n\ncoeffs :: Array{HomogeneousPolynomial{T},1} Vector containing the\n\nHomogeneousPolynomial entries. The i-th component corresponds to the homogeneous polynomial of degree i-1.\n\norder :: Int maximum order of the polynomial expansion.\n\nNote that TaylorN variables are callable. For more information, see evaluate.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.AbstractSeries","page":"API","title":"TaylorSeries.AbstractSeries","text":"AbstractSeries{T<:Number} <: Number\n\nParameterized abstract type for Taylor1, HomogeneousPolynomial and TaylorN.\n\n\n\n\n\n","category":"type"},{"location":"api/#Functions-and-methods","page":"API","title":"Functions and methods","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Taylor1(::Type{T}, ::Int) where {T<:Number}\nHomogeneousPolynomial(::Type{T}, ::Int) where {T<:Number}\nTaylorN(::Type{T}, ::Int; ::Int=get_order()) where {T<:Number}\nset_variables\nget_variables\nshow_params_TaylorN\nshow_monomials\ngetcoeff\nevaluate\nevaluate!\ntaylor_expand\nupdate!\ndifferentiate\nderivative\nintegrate\ngradient\njacobian\njacobian!\nhessian\nhessian!\nconstant_term\nlinear_polynomial\nnonlinear_polynomial\ninverse\nabs\nnorm\nisapprox\nisless\nisfinite\ndisplayBigO\nuse_show_default\nset_taylor1_varname","category":"page"},{"location":"api/#TaylorSeries.Taylor1-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.Taylor1","text":"Taylor1([T::Type=Float64], order::Int)\n\nShortcut to define the independent variable of a Taylor1{T} polynomial of given order. The default type for T is Float64.\n\njulia> Taylor1(16)\n 1.0 t + 𝒪(t¹⁷)\n\njulia> Taylor1(Rational{Int}, 4)\n 1//1 t + 𝒪(t⁵)\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.HomogeneousPolynomial-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.HomogeneousPolynomial","text":"HomogeneousPolynomial([T::Type=Float64], nv::Int])\n\nShortcut to define the nv-th independent HomogeneousPolynomial{T}. The default type for T is Float64.\n\njulia> HomogeneousPolynomial(1)\n 1.0 x₁\n\njulia> HomogeneousPolynomial(Rational{Int}, 2)\n 1//1 x₂\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.TaylorN-Union{Tuple{T}, Tuple{Type{T}, Int64}} where T<:Number","page":"API","title":"TaylorSeries.TaylorN","text":"TaylorN([T::Type=Float64], nv::Int; [order::Int=get_order()])\n\nShortcut to define the nv-th independent TaylorN{T} variable as a polynomial. The order is defined through the keyword parameter order, whose default corresponds to get_order(). The default of type for T is Float64.\n\njulia> TaylorN(1)\n 1.0 x₁ + 𝒪(‖x‖⁷)\n\njulia> TaylorN(Rational{Int},2)\n 1//1 x₂ + 𝒪(‖x‖⁷)\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.set_variables","page":"API","title":"TaylorSeries.set_variables","text":"set_variables([T::Type], names::String; [order=get_order(), numvars=-1])\n\nReturn a TaylorN{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 order is the one defined globally. Changing the order or numvars resets the hash_tables.\n\nIf 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.\n\njulia> set_variables(Int, \"x y z\", order=4)\n3-element Array{TaylorSeries.TaylorN{Int},1}:\n 1 x + 𝒪(‖x‖⁵)\n 1 y + 𝒪(‖x‖⁵)\n 1 z + 𝒪(‖x‖⁵)\n\njulia> set_variables(\"α\", numvars=2)\n2-element Array{TaylorSeries.TaylorN{Float64},1}:\n 1.0 α₁ + 𝒪(‖x‖⁵)\n 1.0 α₂ + 𝒪(‖x‖⁵)\n\njulia> set_variables(\"x\", order=6, numvars=2)\n2-element Array{TaylorSeries.TaylorN{Float64},1}:\n 1.0 x₁ + 𝒪(‖x‖⁷)\n 1.0 x₂ + 𝒪(‖x‖⁷)\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.get_variables","page":"API","title":"TaylorSeries.get_variables","text":"get_variables(T::Type, [order::Int=get_order()])\n\nReturn a TaylorN{T} vector with each entry representing an independent variable. It takes the default _params_TaylorN_ values if set_variables hasn't been changed with the exception that order can be explicitly established by the user without changing internal values for num_vars or variable_names. Omitting T defaults to Float64.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.show_params_TaylorN","page":"API","title":"TaylorSeries.show_params_TaylorN","text":"show_params_TaylorN()\n\nDisplay the current parameters for TaylorN and HomogeneousPolynomial types.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.show_monomials","page":"API","title":"TaylorSeries.show_monomials","text":"show_monomials(ord::Int) --> nothing\n\nList the indices and corresponding of a HomogeneousPolynomial of degree ord.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.getcoeff","page":"API","title":"TaylorSeries.getcoeff","text":"getcoeff(a, n)\n\nReturn the coefficient of order n::Int of a a::Taylor1 polynomial.\n\n\n\n\n\ngetcoeff(a, v)\n\nReturn the coefficient of a::HomogeneousPolynomial, specified by v, which is a tuple (or vector) with the indices of the specific monomial.\n\n\n\n\n\ngetcoeff(a, v)\n\nReturn the coefficient of a::TaylorN, specified by v, which is a tuple (or vector) with the indices of the specific monomial.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.evaluate","page":"API","title":"TaylorSeries.evaluate","text":"evaluate(a, [dx])\n\nEvaluate a Taylor1 polynomial using Horner's rule (hand coded). If dx is omitted, its value is considered as zero. Note that the syntax a(dx) is equivalent to evaluate(a,dx), and a() is equivalent to evaluate(a).\n\n\n\n\n\nevaluate(x, δt)\n\nEvaluates each element of x::AbstractArray{Taylor1{T}}, representing the dependent variables of an ODE, at time δt. Note that the syntax x(δt) is equivalent to evaluate(x, δt), and x() is equivalent to evaluate(x).\n\n\n\n\n\nevaluate(a, x)\n\nSubstitute x::Taylor1 as independent variable in a a::Taylor1 polynomial. Note that the syntax a(x) is equivalent to evaluate(a, x).\n\n\n\n\n\nevaluate(a, [vals])\n\nEvaluate a HomogeneousPolynomial polynomial at vals. If vals is omitted, it's evaluated at zero. Note that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a).\n\n\n\n\n\nevaluate(a, [vals]; sorting::Bool=true)\n\nEvaluate the TaylorN polynomial a at vals. If vals is omitted, it's evaluated at zero. The keyword parameter sorting can be used to avoid sorting (in increasing order by abs2) the terms that are added.\n\nNote that the syntax a(vals) is equivalent to evaluate(a, vals); and a() is equivalent to evaluate(a). No extension exists that incorporates sorting.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.evaluate!","page":"API","title":"TaylorSeries.evaluate!","text":"evaluate!(x, δt, x0)\n\nEvaluates each element of x::AbstractArray{Taylor1{T}}, representing the Taylor expansion for the dependent variables of an ODE at time δt. It updates the vector x0 with the computed values.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.taylor_expand","page":"API","title":"TaylorSeries.taylor_expand","text":"taylor_expand(f, x0; order)\n\nComputes the Taylor expansion of the function f around the point x0.\n\nIf x0 is a scalar, a Taylor1 expansion will be returned. If x0 is a vector, a TaylorN expansion will be computed. If the dimension of x0 (length(x0)) is different from the variables set for TaylorN (get_numvars()), an AssertionError will be thrown.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.update!","page":"API","title":"TaylorSeries.update!","text":"update!(a, x0)\n\nTakes a <: Union{Taylo1,TaylorN} and expands it around the coordinate x0.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.differentiate","page":"API","title":"TaylorSeries.differentiate","text":"differentiate(a)\n\nReturn the Taylor1 polynomial of the differential of a::Taylor1. The order of the result is a.order-1.\n\nThe function derivative is an exact synonym of differentiate.\n\n\n\n\n\ndifferentiate(a, n)\n\nCompute recursively the Taylor1 polynomial of the n-th derivative of a::Taylor1. The order of the result is a.order-n.\n\n\n\n\n\ndifferentiate(n, a)\n\nReturn the value of the n-th differentiate of the polynomial a.\n\n\n\n\n\ndifferentiate(a, r)\n\nPartial differentiation of a::HomogeneousPolynomial series with respect to the r-th variable.\n\n\n\n\n\ndifferentiate(a, r)\n\nPartial differentiation of a::TaylorN series with respect to the r-th variable. The r-th variable may be also specified through its symbol.\n\n\n\n\n\ndifferentiate(a::TaylorN{T}, ntup::NTuple{N,Int})\n\nReturn a TaylorN with the partial derivative of a defined by ntup::NTuple{N,Int}, where the first entry is the number of derivatives with respect to the first variable, the second is the number of derivatives with respect to the second, and so on.\n\n\n\n\n\ndifferentiate(ntup::NTuple{N,Int}, a::TaylorN{T})\n\nReturns the value of the coefficient of a specified by ntup::NTuple{N,Int}, multiplied by the corresponding factorials.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.derivative","page":"API","title":"TaylorSeries.derivative","text":"derivative\n\nAn exact synonym of differentiate.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.integrate","page":"API","title":"TaylorSeries.integrate","text":"integrate(a, [x])\n\nReturn the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if omitted.\n\n\n\n\n\nintegrate(a, r)\n\nIntegrate the a::HomogeneousPolynomial with respect to the r-th variable. The returned HomogeneousPolynomial has no added constant of integration. If the order of a corresponds to get_order(), a zero HomogeneousPolynomial of 0-th order is returned.\n\n\n\n\n\nintegrate(a, r, [x0])\n\nIntegrate the a::TaylorN series with respect to the r-th variable, where x0 the integration constant and must be independent of the r-th variable; if x0 is omitted, it is taken as zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.gradient","page":"API","title":"TaylorSeries.gradient","text":" gradient(f)\n ∇(f)\n\nCompute the gradient of the polynomial f::TaylorN.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.jacobian","page":"API","title":"TaylorSeries.jacobian","text":" jacobian(vf)\n jacobian(vf, [vals])\n\nCompute the jacobian matrix of vf, a vector of TaylorN polynomials, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.jacobian!","page":"API","title":"TaylorSeries.jacobian!","text":" jacobian!(jac, vf)\n jacobian!(jac, vf, [vals])\n\nCompute the jacobian matrix of vf, a vector of TaylorN polynomials evaluated at the vector vals, and write results to jac. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.hessian","page":"API","title":"TaylorSeries.hessian","text":" hessian(f)\n hessian(f, [vals])\n\nReturn the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.hessian!","page":"API","title":"TaylorSeries.hessian!","text":" hessian!(hes, f)\n hessian!(hes, f, [vals])\n\nReturn the hessian matrix (jacobian of the gradient) of f::TaylorN, evaluated at the vector vals, and write results to hes. If vals is omitted, it is evaluated at zero.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.constant_term","page":"API","title":"TaylorSeries.constant_term","text":"constant_term(a)\n\nReturn the constant value (zero order coefficient) for Taylor1 and TaylorN. The fallback behavior is to return a itself if a::Number, or a[1] when a::Vector.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.linear_polynomial","page":"API","title":"TaylorSeries.linear_polynomial","text":"linear_polynomial(a)\n\nReturns the linear part of a as a polynomial (Taylor1 or TaylorN), without the constant term. The fallback behavior is to return a itself.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.nonlinear_polynomial","page":"API","title":"TaylorSeries.nonlinear_polynomial","text":"nonlinear_polynomial(a)\n\nReturns the nonlinear part of a. The fallback behavior is to return zero(a).\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.inverse","page":"API","title":"TaylorSeries.inverse","text":"inverse(f)\n\nReturn the Taylor expansion of f^-1(t), of order N = f.order, for f::Taylor1 polynomial if the first coefficient of f is zero. Otherwise, a DomainError is thrown.