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Increase functionality of RTaylorModel1 (#131)
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* Check monotonicity bounds and exploit dispatch

* Functionality for RTaylorModel1{TaylorN} and tests

* Fixes and add test coverage

* Avoid duplicating code for integrate, fixes, docstrings and tests

* Add centered_dom

* Bump patch version
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@lbenet
lbenet authored Dec 17, 2021
1 parent d1d2125 commit 9789bba
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Showing 10 changed files with 397 additions and 171 deletions.
2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
name = "TaylorModels"
uuid = "314ce334-5f6e-57ae-acf6-00b6e903104a"
repo = "https://github.com/JuliaIntervals/TaylorModels.jl.git"
version = "0.5.1"
version = "0.5.2"

[deps]
IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
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2 changes: 1 addition & 1 deletion src/TaylorModels.jl
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Expand Up @@ -31,7 +31,7 @@ import LinearAlgebra: norm
export TaylorModel1, RTaylorModel1, TaylorModelN, TMSol

export remainder, polynomial, domain, expansion_point, flowpipe, get_xTM,
rpa, fp_rpa, bound_remainder,
rpa, fp_rpa, bound_remainder, centered_dom,
validated_integ, validated_integ2

export linear_dominated_bounder, quadratic_fast_bounder
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36 changes: 23 additions & 13 deletions src/arithmetic.jl
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Expand Up @@ -61,7 +61,7 @@ for TM in tupleTMs
a_order = get_order(a)
b_order = get_order(b)
rnegl_order = a_order + b_order
aux = a.dom - a.x0
aux = centered_dom(a)

# Returned polynomial
res = a.pol * b.pol
Expand All @@ -70,7 +70,7 @@ for TM in tupleTMs
# Remainder of the product
if $TM == TaylorModel1

# Remaing terms of the product as reduced Taylor1 (factored polynomial)
# Remaining terms of the product as reduced Taylor1 (factored polynomial)
rnegl = Taylor1(zero(res[0]), rnegl_order)
for k in order+1:rnegl_order
@inbounds for i = 0:k
Expand All @@ -84,7 +84,7 @@ for TM in tupleTMs
Δ = remainder_product(a, b, aux, Δnegl)
else

# Remaing terms of the product as reduced Taylor1 (factored polynomial)
# Remaining terms of the product as reduced Taylor1 (factored polynomial)
rnegl = Taylor1(zero(res[0]), rnegl_order-order)
for k in order+1:rnegl_order
@inbounds for i = 0:k
Expand Down Expand Up @@ -122,6 +122,7 @@ for TM in tupleTMs
end

# Remainder of the product
# TaylorModel1
function remainder_product(a, b, aux, Δnegl)
Δa = a.pol(aux)
Δb = b.pol(aux)
Expand All @@ -135,12 +136,6 @@ function remainder_product(a::TaylorModel1{TaylorN{T}, S},
# An N-dimensional symmetrical IntervalBox is assumed
# to bound the TaylorN part
auxQ = IntervalBox(-1 .. 1, Val(N))
Δ = remainder_product(a, b, auxT, auxQ, Δnegl)
return Δ
end
function remainder_product(a::TaylorModel1{TaylorN{T}, S},
b::TaylorModel1{TaylorN{T}, S},
auxT, auxQ, Δnegl) where {T, S}
Δa = a.pol(auxT)(auxQ)
Δb = b.pol(auxT)(auxQ)
Δ = Δnegl(auxQ) + Δb * a.rem + Δa * b.rem + a.rem * b.rem
Expand All @@ -153,17 +148,32 @@ function remainder_product(a::TaylorModel1{TaylorModelN{N,T,S},S},
Δ = Δnegl + Δb * a.rem + Δa * b.rem + a.rem * b.rem

# Evaluate at the TMN centered domain
auxN = a[0].dom - a[0].x0
auxN = centered_dom(a[0])
ΔN = Δ(auxN)
return ΔN
end
function remainder_product(a::RTaylorModel1, b::RTaylorModel1, aux, Δnegl, order)
# RTaylorModel1
function remainder_product(a, b, aux, Δnegl, order)
Δa = a.pol(aux)
Δb = b.pol(aux)
V = aux^(order+1)
Δ = Δnegl + Δb * a.rem + Δa * b.rem + a.rem * b.rem * V
return Δ
end
function remainder_product(a::RTaylorModel1{TaylorN{T},S},
b::RTaylorModel1{TaylorN{T},S},
aux, Δnegl, order) where {T, S}
N = get_numvars()
# An N-dimensional symmetrical IntervalBox is assumed
# to bound the TaylorN part
auxQ = symmetric_box(N, T)
Δa = a.pol(aux)(auxQ)
Δb = b.pol(aux)(auxQ)
V = aux^(order+1)
Δ = Δnegl(auxQ) + Δb * a.rem + Δa * b.rem + a.rem * b.rem * V
return Δ
end


