Preliminary work on static analogs for Taylor1

Project.toml

@@ -1,7 +1,7 @@
 name = "TaylorSeries"
 uuid = "6aa5eb33-94cf-58f4-a9d0-e4b2c4fc25ea"
 repo = "https://github.com/JuliaDiff/TaylorSeries.jl.git"
-version = "0.10.3"
+version = "0.11.0"
 
 [deps]
 InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

benchmark/REQUIRE

@@ -0,0 +1,2 @@
+BenchmarkTools 0.3.1
+PkgBenchmark 0.1.1

benchmark/benchmarks.jl

@@ -0,0 +1,57 @@
+using BenchmarkTools, TaylorSeries
+
+const SUITE = BenchmarkGroup()
+
+S = SUITE["Constructors"] = BenchmarkGroup()
+for i in (05,10,20,50)
+    x = rand(i)
+    t = tuple(x...)
+    S["STaylor1(x::Array[len = $i])"] = @benchmarkable STaylor1($x)
+    S["STaylor1(x::Float64, Val($i))"] = @benchmarkable STaylor1($t)
+end
+
+function f(g, x, y, n)
+    t = x
+    for i=1:n
+        t = g(x,y)
+    end
+    t
+end
+function f(g, x, n)
+    t = x
+    for i=1:n
+        t = g(t)
+    end
+    t
+end
+n = 100
+dims = (5,20)
+
+S = SUITE["arithmetic"] = BenchmarkGroup()
+for i in dims
+    r = rand(i)
+    x = Taylor1(r)
+    x2 = Taylor1(r)
+    q = STaylor1(r)
+    q2 = STaylor1(r)
+    y = rand()
+    for g in (+, -, /, *)
+        S["Taylor1{$i,Float64} $(Symbol(g)) Float64"] = @benchmarkable f($g, $x, $y, $n)
+        S["STaylor1{$i,Float64} $(Symbol(g)) Float64"] = @benchmarkable f($g, $q, $y, $n)
+        S["Taylor1{$i,Float64} $(Symbol(g)) Taylor1{$i,Float64}"] = @benchmarkable f($g, $x, $x2, $n)
+        S["STaylor1{$i,Float64} $(Symbol(g)) STaylor1{$i,Float64}"] = @benchmarkable f($g, $q, $q2, $n)
+    end
+end
+
+S = SUITE["functions"] = BenchmarkGroup()
+for i in dims
+    r = rand(i)
+    x = Taylor1(r)
+    q = STaylor1(r)
+    y = rand()
+    for g in (exp, abs, zero, one, real, imag, conj, adjoint, iszero, isnan,
+              isinf, deg2rad, rad2deg)
+        S["$(Symbol(g))(Taylor1{Float64}), len = $i"] = @benchmarkable f($g, $x, $n)
+        S["$(Symbol(g))(STaylor1{$i,Float64}),"] = @benchmarkable f($g, $q, $n)
+    end
+end

benchmark/run_benchmark.jl

@@ -0,0 +1,12 @@
+using PkgBenchmark
+results = benchmarkpkg("TaylorSeries")
+show(results)
+
+#=
+# specify tag and uncommit to benchmark versus prior tagged version
+tag =
+results = judge("TaylorSeries", tag)
+show(results)
+=#
+
+export_markdown("results.md", results)

benchmark/tune.json

@@ -0,0 +1 @@
+[{"Julia":"1.3.1","BenchmarkTools":"0.4.3"},[["BenchmarkGroup",{"data":{"Constructors":["BenchmarkGroup",{"data":{"STaylor1(x::Float64, Val(5))":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":12,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}],"STaylor1(x::Array[length = 10])":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":15,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}],"STaylor1(x::Float64, Val(10))":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":24,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}],"STaylor1(x::Float64, Val(50))":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":22,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}],"STaylor1(x::Array[length = 5])":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":10,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}],"STaylor1(x::Array[length = 50])":["BenchmarkTools.Parameters",{"gctrial":true,"time_tolerance":0.05,"samples":10000,"evals":10,"gcsample":false,"seconds":5.0,"overhead":0.0,"memory_tolerance":0.01}]},"tags":[]}]},"tags":[]}]]]

src/arithmetic.jl

@@ -205,7 +205,6 @@ end
 -(a::Taylor1{T}, b::TaylorN{S}) where {T<:NumberNotSeries,S<:NumberNotSeries} =
     -(promote(a,b)...)
 