\n\nThe algorithm implements Lagrange inversion at t=0 if f(0)=0:\n\nbeginequation*\nf^-1(t) = sum_n=1^N fract^nn left\n fracrm d^n-1rm d z^n-1left(fraczf(z)right)^n\n rightvert_z=0\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.abs","page":"API","title":"Base.abs","text":"abs(a)\n\nFor a Real type returns a if constant_term(a) > 0 and -a if constant_term(a) < 0 for a <:Union{Taylor1,TaylorN}. For a Complex type, such as Taylor1{ComplexF64}, returns sqrt(real(a)^2 + imag(a)^2).\n\nNotice that typeof(abs(a)) <: AbstractSeries and that for a Complex argument a Real type is returned (e.g. typeof(abs(a::Taylor1{ComplexF64})) == Taylor1{Float64}).\n\n\n\n\n\n","category":"function"},{"location":"api/#LinearAlgebra.norm","page":"API","title":"LinearAlgebra.norm","text":"norm(x::AbstractSeries, p::Real)\n\nReturns the p-norm of an x::AbstractSeries, defined by\n\nbeginequation*\nleftVert x rightVert_p = left( sum_k x_k ^p right)^frac1p\nendequation*\n\nwhich returns a non-negative number.\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.isapprox","page":"API","title":"Base.isapprox","text":"isapprox(x::AbstractSeries, y::AbstractSeries; rtol::Real=sqrt(eps), atol::Real=0, nans::Bool=false)\n\nInexact equality comparison between polynomials: returns true if norm(x-y,1) <= atol + rtol*max(norm(x,1), norm(y,1)), where x and y are polynomials. For more details, see Base.isapprox.\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.isless","page":"API","title":"Base.isless","text":"isless(a::Taylor1{<:Real}, b::Real)\nisless(a::TaylorN{<:Real}, b::Real)\n\nCompute isless by comparing the constant_term(a) and b. If they are equal, returns a[nz] < 0, with nz the first non-zero coefficient after the constant term. This defines a total order.\n\nFor many variables, the ordering includes a lexicographical convention in order to be total. We have opted for the simplest one, where the larger variable appears before when the TaylorN variables are defined (e.g., through set_variables).\n\nRefs:\n\nM. Berz, AIP Conference Proceedings 177, 275 (1988); https://doi.org/10.1063/1.37800\nM. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450.\n\n\n\nisless(a::Taylor1{<:Real}, b::Taylor1{<:Real})\nisless(a::TaylorN{<:Real}, b::Taylor1{<:Real})\n\nReturns isless(a - b, zero(b)).\n\n\n\n\n\n","category":"function"},{"location":"api/#Base.isfinite","page":"API","title":"Base.isfinite","text":"isfinite(x::AbstractSeries) -> Bool\n\nTest whether the coefficients of the polynomial x are finite.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.displayBigO","page":"API","title":"TaylorSeries.displayBigO","text":"displayBigO(d::Bool) --> nothing\n\nSet/unset displaying of the big 𝒪 notation in the output of Taylor1 and TaylorN polynomials. The initial value is true.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.use_show_default","page":"API","title":"TaylorSeries.use_show_default","text":"use_Base_show(d::Bool) --> nothing\n\nUse Base.show_default method (default show method in Base), or a custom display. The initial value is false, so customized display is used.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.set_taylor1_varname","page":"API","title":"TaylorSeries.set_taylor1_varname","text":"set_taylor1_varname(var::String)\n\nChange the displayed variable for Taylor1 objects.\n\n\n\n\n\n","category":"function"},{"location":"api/#Internals","page":"API","title":"Internals","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"ParamsTaylor1\nParamsTaylorN\n_InternalMutFuncs\ngenerate_tables\ngenerate_index_vectors\nin_base\nmake_inverse_dict\nresize_coeffs1!\nresize_coeffsHP!\nnumtype\nmul!\nmul!(::HomogeneousPolynomial, ::HomogeneousPolynomial, ::HomogeneousPolynomial)\nmul!(::Vector{Taylor1{T}}, ::Union{Matrix{T},SparseMatrixCSC{T}},::Vector{Taylor1{T}}) where {T<:Number}\ndiv!\npow!\nsquare\nsqr!\naccsqr!\nsqrt!\nexp!\nlog!\nsincos!\ntan!\nasin!\nacos!\natan!\nsinhcosh!\ntanh!\nasinh!\nacosh!\natanh!\ndifferentiate!\n_internalmutfunc_call\n_dict_unary_ops\n_dict_binary_calls\n_dict_unary_calls\n_dict_binary_ops\n_populate_dicts!\n@isonethread","category":"page"},{"location":"api/#TaylorSeries.ParamsTaylor1","page":"API","title":"TaylorSeries.ParamsTaylor1","text":"ParamsTaylor1\n\nDataType holding the current variable name for Taylor1.\n\nField:\n\nvar_name :: String Names of the variables\n\nThese parameters can be changed using set_taylor1_varname\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.ParamsTaylorN","page":"API","title":"TaylorSeries.ParamsTaylorN","text":"ParamsTaylorN\n\nDataType holding the current parameters for TaylorN and HomogeneousPolynomial.\n\nFields:\n\norder :: Int Order (degree) of the polynomials\nnum_vars :: Int Number of variables\nvariable_names :: Vector{String} Names of the variables\nvariable_symbols :: Vector{Symbol} Symbols of the variables\n\nThese parameters can be changed using set_variables\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries._InternalMutFuncs","page":"API","title":"TaylorSeries._InternalMutFuncs","text":"_InternalMutFuncs\n\nContains parameters and expressions that allow a simple programmatic construction for calling the internal mutating functions.\n\n\n\n\n\n","category":"type"},{"location":"api/#TaylorSeries.generate_tables","page":"API","title":"TaylorSeries.generate_tables","text":"generate_tables(num_vars, order)\n\nReturn the hash tables coeff_table, index_table, size_table and pos_table. Internally, these are treated as const.\n\nHash tables\n\ncoeff_table :: Array{Array{Array{Int,1},1},1}\n\nThe i+1-th component contains a vector with the vectors of all the possible combinations of monomials of a HomogeneousPolynomial of order i.\n\nindex_table :: Array{Array{Int,1},1}\n\nThe i+1-th component contains a vector of (hashed) indices that represent the distinct monomials of a HomogeneousPolynomial of order (degree) i.\n\nsize_table :: Array{Int,1}\n\nThe i+1-th component contains the number of distinct monomials of the HomogeneousPolynomial of order i, equivalent to length(coeff_table[i]).\n\npos_table :: Array{Dict{Int,Int},1}\n\nThe i+1-th component maps the hash index to the (lexicographic) position of the corresponding monomial in coeffs_table.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.generate_index_vectors","page":"API","title":"TaylorSeries.generate_index_vectors","text":"generate_index_vectors(num_vars, degree)\n\nReturn a vector of index vectors with num_vars (number of variables) and degree.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.in_base","page":"API","title":"TaylorSeries.in_base","text":"in_base(order, v)\n\nConvert vector v of non-negative integers to base oorder, where oorder is the next odd integer of order.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.make_inverse_dict","page":"API","title":"TaylorSeries.make_inverse_dict","text":"make_inverse_dict(v)\n\nReturn a Dict with the enumeration of v: the elements of v point to the corresponding index.\n\nIt is used to construct pos_table from index_table.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.resize_coeffs1!","page":"API","title":"TaylorSeries.resize_coeffs1!","text":"resize_coeffs1!{T nothing\n\nUpdate the k-th expansion coefficient c[k] of c = a * b, where all c, a, and b are either Taylor1 or TaylorN. Note that for TaylorN the result of a * b is accumulated in c[k].\n\nThe coefficients are given by\n\nc_k = sum_j=0^k a_j b_k-j\n\n\n\n\n\n","category":"function"},{"location":"api/#LinearAlgebra.mul!-Tuple{HomogeneousPolynomial, HomogeneousPolynomial, HomogeneousPolynomial}","page":"API","title":"LinearAlgebra.mul!","text":"mul!(c, a, b) --> nothing\n\nAccumulates in c[k] the result of a*b with minimum allocation; all arguments are HomogeneousPolynomial.\n\n\n\n\n\n","category":"method"},{"location":"api/#LinearAlgebra.mul!-Union{Tuple{T}, Tuple{Array{Taylor1{T}, 1}, Union{SparseArrays.SparseMatrixCSC{T}, Matrix{T}}, Array{Taylor1{T}, 1}}} where T<:Number","page":"API","title":"LinearAlgebra.mul!","text":"mul!(Y, A, B)\n\nMultiply A*B and save the result in Y.\n\n\n\n\n\n","category":"method"},{"location":"api/#TaylorSeries.div!","page":"API","title":"TaylorSeries.div!","text":"div!(c, a, b, k::Int)\n\nCompute the k-th expansion coefficient c[k] of c = a / b, where all c, a and b are either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nc_k = frac1b_0 big(a_k - sum_j=0^k-1 c_j b_k-jbig)\n\nFor Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.pow!","page":"API","title":"TaylorSeries.pow!","text":"pow!(c, a, r::Real, k::Int)\n\nUpdate the k-th expansion coefficient c[k] of c = a^r, for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nc_k = frac1k a_0 sum_j=0^k-1 big(r(k-j) -jbig)a_k-j c_j\n\nFor Taylor1 polynomials, a similar formula is implemented which exploits k_0, the order of the first non-zero coefficient of a.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.square","page":"API","title":"TaylorSeries.square","text":"square(a::AbstractSeries) --> typeof(a)\n\nReturn a^2; see TaylorSeries.sqr!.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sqr!","page":"API","title":"TaylorSeries.sqr!","text":"sqr!(c, a, k::Int) --> nothing\n\nUpdate the k-th expansion coefficient c[k] of c = a^2, for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\nc_k = 2 sum_j=0^(k-1)2 a_k-j a_j\n text if k is odd \nc_k = 2 sum_j=0^(k-2)2 a_k-j a_j + (a_k2)^2\n text if k is even\nendaligned\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.accsqr!","page":"API","title":"TaylorSeries.accsqr!","text":"accsqr!(c, a)\n\nReturns c += a*a with no allocation; all parameters are HomogeneousPolynomial.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sqrt!","page":"API","title":"TaylorSeries.sqrt!","text":"sqrt!(c, a, k::Int, k0::Int=0)\n\nCompute the k-th expansion coefficient c[k] of c = sqrt(a) for bothc and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\nc_k = frac12 c_0 big( a_k - 2 sum_j=1^(k-1)2 c_k-jc_jbig)\n text if k is odd \nc_k = frac12 c_0 big( a_k - 2 sum_j=1^(k-2)2 c_k-jc_j\n - (c_k2)^2big) text if k is even\nendaligned\n\nFor Taylor1 polynomials, k0 is the order of the first non-zero coefficient, which must be even.\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.exp!","page":"API","title":"TaylorSeries.exp!","text":"exp!(c, a, k) --> nothing\n\nUpdate the k-th expansion coefficient c[k+1] of c = exp(a) for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = frac1k sum_j=0^k-1 (k-j) a_k-j c_j\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.log!","page":"API","title":"TaylorSeries.log!","text":"log!(c, a, k) --> nothing\n\nUpdate the k-th expansion coefficient c[k+1] of c = log(a) for both c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = frac1a_0 big(a_k - frac1k sum_j=0^k-1 j a_k-j c_j big)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sincos!","page":"API","title":"TaylorSeries.sincos!","text":"sincos!(s, c, a, k) --> nothing\n\nUpdate the k-th expansion coefficients s[k+1] and c[k+1] of s = sin(a) and c = cos(a) simultaneously, for s, c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\ns_k = frac1ksum_j=0^k-1 (k-j) a_k-j c_j \nc_k = -frac1ksum_j=0^k-1 (k-j) a_k-j s_j\nendaligned\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.tan!","page":"API","title":"TaylorSeries.tan!","text":"tan!(c, a, p, k::Int) --> nothing\n\nUpdate the k-th expansion coefficients c[k+1] of c = tan(a), for c and a either Taylor1 or TaylorN; p = c^2 and is passed as an argument for efficiency.\n\nThe coefficients are given by\n\nbeginequation*\nc_k = a_k + frac1k sum_j=0^k-1 (k-j) a_k-j p_j\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.asin!","page":"API","title":"TaylorSeries.asin!","text":"asin!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = asin(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 sqrtr_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.acos!","page":"API","title":"TaylorSeries.acos!","text":"acos!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = acos(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = - frac1 r_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.atan!","page":"API","title":"TaylorSeries.atan!","text":"atan!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = atan(a), for c and a either Taylor1 or TaylorN; r = 1+a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1r_0big(a_k - frac1k sum_j=1^k-1 j r_k-j c_jbig)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.sinhcosh!","page":"API","title":"TaylorSeries.sinhcosh!","text":"sinhcosh!(s, c, a, k)\n\nUpdate the k-th expansion coefficients s[k+1] and c[k+1] of s = sinh(a) and c = cosh(a) simultaneously, for s, c and a either Taylor1 or TaylorN.\n\nThe coefficients are given by\n\nbeginaligned\ns_k = frac1k sum_j=0^k-1 (k-j) a_k-j c_j \nc_k = frac1k sum_j=0^k-1 (k-j) a_k-j s_j\nendaligned\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.tanh!","page":"API","title":"TaylorSeries.tanh!","text":"tanh!(c, a, p, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = tanh(a), for c and a either Taylor1 or TaylorN; p = a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = a_k - frac1k sum_j=0^k-1 (k-j) a_k-j p_j\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.asinh!","page":"API","title":"TaylorSeries.asinh!","text":"asinh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = asinh(a), for c and a either Taylor1 or TaylorN; r = sqrt(1-c^2) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 sqrtr_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.acosh!","page":"API","title":"TaylorSeries.acosh!","text":"acosh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = acosh(a), for c and a either Taylor1 or TaylorN; r = sqrt(c^2-1) and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1 r_0 \n big( a_k - frac1k sum_j=1^k-1 j r_k-j c_j big)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.atanh!","page":"API","title":"TaylorSeries.atanh!","text":"atanh!(c, a, r, k)\n\nUpdate the k-th expansion coefficients c[k+1] of c = atanh(a), for c and a either Taylor1 or TaylorN; r = 1-a^2 and is passed as an argument for efficiency.\n\nbeginequation*\nc_k = frac1r_0big(a_k + frac1k sum_j=1^k-1 j r_k-j c_jbig)\nendequation*\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.differentiate!","page":"API","title":"TaylorSeries.differentiate!","text":"differentiate!(res, a) --> nothing\n\nIn-place version of differentiate. Compute the Taylor1 polynomial of the differential of a::Taylor1 and return it as res (order of res remains unchanged).\n\n\n\n\n\ndifferentiate!(p, a, k) --> nothing\n\nUpdate in-place the k-th expansion coefficient p[k] of p = differentiate(a) for both p and a Taylor1.\n\nThe coefficients are given by\n\np_k = (k+1) a_k+1\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries._internalmutfunc_call","page":"API","title":"TaylorSeries._internalmutfunc_call","text":"_internalmutfunc_call( fn :: _InternalMutFuncs )\n\nCreates the appropriate call to the internal mutating function defined by the _InternalMutFuncs object. This is used to construct _dict_unary_calls and _dict_binary_calls. The call contains the prefix TaylorSeries..\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries._dict_unary_ops","page":"API","title":"TaylorSeries._dict_unary_ops","text":"_dict_binary_ops\n\nDict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to unary operations.\n\nThe keys correspond to the function symbols.\n\nThe arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1, for the required argument, possibly :_aux when there is an auxiliary expression needed, and :_k for the computed order of :_res. When an auxiliary expression is required, an Expr defining its calling pattern is added as the last entry of the vector.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_binary_calls","page":"API","title":"TaylorSeries._dict_binary_calls","text":"_dict_binary_calls::Dict{Symbol, NTuple{2,Expr}}\n\nDictionary with the expressions that define the internal binary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.\n\nEvaluating the entries generates symbols that represent the actual calls to the internal mutating functions.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_unary_calls","page":"API","title":"TaylorSeries._dict_unary_calls","text":"_dict_unary_calls::Dict{Symbol, NTuple{2,Expr}}\n\nDictionary with the expressions that define the internal unary functions and the auxiliary functions, whenever they exist. The keys correspond to those functions, passed as symbols, with the defined internal mutating functions.\n\nEvaluating the entries generates expressions that represent the actual calls to the internal mutating functions.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._dict_binary_ops","page":"API","title":"TaylorSeries._dict_binary_ops","text":"_dict_binary_ops\n\nDict{Symbol, Array{Any,1}} with the information to construct the _InternalMutFuncs related to binary operations.\n\nThe keys correspond to the function symbols.\n\nThe arguments of the array are the function name (e.g. add!), a tuple with the function arguments, and an Expr with the calling pattern. The convention for the arguments of the functions and the calling pattern is to use :_res for the (mutated) result, :_arg1 and _arg2 for the required arguments, and :_k for the computed order of :_res.\n\n\n\n\n\n","category":"constant"},{"location":"api/#TaylorSeries._populate_dicts!","page":"API","title":"TaylorSeries._populate_dicts!","text":"_populate_dicts!()\n\nFunction that populates the internal dictionaries _dict_unary_calls and _dict_binary_calls\n\n\n\n\n\n","category":"function"},{"location":"api/#TaylorSeries.@isonethread","page":"API","title":"TaylorSeries.@isonethread","text":"@isonethread (expr)\n\nInternal macro used to check the number of threads in use, to prevent a data race that modifies coeff_table when using differentiate or integrate; see https://github.com/JuliaDiff/TaylorSeries.jl/issues/318.\n\nThis macro is inspired by the macro @threaded; see https://github.com/trixi-framework/Trixi.jl/blob/main/src/auxiliary/auxiliary.jl; and https://github.com/trixi-framework/Trixi.jl/pull/426/files.\n\n\n\n\n\n","category":"macro"},{"location":"api/#Index","page":"API","title":"Index","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"Pages = [\"api.md\"]\nModule = [\"TaylorSeries\"]\nOrder = [:type, :function]","category":"page"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"examples/#Expanding-exp(x)-with-taylor_expand()","page":"Examples","title":"Expanding exp(x) with taylor_expand()","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The taylor_expand function takes the function to expand as it's first argument, and the point to expand about as the second argument. A keyword argument order determines which order to expand to:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n\ntaylor_expand(exp, 0, order=2)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"And voìla! It really is that simple to calculate a simple taylor polynomial. The next example is slightly more complicated.","category":"page"},{"location":"examples/#Expanding-exp(x)-with-a-symbolic-object","page":"Examples","title":"Expanding exp(x) with a symbolic object","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"An alternative way to compute the single-variable taylor expansion for a function is by defining a variable of type Taylor1, and using it in the function you wish to expand. The argument given to the Taylor1 constructor is the order to expand to:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n\nx = Taylor1(2)\n\nexp(x)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Let's also get rid of the printed error for the next few examples, and set the printed independent variable to x:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"displayBigO(false)\n\nset_taylor1_varname(\"x\")\n\nexp(x)","category":"page"},{"location":"examples/#Changing-point-to-expand-about","page":"Examples","title":"Changing point to expand about","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"A variable constructed with Taylor1() automatically expands about the point x=0. But what if you want to use the symbolic object to expand about a point different from zero? Because expanding exp(x) about x=1 is exactly the same as expanding exp(x+1) about x=0, simply replace the x in your expression with x+1 to expand about x=1:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"p = exp(x+1)\n\np(0.01)\n\nexp(1.01)","category":"page"},{"location":"examples/#More-examples","page":"Examples","title":"More examples","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"You can even use custum functions","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"f(a) = 1/(a+1)\n\nf(x)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Functions can be nested","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sin(f(x))","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"and complicated further in a modular way","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sin(exp(x+2))/(x+2)+cos(x+2)+f(x+2)","category":"page"},{"location":"examples/#Four-square-identity","page":"Examples","title":"Four-square identity","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The first example shows that the four-square identity holds:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"(a_1^2 + a_2^2 + a_3^2 + a_4^2)cdot(b_1^2 + b_2^2 + b_3^2 + b_4^2) = \n qquad (a_1 b_1 - a_2 b_2 - a_3 b_3 -a_4 b_4)^2 +\n (a_1 b_2 - a_2 b_1 - a_3 b_4 -a_4 b_3)^2 + \n qquad (a_1 b_3 - a_2 b_4 - a_3 b_1 -a_4 b_2)^2 +\n (a_1 b_4 - a_2 b_3 - a_3 b_2 -a_4 b_1)^2","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"which was originally proved by Euler. The code can also be found in this test of the package.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"First, we reset the maximum degree of the polynomial to 4, since the RHS of the equation has a priori terms of fourth order, and define the 8 independent variables.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\n# Define the variables α₁, ..., α₄, β₁, ..., β₄\nmake_variable(name, index::Int) = string(name, TaylorSeries.subscriptify(index))\nvariable_names = [make_variable(\"α\", i) for i in 1:4]\nappend!(variable_names, [make_variable(\"β\", i) for i in 1:4])\n# Create the TaylorN variables (order=4, numvars=8)\na1, a2, a3, a4, b1, b2, b3, b4 = set_variables(variable_names, order=4)\na1 # variable a1","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Now we compute each term appearing in Eq. (\\ref{eq:Euler})","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"# left-hand side\nlhs1 = a1^2 + a2^2 + a3^2 + a4^2 ;\nlhs2 = b1^2 + b2^2 + b3^2 + b4^2 ;\nlhs = lhs1 * lhs2;\n# right-hand side\nrhs1 = (a1*b1 - a2*b2 - a3*b3 - a4*b4)^2 ;\nrhs2 = (a1*b2 + a2*b1 + a3*b4 - a4*b3)^2 ;\nrhs3 = (a1*b3 - a2*b4 + a3*b1 + a4*b2)^2 ;\nrhs4 = (a1*b4 + a2*b3 - a3*b2 + a4*b1)^2 ;\nrhs = rhs1 + rhs2 + rhs3 + rhs4;","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"We now compare the two sides of the identity,","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"lhs == rhs","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"The identity is satisfied. square","category":"page"},{"location":"examples/#Fateman-test","page":"Examples","title":"Fateman test","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Richard J. Fateman, from Berkeley, proposed as a stringent test of polynomial multiplication the evaluation of scdot(s+1), where s = (1+x+y+z+w)^20. This is implemented in the function fateman1 below. We shall also consider the form s^2+s in fateman2, which involves fewer operations (and makes a fairer comparison to what Mathematica does).","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using TaylorSeries\nconst order = 20\nconst x, y, z, w = set_variables(Int128, \"x\", numvars=4, order=2order)\nfunction fateman1(degree::Int)\n T = Int128\n s = one(T) + x + y + z + w\n s = s^degree\n s * ( s + one(T) )\nend","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"(In the following lines, which are run when the documentation is built, by some reason the timing appears before the command executed.)