-
 ## Multiplication ##
 for T in (:Taylor1, :HomogeneousPolynomial, :TaylorN)
 

src/auxiliary.jl

@@ -103,7 +103,6 @@ function setindex!(a::Taylor1{T}, x::Array{T,1}, u::StepRange{Int,Int}) where {T
     end
 end
 
-
 """
     getcoeff(a, v)
 
@@ -237,7 +236,6 @@ for T in (:Taylor1, :HomogeneousPolynomial, :TaylorN)
     end
 end
 
-
 ## fixorder ##
 for T in (:Taylor1, :TaylorN)
     @eval begin
@@ -271,7 +269,6 @@ function Base.findlast(a::Taylor1{T}) where {T<:Number}
     return last-1
 end
 
-
 ## copyto! ##
 # Inspired from base/abstractarray.jl, line 665
 for T in (:Taylor1, :HomogeneousPolynomial, :TaylorN)
@@ -317,3 +314,46 @@ linear_polynomial(a::TaylorN) = TaylorN([a[1]])
 linear_polynomial(a::Vector{T}) where {T<:Number} = linear_polynomial.(a)
 
 linear_polynomial(a::Number) = a
+
+
+#=
+Tuple operations taken from ForwardDiff.jl
+
+Copyright (c) 2015: Jarrett Revels, Theodore Papamarkou, Miles Lubin, and other contributors
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
+The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+=#
+function tupexpr(f, N)
+    ex = Expr(:tuple, [f(i) for i=1:N]...)
+    return quote
+        $(Expr(:meta, :inline))
+        @inbounds return $ex
+    end
+end
+
+@generated function scale_tuple(tup::NTuple{N}, x) where N
+    return tupexpr(i -> :(tup[$i] * x), N)
+end
+
+@generated function div_tuple_by_scalar(tup::NTuple{N}, x) where N
+    return tupexpr(i -> :(tup[$i] / x), N)
+end
+
+@generated function add_tuples(a::NTuple{N}, b::NTuple{N})  where N
+    return tupexpr(i -> :(a[$i] + b[$i]), N)
+end
+
+@generated function sub_tuples(a::NTuple{N}, b::NTuple{N})  where N
+    return tupexpr(i -> :(a[$i] - b[$i]), N)
+end
+
+@generated function minus_tuple(tup::NTuple{N}) where N
+    return tupexpr(i -> :(-tup[$i]), N)
+end
+
+@generated function mul_tuples(a::NTuple{N}, b::NTuple{N}, afactor, bfactor) where N
+    return tupexpr(i -> :((afactor * a[$i]) + (bfactor * b[$i])), N)
+end

src/broadcasting.jl

@@ -12,7 +12,7 @@ import .Broadcast: BroadcastStyle, Broadcasted, broadcasted
 
 # BroadcastStyle definitions and basic precedence rules
 struct Taylor1Style{T} <: Base.Broadcast.AbstractArrayStyle{0} end
-Taylor1Style{T}(::Val{N}) where {T, N}= Base.Broadcast.DefaultArrayStyle{N}()
+Taylor1Style{T}(::Val{N}) where {T, N} = Base.Broadcast.DefaultArrayStyle{N}()
 BroadcastStyle(::Type{<:Taylor1{T}}) where {T} = Taylor1Style{T}()
 BroadcastStyle(::Taylor1Style{T}, ::Base.Broadcast.DefaultArrayStyle{0}) where {T} = Taylor1Style{T}()
 BroadcastStyle(::Taylor1Style{T}, ::Base.Broadcast.DefaultArrayStyle{1}) where {T} = Base.Broadcast.DefaultArrayStyle{1}()

src/constructors.jl

@@ -71,7 +71,6 @@ julia> Taylor1(Rational{Int}, 4)
 Taylor1(::Type{T}, order::Int) where {T<:Number} = Taylor1( [zero(T), one(T)], order)
 Taylor1(order::Int) = Taylor1(Float64, order)
 