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"@time fateman1(0);\n# hide\n@time f1 = fateman1(20);","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Another implementation of the same, but exploiting optimizations related to ^2 yields:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"function fateman2(degree::Int)\n T = Int128\n s = one(T) + x + y + z + w\n s = s^degree\n s^2 + s\nend\nfateman2(0);\n@time f2 = fateman2(20); # the timing appears above","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"We note that the above functions use expansions in Int128. This is actually required, since some coefficients are larger than typemax(Int):","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"getcoeff(f2, (1,6,7,20)) # coefficient of x y^6 z^7 w^{20}\nans > typemax(Int)\nlength(f2)\nsum(TaylorSeries.size_table)\nset_variables(\"x\", numvars=2, order=6) # hide","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"These examples show that fateman2 is nearly twice as fast as fateman1, and that the series has 135751 monomials in 4 variables.","category":"page"},{"location":"examples/#Benchmarks","page":"Examples","title":"Benchmarks","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"The functions described above have been compared against Mathematica v11.1. The relevant files used for benchmarking can be found here. Running on a MacPro with Intel-Xeon processors 2.7GHz, we obtain that Mathematica requires on average (5 runs) 3.075957 seconds for the computation, while for fateman1 and fateman2 above we obtain 2.15408 and 1.08337, respectively.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then, with the current version of TaylorSeries.jl and using Julia v0.7.0, our implementation of fateman1 is about 30%-40% faster. (The original test by Fateman corresponds to fateman1 above, which avoids some optimizations related to squaring; the implementation in Mathematica is done such that this optimization does not occur.)","category":"page"},{"location":"userguide/#User-guide","page":"User guide","title":"User guide","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"CurrentModule = TaylorSeries","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"For some simple examples, head over to the examples section. For a detailed guide, keep reading.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"TaylorSeries.jl is a basic polynomial algebraic manipulator in one or more variables; these two cases are treated separately. Three new types are defined, Taylor1, HomogeneousPolynomial and TaylorN, which correspond to expansions in one independent variable, homogeneous polynomials of various variables, and the polynomial series in many independent variables, respectively. These types are subtypes of AbstractSeries, which in turn is a subtype of Number, and are defined parametrically.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The package is loaded as usual:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"using TaylorSeries","category":"page"},{"location":"userguide/#One-independent-variable","page":"User guide","title":"One independent variable","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor expansions in one variable are represented by the Taylor1 type, which consists of a vector of coefficients (fieldname coeffs) and the maximum order considered for the expansion (fieldname order). The coefficients are arranged in ascending order with respect to the degree of the monomial, so that coeffs[1] is the constant term, coeffs[2] gives the first order term (t^1), etc. Yet, it is possible to have the natural ordering with respect to the degree; see below. This is a dense representation of the polynomial. The order of the polynomial can be omitted in the constructor, which is then fixed by the length of the vector of coefficients. If the length of the vector does not correspond with the order, order is used, which effectively truncates polynomial to degree order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor1([1, 2, 3],4) # Polynomial of order 4 with coefficients 1, 2, 3\nTaylor1([0.0, 1im]) # Also works with complex numbers\nTaylor1(ones(8), 2) # Polynomial truncated to order 2\nshift_taylor(a) = a + Taylor1(typeof(a),5) ## a + taylor-polynomial of order 5\nt = shift_taylor(0.0) # Independent variable `t`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"warning: Warning\nThe information about the maximum order considered is displayed using a big-𝒪 notation. The convention followed when different orders are combined, and when certain functions are used in a way that they reduce the order (like differentiate), is to be consistent with the mathematics and the big-𝒪 notation, i.e., to propagate the lowest order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"In some cases, it is desirable to not display the big-𝒪 notation. The function displayBigO allows to control whether it is displayed or not.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"displayBigO(false) # turn-off displaying big O notation\nt\ndisplayBigO(true) # turn it on\nt","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similarly, it is possible to control some aspects of the format of the displayed series through the function use_show_default; use_show_default(true) uses the Base.show_default, while use_show_default(false) uses the custom display form (default).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"use_show_default(true) # use Base.show method\nt\nuse_show_default(false) # use custom `show`\nt","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The definition of shift_taylor(a) uses the method Taylor1([::Type{Float64}], order::Int), which is a shortcut to define the independent variable of a Taylor expansion, of given type and order (the default is Float64). Defining the independent variable in advance is one of the easiest ways to use the package.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with the Taylor1 type, including promotions that involve Numbers. The operations return a valid Taylor expansion, consistent with the order of the series. This is apparent in the penultimate example below, where the fist non-zero coefficient is beyond the order of the expansion, and hence the result is zero.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t*(3t+2.5)\n1/(1-t)\nt*(t^2-4)/(t+2)\ntI = im*t\n(1-t)^3.2\n(1+t)^t\nTaylor1(3) + Taylor1(5) == 2Taylor1(3) # big-𝒪 convention applies\nt^6 # t is of order 5\nt^2 / t # The result is of order 4, instead of 5","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that the last example returns a Taylor1 series of order 4, instead of order 5; this is be consistent with the number of known coefficients of the returned series, since the result corresponds to factorize t in the numerator to cancel the same factor in the denominator.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Taylor1 is also equiped with a total order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. This is illustrated by:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"1 > t > 2*t^2 > 100*t^3 >= 0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"If no valid Taylor expansion can be computed an error is thrown, for instance, when a derivative is not defined at the expansion point, or it simply diverges.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"1/t\nt^3.2\nabs(t)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Several elementary functions have been implemented; their coefficients are computed recursively. At the moment of this writing, these functions are exp, log, sqrt, the trigonometric functions sin, cos and tan, their inverses, as well as the hyperbolic functions sinh, cosh and tanh and their inverses; more functions will be added in the future. Note that this way of obtaining the Taylor coefficients is not a lazy way, in particular for many independent variables. Yet, the implementation is efficient enough, especially for the integration of ordinary differential equations, which is among the applications we have in mind (see TaylorIntegration.jl).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exp(t)\nlog(1-t)\nsqrt(1 + t)\nimag(exp(tI)')\nreal(exp(Taylor1([0.0,1im],17))) - cos(Taylor1([0.0,1.0],17)) == 0.0\nconvert(Taylor1{Rational{Int}}, exp(t))","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Again, errors are thrown whenever it is necessary.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"sqrt(t)\nlog(t)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To obtain a specific coefficient, getcoeff can be used. Another alternative is to request the specific degree using the vector notation, where the index corresponds to the degree of the term.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"expon = exp(t)\ngetcoeff(expon, 0) == expon[0]\nrationalize(expon[3])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that certain arithmetic operations, or the application of some functions, may change the order of the result, as mentioned above.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t # order 5 independent variable\nt^2/t # returns an order 4 series expansion\nsqrt(t^2) # returns an order 2 series expansion\n(t^4)^(1/4) # order 1 series","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Differentiating and integrating is straightforward for polynomial expansions in one variable, using differentiate and integrate. (The function derivative is a synonym of differentiate.) These functions return the corresponding Taylor1 expansions. The order of the derivative of a Taylor1 is reduced by 1. For the integral, an integration constant may be set by the user (the default is zero); the order of the integrated polynomial for the integral is kept unchanged. The value of the n-th (n ge 0) derivative is obtained using differentiate(n,a), where a is a Taylor series; likewise, the Taylor1 polynomial of the n-th derivative is obtained as differentiate(a,n); the resulting polynomial is of order get_order(a)-n.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate(exp(t)) # exp(t) is of order 5; the derivative is of order 4\nintegrate(exp(t)) # the resulting TaylorSeries is of order 5\nintegrate( exp(t), 1.0)\nintegrate( differentiate( exp(-t)), 1.0 ) == exp(-t)\ndifferentiate(1, exp(shift_taylor(1.0))) == exp(1.0)\ndifferentiate(5, exp(shift_taylor(1.0))) == exp(1.0) # 5-th differentiate of `exp(1+t)`\nderivative(exp(1+t), 3) # Taylor1 polynomial of the 3-rd derivative of `exp(1+t)`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To evaluate a Taylor series at a given point, Horner's rule is used via the function evaluate(a, dt). Here, dt is the increment from the point t_0 around which the Taylor expansion of a is calculated, i.e., the series is evaluated at t = t_0 + dt. Omitting dt corresponds to dt = 0; see evaluate.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate(exp(shift_taylor(1.0))) - ℯ # exp(t) around t0=1 (order 5), evaluated there (dt=0)\nevaluate(exp(t), 1) - ℯ # exp(t) around t0=0 (order 5), evaluated at t=1\nevaluate(exp( Taylor1(17) ), 1) - ℯ # exp(t) around t0=0, order 17\ntBig = Taylor1(BigFloat, 50) # Independent variable with BigFloats, order 50\neBig = evaluate( exp(tBig), one(BigFloat) )\nℯ - eBig","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Another way to evaluate the value of a Taylor1 polynomial p at a given value x, is to call p as if it was a function, i.e., p(x):","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"t = Taylor1(15)\np = sin(t)\nevaluate(p, pi/2) # value of p at pi/2 using `evaluate`\np(pi/2) # value of p at pi/2 by evaluating p as a function\np(pi/2) == evaluate(p, pi/2)\np(0.0)\np() == p(0.0) # p() is a shortcut to obtain the 0-th order coefficient of `p`","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Note that the syntax p(x) is equivalent to evaluate(p, x), whereas p() is equivalent to evaluate(p).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Useful shortcuts are taylor_expand and update!. The former returns the expansion of a function around a given value t0, mimicking the use of shift_taylor above. In turn, update! provides an in-place update of a given Taylor polynomial, that is, it shifts it further by the provided amount.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"p = taylor_expand( x -> sin(x), pi/2, order=16) # 16-th order expansion of sin(t) around pi/2\nupdate!(p, 0.025) # updates the expansion given by p, by shifting it further by 0.025\np","category":"page"},{"location":"userguide/#Many-variables","page":"User guide","title":"Many variables","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"A polynomial in N1 variables can be represented in (at least) two ways: As a vector whose coefficients are homogeneous polynomials of fixed degree, or as a vector whose coefficients are polynomials in N-1 variables. The current implementation of TaylorSeries.jl corresponds to the first option, though some infrastructure has been built that permits to develop the second one. An elegant (lazy) implementation of the second representation was discussed here.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The structure TaylorN is constructed as a vector of parameterized homogeneous polynomials defined by the type HomogeneousPolynomial, which in turn is an ordered vector of coefficients of given order (degree). This implementation imposes the user to specify the (maximum) order considered and the number of independent variables at the beginning, which can be conveniently done using set_variables. A vector of the resulting Taylor variables is returned:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\")\ntypeof(x)\nx.order\nx.coeffs","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As shown, the resulting objects are of TaylorN{Float64} type. There is an optional order keyword argument in set_variables, used to specify the maximum order of the TaylorN polynomials. Note that one can specify the variables using a vector of symbols.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables([:x, :y], order=10)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similarly, subindexed variables are also available by specifying a single variable name and the optional keyword argument numvars:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables(\"α\", numvars=3)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Alternatively to set_variables, get_variables can be used if one does not want to change internal dictionaries. get_variables returns a vector of TaylorN independent variables of a desired order (lesser than get_order so the internals doesn't have to change) with the length and variable names defined by set_variables initially.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"get_variables(2) # vector of independent variables of order 2","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"warning: Warning\nAn OverflowError is thrown when the construction of the internal tables is not fully consistent, avoiding silent errors; see issue #85.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The function show_params_TaylorN displays the current values of the parameters, in an info block.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"show_params_TaylorN()","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Internally, changing the parameters (maximum order and number of variables) redefines the hash-tables that translate the index of the coefficients of a HomogeneousPolynomial of given order into the corresponding multi-variable monomials, or the other way around. Fixing these values from the start is imperative; the initial (default) values are order = 6 and num_vars=2.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The easiest way to construct a TaylorN object is by defining the independent variables. This can be done using set_variables as above, or through the method TaylorN{T<:Number}(::Type{T}, nv::Int) for the nv independent TaylorN{T} variable; the order can be also specified using the optional keyword argument order.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", numvars=2, order=6);\nx\nTaylorN(1, order=4) # variable 1 of order 4\nTaylorN(Int, 2) # variable 2, type Int, order=get_order()=6","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Other ways of constructing TaylorN polynomials involve using HomogeneousPolynomial objects directly, which is uncomfortable.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"set_variables(:x, numvars=2); # symbols can be used\nHomogeneousPolynomial([1,-1])\nTaylorN([HomogeneousPolynomial([1,0]), HomogeneousPolynomial([1,2,3])],4)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The Taylor expansions are implemented around 0 for all variables; if the expansion is needed around a different value, the trick is a simple translation of the corresponding independent variable, i.e. x to x+a.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As before, the usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with TaylorN objects, including the appropriate promotions to deal with the usual numeric types. Note that some of the arithmetic operations have been extended for HomogeneousPolynomial, whenever the result is a HomogeneousPolynomial; division, for instance, is not extended. The same convention used for Taylor1 objects is used when combining TaylorN polynomials of different order. Both HomogeneousPolynomial and TaylorN are equiped with a total lexicographical order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, \"Automatic Differentiation as Nonarchimedean Analysis\", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. Essentially, the lexicographic order works as follows: smaller order monomials are larger than higher order monomials; when the order is the same, larger monomials appear before in the hash-tables; the function show_monomials.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", order=10);\n\nshow_monomials(2)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Then, the following then holds:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"0 < 1e8 * y^2 < x*y < x^2 < y < x/1e8 < 1.0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The elementary functions have also been implemented, again by computing their coefficients recursively:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy = exp(x+y)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The function getcoeff gives the normalized coefficient of the polynomial that corresponds to the monomial specified by the tuple or vector v containing the powers. For instance, for the polynomial exy above, the coefficient of the monomial x^3 y^5 is obtained using getcoeff(exy, (3,5)) or getcoeff(exy, [3,5]).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"getcoeff(exy, (3,5))\nrationalize(ans)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Similar to Taylor1, vector notation can be used to request specific coefficients of HomogeneousPolynomial or TaylorN objects. For TaylorN objects, the index refers to the degree of the HomogeneousPolynomial. In the case of HomogeneousPolynomial the index refers to the position of the hash table. The function show_monomials can be used to obtain the coefficient a specific monomial, given the degree of the HomogeneousPolynomial.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy[8] # get the 8th order term\nshow_monomials(8)\nexy[8][6] # get the 6th coeff of the 8th order term","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Partial differentiation is also implemented for TaylorN objects, through the function differentiate, specifying the number of the variable, or its symbol, as the second argument.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"p = x^3 + 2x^2 * y - 7x + 2\nq = y - x^4\ndifferentiate( p, 1 ) # partial derivative with respect to 1st variable\ndifferentiate( q, :y ) # partial derivative with respect to :y","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"If we ask for the partial derivative with respect to a non-defined variable, an error is thrown.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate( q, 3 ) # error, since we are dealing with 2 variables","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"To obtain more specific partial derivatives we have two specialized methods that involve a tuple, which represents the number of derivatives with respect to each variable (so the tuple's length has to be the same as the actual number of variables). These methods either return the TaylorN object in question, or the coefficient corresponding to the specified tuple, normalized by the factorials defined by the tuple. The latter is in essence the 0-th order coefficient of the former.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"differentiate(p, (2,1)) # two derivatives on :x and one on :y\ndifferentiate((2,1), p) # 0-th order coefficient of the previous expression\ndifferentiate(p, (1,1)) # one derivative on :x and one on :y\ndifferentiate((1,1), p) # 0-th order coefficient of the previous expression","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Integration with respect to the r-th variable for HomogeneousPolynomials and TaylorN objects is obtained using integrate. Note that integrate for TaylorN objects allows to specify a constant of integration, which must be independent from the integrated variable. Again, the integration variable may be specified by its symbol.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"integrate( differentiate( p, 1 ), 1) # integrate with respect to the first variable\nintegrate( differentiate( p, 1 ), :x, 2) # integration with respect to :x, constant of integration is 2\nintegrate( differentiate( q, 2 ), :y, -x^4) == q\nintegrate( differentiate( q, 2 ), 2, y)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate can also be used for TaylorN objects, using it on vectors of numbers (Real or Complex); the length of the vector must coincide with the number of independent variables. evaluate also allows to specify only one variable and a value.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"evaluate(exy, [.1,.02]) == exp(0.12)\nevaluate(exy, :x, 0.0) == exp(y) # evaluate `exy` for :x -> 0","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Analogously to Taylor1, another way to obtain the value of a TaylorN polynomial p at a given point x, is to call it as if it were a function: the syntax p(x) for p::TaylorN is equivalent to evaluate(p,x), and p() is equivalent to evaluate(p).","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy([.1,.02]) == exp(0.12)\nexy(:x, 0.0)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Internally, evaluate for TaylorN considers separately the contributions of all HomogeneousPolynomials by order, which are finally added up after sorting them in place (which is the default) in increasing order by abs2. This is done in order to use as many significant figures as possible of all terms in the final sum, which then should yield a more accurate result. This default can be changed to a non-sorting sum thought, which may be more performant or useful for certain subtypes of Number which, for instance, do not have isless defined. See this issue for a motivating example. This can be done using the keyword sorting in evaluate, which expects a Bool, or using a that boolean as the first argument in the function-like evaluation.