-
 ######################### HomogeneousPolynomial
 """
     HomogeneousPolynomial{T<:Number} <: AbstractSeries{T}

src/conversion.jl

@@ -28,7 +28,6 @@ convert(::Type{Taylor1{T}}, b::T)  where {T<:Number} = Taylor1([b], 0)
 
 convert(::Type{Taylor1}, a::T) where {T<:Number} = Taylor1(a, 0)
 
-
 convert(::Type{HomogeneousPolynomial{T}}, a::HomogeneousPolynomial) where {T<:Number} =
     HomogeneousPolynomial(convert(Array{T,1}, a.coeffs), a.order)
 
@@ -171,7 +170,6 @@ promote_rule(::Type{Taylor1{T}}, ::Type{S}) where {T<:Number, S<:Number} =
 promote_rule(::Type{Taylor1{Taylor1{T}}}, ::Type{Taylor1{T}}) where {T<:Number} =
     Taylor1{Taylor1{T}}
 
-
 promote_rule(::Type{HomogeneousPolynomial{T}},
     ::Type{HomogeneousPolynomial{S}}) where {T<:Number, S<:Number} =
         HomogeneousPolynomial{promote_type(T,S)}

src/evaluate.jl

@@ -10,9 +10,10 @@
 """
     evaluate(a, [dx])
 
-Evaluate a `Taylor1` polynomial using Horner's rule (hand coded). If `dx` is
-ommitted, 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)`.
+Evaluate a `Taylor1` (or `STaylor1{N,T}`) polynomial using Horner's rule (hand
+coded). If `dx` is ommitted, 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)`.
 """
 function evaluate(a::Taylor1{T}, dx::T) where {T<:Number}
     @inbounds suma = a[end]
@@ -33,10 +34,10 @@ evaluate(a::Taylor1{T}) where {T<:Number} = a[0]
 """
     evaluate(x, δt)
 
-Evaluates each element of `x::Union{ Vector{Taylor1{T}}, Matrix{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)`.
+Evaluates each element of `x::Union{Vector{Taylor1{T}}, Matrix{Taylor1{T}},
+Vector{STaylor1{N,T}}, Matrix{STaylor1{N,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)`.
 """
 evaluate(x::Union{Array{Taylor1{T}}, SubArray{Taylor1{T}}}, δt::S) where
     {T<:Number, S<:Number} = evaluate.(x, δt)
@@ -110,7 +111,6 @@ evaluate(p::Taylor1{T}, x::Array{S}) where {T<:Number, S<:Number} =
 
 #function-like behavior for Taylor1
 (p::Taylor1)(x) = evaluate(p, x)
-
 (p::Taylor1)() = evaluate(p)
 
 #function-like behavior for Vector{Taylor1}

src/functions.jl

@@ -147,7 +147,6 @@ for T in (:Taylor1, :TaylorN)
     end
 end
 
-
 # Recursive functions (homogeneous coefficients)
 for T in (:Taylor1, :TaylorN)
     @eval begin
@@ -422,7 +421,6 @@ for T in (:Taylor1, :TaylorN)
     end
 end
 
-
 @doc doc"""
     inverse(f)
 

src/intervals.jl

@@ -146,3 +146,5 @@ function _normalize(a::TaylorN, I::IntervalBox{N,T}, ::Val{false}) where {N,T}
     end
     return a(x)
 end
+
+square(x::Interval{T}) where {T <: Real} = pow(x,2)

src/other_functions.jl

@@ -21,7 +21,6 @@ for T in (:Taylor1, :HomogeneousPolynomial, :TaylorN)
     @eval isnan(a::$T) = any( isnan.(a.coeffs) )
 end
 
-
 ## Division functions: rem and mod ##
 for op in (:mod, :rem)
     for T in (:Taylor1, :TaylorN)

src/power.jl

@@ -150,7 +150,6 @@ function ^(a::TaylorN, r::S) where {S<:Real}
     return c
 end
 
-
 # Homogeneous coefficients for real power
 @doc doc"""
     pow!(c, a, r::Real, k::Int)
@@ -283,6 +282,8 @@ function square(a::HomogeneousPolynomial)
     return res
 end
 