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"exy([.1,.02]) # default is `sorting=true`\nevaluate(exy, [.1,.02]; sorting=false)\nexy(false, [.1,.02])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"In the examples shown above, the first entry corresponds to the default case (sorting=true), which yields the same result as exp(0.12), and the remaining two illustrate turning off sorting the terms. Note that the results are not identical, since floating point addition is not associative, which may introduce rounding errors.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The functions taylor_expand and update! work as well for TaylorN.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"xysq = x^2 + y^2\nupdate!(xysq, [1.0, -2.0]) # expand around (1,-2)\nxysq\nupdate!(xysq, [-1.0, 2.0]) # shift-back\nxysq == x^2 + y^2","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Functions to compute the gradient, Jacobian and Hessian have also been implemented; note that these functions are not exported, so its use require the prefix TaylorSeries. Using the polynomials p = x^3 + 2x^2 y - 7 x + 2 and q = y-x^4 defined above, we may use TaylorSeries.gradient (or ∇); the results are of type Array{TaylorN{T},1}. To compute the Jacobian and Hessian of a vector field evaluated at a point, we use respectively TaylorSeries.jacobian and TaylorSeries.hessian:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"∇(p)\nTaylorSeries.gradient( q )\nr = p-q-2*p*q\nTaylorSeries.hessian(ans)\nTaylorSeries.jacobian([p,q], [2,1])\nTaylorSeries.hessian(r, [1.0,1.0])","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"Other specific applications are described in the Examples.","category":"page"},{"location":"userguide/#Mixtures","page":"User guide","title":"Mixtures","text":"","category":"section"},{"location":"userguide/","page":"User guide","title":"User guide","text":"As mentioned above, Taylor1{T}, HomogeneousPolynomial{T} and TaylorN{T} are parameterized structures such that T<:AbstractSeries, the latter is a subtype of Number. Then, we may actually define Taylor expansions in N+1 variables, where one of the variables (the Taylor1 variable) is somewhat special.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"x, y = set_variables(\"x y\", order=3)\nt1N = Taylor1([zero(x), one(x)], 5)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"The last line defines a Taylor1{TaylorN{Float64}} variable, which is of order 5 in t and order 3 in x and y. Then, we can evaluate functions involving such polynomials:","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"cos(2.1+x+t1N)","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"This kind of expansions are of interest when studying the dependence of parameters, for instance in the context of bifurcation theory or when considering the dependence of the solution of a differential equation on the initial conditions, around a given solution. In this case, x and y represent small variations around a given value of the parameters, or around some specific initial condition. Such constructions are exploited in the package TaylorIntegration.jl.","category":"page"},{"location":"userguide/","page":"User guide","title":"User guide","text":"CurrentModule = nothing","category":"page"},{"location":"background/#Background","page":"Background","title":"Background","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"","category":"page"},{"location":"background/#Introduction","page":"Background","title":"Introduction","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"TaylorSeries.jl is an implementation of high-order automatic differentiation, as presented in the book by W. Tucker [1]. The general idea is the following.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The Taylor series expansion of an analytical function f(t) with one independent variable t around t_0 can be written as","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f(t) = f_0 + f_1 (t-t_0) + f_2 (t-t_0)^2 + cdots + f_k (t-t_0)^k + cdots","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"where f_0=f(t_0), and the Taylor coefficients f_k = f_k(t_0) are the k-th normalized derivatives at t_0:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f_k = frac1k fracrm d^k f rm d t^k(t_0)","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Thus, computing the high-order derivatives of f(t) is equivalent to computing its Taylor expansion.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"In the case of many independent variables the same statements hold, though things become more subtle. Following Alex Haro's approach [2], the Taylor expansion is an infinite sum of homogeneous polynomials in the d independent variables x_1 x_2 dots x_d, which takes the form","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"f_k (mathbfx_0) = sum_m_1+cdots+m_d = k f_m_1dotsm_d \n(x_1-x_0_1)^m_1 cdots (x_d-x_0_d)^m_d =\nsum_mathbfm=k f_mathbfm (mathbfx-mathbfx_0)^mathbfm","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"Here, mathbfmin mathbbN^d is a multi-index of the k-th order homogeneous polynomial and mathbfx=(x_1x_2ldotsx_d) are the d independent variables.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"In both cases, a Taylor series expansion can be represented by a vector containing its coefficients. The difference between the cases of one or more independent variables is that the coefficients are real or complex numbers in the former case, but homogeneous polynomials in the latter case. This motivates the construction of the Taylor1 and TaylorN types.","category":"page"},{"location":"background/#Arithmetic-operations","page":"Background","title":"Arithmetic operations","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Arithmetic operations involving Taylor series can be expressed as operations on the coefficients:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"(f(x) pm g(x))_k = f_k pm g_k \n(f(x) cdot g(x))_k = sum_i=0^k f_i g_k-i \nBig( fracf(x)g(x) Big)_k = frac1g_0 Big f_k -\nsum_i=0^k-1 big(fracf(x)g(x)big)_i g_k-i Big ","category":"page"},{"location":"background/#Elementary-functions-of-polynomials","page":"Background","title":"Elementary functions of polynomials","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"Consider a function y(t) that satisfies the ordinary differential equation doty = f(y), y(t_0)=y_0, where t is the independent variable. Writing y(t) and f(t) as Taylor polynomials of t, substituting these in the differential equation and equating equal powers of the independent variable leads to the recursion relation","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"y_n+1 = fracf_nn+1","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The last equation and the corresponding initial condition y(t_0)=y_0 define a recurrence relation for the Taylor coefficients of y(t) around t_0.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The following are examples of such recurrence relations for some elementary functions:","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"p(t) =(f(t))^alpha qquad\n p_k = frac1k f_0sum_j=0^k-1big( alpha(k-j)-jbig)\n f_k-j p_j \ne(t) = exp(f(t)) qquad\n e_k = frac1ksum_j=0^k-1 (k-j) f_k-j e_j \nl(t) = log(f(t)) qquad\n l_k = frac1f_0big( f_k - frac1ksum_j=1^k-1 j\n f_k-j l_j big) \ns(t) = sin(f(t)) qquad\n s_k = frac1ksum_j=0^k-1 (k-j) f_k-j c_j \nc(t) = cos(f(t)) qquad\n c_k = -frac1ksum_j=0^k-1 (k-j) f_k-j s_j","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"The recursion relations for s(t) = sinbig(f(t)big) and c(t) = cosbig(f(t)big) depend on each other; this reflects the fact that they are solutions of a second-order differential equation.","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"All these relations hold for Taylor expansions in one and more independent variables; in the latter case, the Taylor coefficients f_k are homogeneous polynomials of degree k; see [2].","category":"page"},{"location":"background/#refs","page":"Background","title":"References","text":"","category":"section"},{"location":"background/","page":"Background","title":"Background","text":"[1] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press (2011).","category":"page"},{"location":"background/","page":"Background","title":"Background","text":"[2] A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, preprint.","category":"page"},{"location":"#TaylorSeries.jl","page":"Home","title":"TaylorSeries.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A Julia package for Taylor expansions in one or more independent variables.","category":"page"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"#Authors","page":"Home","title":"Authors","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM).\nDavid P. Sanders, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM).","category":"page"},{"location":"#Citing","page":"Home","title":"Citing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you find useful this package, please cite the paper:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Benet, L., & Sanders, D. P. (2019). TaylorSeries.jl: Taylor expansions in one and several variables in Julia. Journal of Open Source Software, 4(36), 1–4. https://doi.org/10.5281/zenodo.2601941","category":"page"},{"location":"#License","page":"Home","title":"License","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorSeries is licensed under the MIT \"Expat\" license; see LICENSE for the full license text.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"TaylorSeries.jl is a registered package, and is simply installed by running","category":"page"},{"location":"","page":"Home","title":"Home","text":"pkg> add(\"TaylorSeries\")","category":"page"},{"location":"#Related-packages","page":"Home","title":"Related packages","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Polynomials.jl: Polynomial manipulations\nPowerSeries.jl: Truncated power series for Julia\nMultivariatePolynomials.jl: Multivariate polynomials interface\nAbstractAlgebra.jl: Generic abstract algebra functionality in pure Julia\nForwardDiff.jl: Forward Mode Automatic Differentiation for Julia\nReverseDiff.jl: Reverse Mode Automatic Differentiation for Julia\nHyperDualNumbers.jl: Julia implementation of HyperDualNumbers","category":"page"},{"location":"#Acknowledgments","page":"Home","title":"Acknowledgments","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This project began (using Python) during a Masters' course in the postgraduate programs in Physics and in Mathematics at UNAM, during the second half of 2013. We thank the participants of the course for putting up with the half-baked material and contributing energy and ideas.","category":"page"},{"location":"","page":"Home","title":"Home","text":"We acknowledge financial support from DGAPA-UNAM PAPIME grants PE-105911 and PE-107114, and PAPIIT grants IG-101113 and IG-100616. LB acknowledges support through a Cátedra Marcos Moshinsky (2013).","category":"page"}] } diff --git a/previews/PR333/userguide/index.html b/previews/PR333/userguide/index.html index 86d67650..0a9194e9 100644 --- a/previews/PR333/userguide/index.html +++ b/previews/PR333/userguide/index.html @@ -1,5 +1,5 @@ -User guide · TaylorSeries.jl
      Edit on GitHub

      User guide

      For some simple examples, head over to the examples section. For a detailed guide, keep reading.

      TaylorSeries.jl is a basic polynomial algebraic manipulator in one or more variables; these two cases are treated separately. Three new types are defined, Taylor1, HomogeneousPolynomial and TaylorN, which correspond to expansions in one independent variable, homogeneous polynomials of various variables, and the polynomial series in many independent variables, respectively. These types are subtypes of AbstractSeries, which in turn is a subtype of Number, and are defined parametrically.

      The package is loaded as usual:

      julia> using TaylorSeries

      One independent variable

      Taylor expansions in one variable are represented by the Taylor1 type, which consists of a vector of coefficients (fieldname coeffs) and the maximum order considered for the expansion (fieldname order). The coefficients are arranged in ascending order with respect to the degree of the monomial, so that coeffs[1] is the constant term, coeffs[2] gives the first order term (t^1), etc. Yet, it is possible to have the natural ordering with respect to the degree; see below. This is a dense representation of the polynomial. The order of the polynomial can be omitted in the constructor, which is then fixed by the length of the vector of coefficients. If the length of the vector does not correspond with the order, order is used, which effectively truncates polynomial to degree order.

      julia> Taylor1([1, 2, 3],4) # Polynomial of order 4 with coefficients 1, 2, 3 1 + 2 x + 3 x²
      julia> Taylor1([0.0, 1im]) # Also works with complex numbers ( 0.0 + 1.0im ) x
      julia> Taylor1(ones(8), 2) # Polynomial truncated to order 2 1.0 + 1.0 x + 1.0 x²
      julia> shift_taylor(a) = a + Taylor1(typeof(a),5)  ## a + taylor-polynomial of order 5shift_taylor (generic function with 1 method)
      julia> t = shift_taylor(0.0) # Independent variable `t` 1.0 x
      Warning

      The information about the maximum order considered is displayed using a big-𝒪 notation. The convention followed when different orders are combined, and when certain functions are used in a way that they reduce the order (like differentiate), is to be consistent with the mathematics and the big-𝒪 notation, i.e., to propagate the lowest order.

      In some cases, it is desirable to not display the big-𝒪 notation. The function displayBigO allows to control whether it is displayed or not.

      julia> displayBigO(false) # turn-off displaying big O notationfalse
      julia> t 1.0 x
      julia> displayBigO(true) # turn it ontrue
      julia> t 1.0 x + 𝒪(x⁶)

      Similarly, it is possible to control some aspects of the format of the displayed series through the function use_show_default; use_show_default(true) uses the Base.show_default, while use_show_default(false) uses the custom display form (default).

      julia> use_show_default(true) # use Base.show methodtrue
      julia> tTaylor1{Float64}([0.0, 1.0, 0.0, 0.0, 0.0, 0.0], 5)
      julia> use_show_default(false) # use custom `show`false
      julia> t 1.0 x + 𝒪(x⁶)

      The definition of shift_taylor(a) uses the method Taylor1([::Type{Float64}], order::Int), which is a shortcut to define the independent variable of a Taylor expansion, of given type and order (the default is Float64). Defining the independent variable in advance is one of the easiest ways to use the package.

      The usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with the Taylor1 type, including promotions that involve Numbers. The operations return a valid Taylor expansion, consistent with the order of the series. This is apparent in the penultimate example below, where the fist non-zero coefficient is beyond the order of the expansion, and hence the result is zero.

      julia> t*(3t+2.5) 2.5 x + 3.0 x² + 𝒪(x⁶)
      julia> 1/(1-t) 1.0 + 1.0 x + 1.0 x² + 1.0 x³ + 1.0 x⁴ + 1.0 x⁵ + 𝒪(x⁶)
      julia> t*(t^2-4)/(t+2) - 2.0 x + 1.0 x² + 𝒪(x⁶)
      julia> tI = im*t ( 0.0 + 1.0im ) x + 𝒪(x⁶)
      julia> (1-t)^3.2 1.0 - 3.2 x + 3.5200000000000005 x² - 1.4080000000000004 x³ + 0.07040000000000009 x⁴ + 0.011264000000000012 x⁵ + 𝒪(x⁶)
      julia> (1+t)^t 1.0 + 1.0 x² - 0.5 x³ + 0.8333333333333333 x⁴ - 0.75 x⁵ + 𝒪(x⁶)
      julia> Taylor1(3) + Taylor1(5) == 2Taylor1(3)  # big-𝒪 convention appliestrue
      julia> t^6  # t is of order 5 0.0 + 𝒪(x⁶)
      julia> t^2 / t # The result is of order 4, instead of 5 1.0 x + 𝒪(x⁵)

      Note that the last example returns a Taylor1 series of order 4, instead of order 5; this is be consistent with the number of known coefficients of the returned series, since the result corresponds to factorize t in the numerator to cancel the same factor in the denominator.