+#two(x::T) = convert(T, x)
+square(x::T) where {T <: Real} = x^2
 
 # Homogeneous coefficients for square
 @doc doc"""

src/static.jl

@@ -0,0 +1,63 @@
+==(a::STaylor1, b::STaylor1) = (a.coeffs == b.coeffs)
+
+iszero(a::STaylor1) = all(iszero, a.coeffs)
+
+zero(::STaylor1{N,T}) where {N, T<:Number} = STaylor1(zero(T), Val(N-1))
+one(::STaylor1{N,T}) where {N, T<:Number} = STaylor1(one(T), Val(N-1))
+
+@inline +(a::STaylor1{N,T}, b::STaylor1{N,T}) where {N, T<:Number} = STaylor1(a.coeffs .+ b.coeffs)
+@inline -(a::STaylor1{N,T}, b::STaylor1{N,T}) where {N, T<:Number} = STaylor1(a.coeffs .- b.coeffs)
+@inline +(a::STaylor1) = a
+@inline -(a::STaylor1) = STaylor1(minus_tuple(a.coeffs))
+
+function +(a::STaylor1{N,T}, b::T) where {N, T<:Number}
+    STaylor1{N,T}(ntuple(i -> i == 1 ? a.coeffs[1] + b : a.coeffs[i], Val(N)))
+end
+function +(b::T, a::STaylor1{N,T}) where {N, T<:Number}
+    STaylor1{N,T}(ntuple(i -> i == 1 ? a.coeffs[1] + b : a.coeffs[i], Val(N)))
+end
+function -(a::STaylor1{N,T}, b::T) where {N, T<:Number}
+    STaylor1{N,T}(ntuple(i -> i == 1 ? a.coeffs[1] - b : a.coeffs[i], Val(N)))
+end
+-(b::T, a::STaylor1{N,T}) where {N, T<:Number}  = b + (-a)
+
+#+(a::STaylor1{N,T}, b::S) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = +(promote(a,b)...)
+#+(a::STaylor1{N,T}, b::STaylor1{N,S}) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = +(promote(a,b)...)
+#+(a::STaylor1{N,T}, b::S) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = +(promote(a,b)...)
+#+(b::S, a::STaylor1{N,T}) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = +(promote(b,a)...)
+
+#-(a::STaylor1{N,T}, b::STaylor1{N,S}) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = -(promote(a,b)...)
+#-(a::STaylor1{N,T}, b::S) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = -(promote(a,b)...)
+#-(b::S, a::STaylor1{N,T}) where {N, T<:NumberNotSeries, S<:NumberNotSeries} = -(promote(b,a)...)
+
+@generated function *(x::STaylor1{N,T}, y::STaylor1{N,T}) where {T<:Number,N}
+     ex_calc = quote end
+     append!(ex_calc.args, Any[nothing for i in 1:N])
+     syms = Symbol[Symbol("c$i") for i in 1:N]
+     for j = 1:N
+         ex_line = :(x.coeffs[1]*y.coeffs[$j])
+         for k = 2:j
+             ex_line = :($ex_line + x.coeffs[$k]*y.coeffs[$(j-k+1)])
+         end
+         sym = syms[j]
+         ex_line = :($sym = $ex_line)
+         ex_calc.args[j] = ex_line
+     end
+     exout = :(($(syms[1]),))
+     for i = 2:N
+         push!(exout.args, syms[i])
+     end
+     return quote
+                Base.@_inline_meta
+                $ex_calc
+                return STaylor1{N,T}($exout)
+             end
+end
+
+
+function *(a::STaylor1{N,T}, b::T) where {N, T<:Number}
+    STaylor1{N,T}(b .* a.coeffs)
+end
+function *(b::T, a::STaylor1{N,T}) where {N, T<:Number}
+    STaylor1{N,T}(b .* a.coeffs)
+end

test/onevariable.jl

@@ -523,6 +523,17 @@ eeuler = Base.MathConstants.e
     @test Taylor1{Int}(false) == Taylor1([0])
 end
 
+function test_vs_Taylor1(x,y)
+    flag = true
+    for i in 0:2
+        if x[i] !== y[i]
+            flag = false
+            break
+        end
+    end
+    flag
+end
+
 @testset "Test `inv` for `Matrix{Taylor1{Float64}}``" begin
     t = Taylor1(5)
     a = Diagonal(rand(0:10,3)) + rand(3, 3)