      Taylor1 is also equiped with a total order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, "Automatic Differentiation as Nonarchimedean Analysis", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. This is illustrated by:

      julia> 1 > t > 2*t^2 > 100*t^3 >= 0true

      If no valid Taylor expansion can be computed an error is thrown, for instance, when a derivative is not defined at the expansion point, or it simply diverges.

      julia> 1/tERROR: ArgumentError: Division does not define a Taylor1 polynomial;
+User guide · TaylorSeries.jl
      GitHub
      User guide
      For some simple examples, head over to the examples section. For a detailed guide, keep reading.
      TaylorSeries.jl is a basic polynomial algebraic manipulator in one or more variables; these two cases are treated separately. Three new types are defined, Taylor1, HomogeneousPolynomial and TaylorN, which correspond to expansions in one independent variable, homogeneous polynomials of various variables, and the polynomial series in many independent variables, respectively. These types are subtypes of AbstractSeries, which in turn is a subtype of Number, and are defined parametrically.
      The package is loaded as usual:
      julia> using TaylorSeries
      One independent variable
      Taylor expansions in one variable are represented by the Taylor1 type, which consists of a vector of coefficients (fieldname coeffs) and the maximum order considered for the expansion (fieldname order). The coefficients are arranged in ascending order with respect to the degree of the monomial, so that coeffs[1] is the constant term, coeffs[2] gives the first order term (t^1), etc. Yet, it is possible to have the natural ordering with respect to the degree; see below. This is a dense representation of the polynomial. The order of the polynomial can be omitted in the constructor, which is then fixed by the length of the vector of coefficients. If the length of the vector does not correspond with the order, order is used, which effectively truncates polynomial to degree order.
      julia> Taylor1([1, 2, 3],4) # Polynomial of order 4 with coefficients 1, 2, 3 1 + 2 x + 3 x²
      julia> Taylor1([0.0, 1im]) # Also works with complex numbers ( 0.0 + 1.0im ) x
      julia> Taylor1(ones(8), 2) # Polynomial truncated to order 2 1.0 + 1.0 x + 1.0 x²
      julia> shift_taylor(a) = a + Taylor1(typeof(a),5)  ## a + taylor-polynomial of order 5shift_taylor (generic function with 1 method)
      julia> t = shift_taylor(0.0) # Independent variable `t` 1.0 x
      Warning
      The information about the maximum order considered is displayed using a big-𝒪 notation. The convention followed when different orders are combined, and when certain functions are used in a way that they reduce the order (like differentiate), is to be consistent with the mathematics and the big-𝒪 notation, i.e., to propagate the lowest order.
      In some cases, it is desirable to not display the big-𝒪 notation. The function displayBigO allows to control whether it is displayed or not.
      julia> displayBigO(false) # turn-off displaying big O notationfalse
      julia> t 1.0 x
      julia> displayBigO(true) # turn it ontrue
      julia> t 1.0 x + 𝒪(x⁶)
      Similarly, it is possible to control some aspects of the format of the displayed series through the function use_show_default; use_show_default(true) uses the Base.show_default, while use_show_default(false) uses the custom display form (default).
      julia> use_show_default(true) # use Base.show methodtrue
      julia> tTaylor1{Float64}([0.0, 1.0, 0.0, 0.0, 0.0, 0.0], 5)
      julia> use_show_default(false) # use custom `show`false
      julia> t 1.0 x + 𝒪(x⁶)
      The definition of shift_taylor(a) uses the method Taylor1([::Type{Float64}], order::Int), which is a shortcut to define the independent variable of a Taylor expansion, of given type and order (the default is Float64). Defining the independent variable in advance is one of the easiest ways to use the package.
      The usual arithmetic operators (+, -, *, /, ^, ==) have been extended to work with the Taylor1 type, including promotions that involve Numbers. The operations return a valid Taylor expansion, consistent with the order of the series. This is apparent in the penultimate example below, where the fist non-zero coefficient is beyond the order of the expansion, and hence the result is zero.
      julia> t*(3t+2.5) 2.5 x + 3.0 x² + 𝒪(x⁶)
      julia> 1/(1-t) 1.0 + 1.0 x + 1.0 x² + 1.0 x³ + 1.0 x⁴ + 1.0 x⁵ + 𝒪(x⁶)
      julia> t*(t^2-4)/(t+2) - 2.0 x + 1.0 x² + 𝒪(x⁶)
      julia> tI = im*t ( 0.0 + 1.0im ) x + 𝒪(x⁶)
      julia> (1-t)^3.2 1.0 - 3.2 x + 3.5200000000000005 x² - 1.4080000000000004 x³ + 0.07040000000000009 x⁴ + 0.011264000000000012 x⁵ + 𝒪(x⁶)
      julia> (1+t)^t 1.0 + 1.0 x² - 0.5 x³ + 0.8333333333333333 x⁴ - 0.75 x⁵ + 𝒪(x⁶)
      julia> Taylor1(3) + Taylor1(5) == 2Taylor1(3)  # big-𝒪 convention appliestrue
      julia> t^6  # t is of order 5 0.0 + 𝒪(x⁶)
      julia> t^2 / t # The result is of order 4, instead of 5 1.0 x + 𝒪(x⁵)
      Note that the last example returns a Taylor1 series of order 4, instead of order 5; this is be consistent with the number of known coefficients of the returned series, since the result corresponds to factorize t in the numerator to cancel the same factor in the denominator.
      Taylor1 is also equiped with a total order, provided by overloading isless. The ordering is consistent with the interpretation that there are infinitessimal elements in the algebra; for details see M. Berz, "Automatic Differentiation as Nonarchimedean Analysis", Computer Arithmetic and Enclosure Methods, (1992), Elsevier, 439-450. This is illustrated by:
      julia> 1 > t > 2*t^2 > 100*t^3 >= 0true
      If no valid Taylor expansion can be computed an error is thrown, for instance, when a derivative is not defined at the expansion point, or it simply diverges.
      julia> 1/tERROR: ArgumentError: Division does not define a Taylor1 polynomial;
 order k=0 => coeff[0]=Inf.
      julia> t^3.2ERROR: DomainError with  1.0 x + 𝒪(x⁶):
 The 0-th order Taylor1 coefficient must be non-zero
 to raise the Taylor1 polynomial to a non-integer exponent.
      julia> abs(t)ERROR: DomainError with  1.0 x + 𝒪(x⁶):
@@ -51,4 +51,4 @@
  -26.0  20.0
   20.0  -8.0
      Other specific applications are described in the Examples.
      Mixtures
      As mentioned above, Taylor1{T}, HomogeneousPolynomial{T} and TaylorN{T} are parameterized structures such that T<:AbstractSeries, the latter is a subtype of Number. Then, we may actually define Taylor expansions in $N+1$ variables, where one of the variables (the Taylor1 variable) is somewhat special.
      julia> x, y = set_variables("x y", order=3)2-element Vector{TaylorN{Float64}}:
   1.0 x + 𝒪(‖x‖⁴)
-  1.0 y + 𝒪(‖x‖⁴)
      julia> t1N = Taylor1([zero(x), one(x)], 5) ( 1.0 + 𝒪(‖x‖⁴)) x + 𝒪(x⁶)
      The last line defines a Taylor1{TaylorN{Float64}} variable, which is of order 5 in t and order 3 in x and y. Then, we can evaluate functions involving such polynomials:
      julia> cos(2.1+x+t1N)  - 0.5048461045998576 - 0.8632093666488737 x + 0.2524230522999288 x² + 0.14386822777481229 x³ + 𝒪(‖x‖⁴) + ( - 0.8632093666488737 + 0.5048461045998576 x + 0.43160468332443686 x² - 0.0841410174333096 x³ + 𝒪(‖x‖⁴)) x + ( 0.2524230522999288 + 0.43160468332443686 x - 0.1262115261499644 x² - 0.07193411388740614 x³ + 𝒪(‖x‖⁴)) x² + ( 0.14386822777481229 - 0.0841410174333096 x - 0.07193411388740614 x² + 0.0140235029055516 x³ + 𝒪(‖x‖⁴)) x³ + ( - 0.0210352543583274 - 0.03596705694370307 x + 0.0105176271791637 x² + 0.005994509490617178 x³ + 𝒪(‖x‖⁴)) x⁴ + ( - 0.007193411388740615 + 0.00420705087166548 x + 0.0035967056943703073 x² - 0.00070117514527758 x³ + 𝒪(‖x‖⁴)) x⁵ + 𝒪(x⁶)
      This kind of expansions are of interest when studying the dependence of parameters, for instance in the context of bifurcation theory or when considering the dependence of the solution of a differential equation on the initial conditions, around a given solution. In this case, x and y represent small variations around a given value of the parameters, or around some specific initial condition. Such constructions are exploited in the package TaylorIntegration.jl.
      
+  1.0 y + 𝒪(‖x‖⁴)
      julia> t1N = Taylor1([zero(x), one(x)], 5) ( 1.0 + 𝒪(‖x‖⁴)) x + 𝒪(x⁶)

      The last line defines a Taylor1{TaylorN{Float64}} variable, which is of order 5 in t and order 3 in x and y. Then, we can evaluate functions involving such polynomials:

      julia> cos(2.1+x+t1N)  - 0.5048461045998576 - 0.8632093666488737 x + 0.2524230522999288 x² + 0.14386822777481229 x³ + 𝒪(‖x‖⁴) + ( - 0.8632093666488737 + 0.5048461045998576 x + 0.43160468332443686 x² - 0.0841410174333096 x³ + 𝒪(‖x‖⁴)) x + ( 0.2524230522999288 + 0.43160468332443686 x - 0.1262115261499644 x² - 0.07193411388740614 x³ + 𝒪(‖x‖⁴)) x² + ( 0.14386822777481229 - 0.0841410174333096 x - 0.07193411388740614 x² + 0.0140235029055516 x³ + 𝒪(‖x‖⁴)) x³ + ( - 0.0210352543583274 - 0.03596705694370307 x + 0.0105176271791637 x² + 0.005994509490617178 x³ + 𝒪(‖x‖⁴)) x⁴ + ( - 0.007193411388740615 + 0.00420705087166548 x + 0.0035967056943703073 x² - 0.00070117514527758 x³ + 𝒪(‖x‖⁴)) x⁵ + 𝒪(x⁶)

      This kind of expansions are of interest when studying the dependence of parameters, for instance in the context of bifurcation theory or when considering the dependence of the solution of a differential equation on the initial conditions, around a given solution. In this case, x and y represent small variations around a given value of the parameters, or around some specific initial condition. Such constructions are exploited in the package TaylorIntegration.jl.