# Division
function /(a::TaylorModel1, b::TaylorModel1)
Expand Down Expand Up @@ -207,7 +217,7 @@ function truncate_taylormodel(a::RTaylorModel1, m::Integer)

apol = Taylor1(copy(a.pol.coeffs[1:m+1]))
bpol = Taylor1(copy(a.pol.coeffs))
aux = a.dom - a.x0
aux = centered_dom(a)
Δnegl = bound_truncation(RTaylorModel1, bpol, aux, m)
Δ = Δnegl + a.rem * (aux)^(order-m)
return RTaylorModel1( apol, Δ, a.x0, a.dom )
Expand Down Expand Up @@ -253,7 +263,7 @@ function *(a::TaylorModelN, b::TaylorModelN)
b_order = get_order(b)
rnegl_order = a_order + b_order
@assert rnegl_order ≤ get_order()
aux = a.dom - a.x0
aux = centered_dom(a)

# Returned polynomial
res = a.pol * b.pol
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4 changes: 2 additions & 2 deletions src/auxiliary.jl
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Expand Up @@ -55,7 +55,7 @@ for TM in tupleTMs
apol, bpol = TaylorSeries.fixorder(apol0, bpol0)

# Bound for the neglected part of the polynomial
dom = a.dom - a.x0
dom = centered_dom(a)
Δa = bound_truncation($TM, apol0, dom, order) + remainder(a)
Δb = bound_truncation($TM, bpol0, dom, order) + remainder(b)

Expand Down Expand Up @@ -86,7 +86,7 @@ function fixorder(a::TaylorModelN, b::TaylorModelN)
apol, bpol = TaylorSeries.fixorder(apol0, bpol0)

# Bound for the neglected part of the polynomial
dom = a.dom - a.x0
dom = centered_dom(a)
Δa = bound_truncation(TaylorModelN, apol0, dom, order) + remainder(a)
Δb = bound_truncation(TaylorModelN, bpol0, dom, order) + remainder(b)

Expand Down
143 changes: 96 additions & 47 deletions src/bounds.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -18,43 +18,16 @@ function bound_remainder(::Type{TaylorModel1}, f::Function, polf::Taylor1, polfI

_order = get_order(polf) + 1
fTIend = polfI[_order]
if (sup(fTIend) < 0 || inf(fTIend) > 0)
# Absolute remainder is monotonic
a = Interval(inf(I))
b = Interval(sup(I))
Δlo = f(a) - polf(a-x0)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δhi = f(b) - polf(b-x0)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δx0 = f(x0) - polf[0]
Δ = hull(Δlo, Δx0, Δhi)
else
# Lagrange bound
Δ = fTIend * (I-x0)^_order
end
return Δ
bb = sup(fTIend) < 0 || inf(fTIend) > 0
return _monot_bound_remainder(TaylorModel1, Val(bb), f, polf, polfI, x0, I)
end
function bound_remainder(::Type{TaylorModel1}, f::Function, polf::Taylor1{TaylorN{T}}, polfI::Taylor1, x0, I::Interval) where {T}
# The domain of the TaylorN part is assumed to be
# the normalized symmetric box
_order = get_order(polf) + 1
fTIend = polfI[_order]
if (sup(fTIend) < 0 || inf(fTIend) > 0)
# Absolute remainder is monotonic
a = Interval(inf(I))
b = Interval(sup(I))
symIbox = IntervalBox(-1 .. 1, get_numvars())
Δlo = (f(a) - polf(a-x0))(symIbox)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δhi = (f(b) - polf(b-x0))(symIbox)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δx0 = (f(x0) - polf[0])(symIbox)
Δ = hull(Δlo, Δx0, Δhi)
else
# Lagrange bound
Δ = fTIend * (I-x0)^_order
end
return Δ
bb = sup(fTIend) < 0 || inf(fTIend) > 0
return _monot_bound_remainder(TaylorModel1, Val(bb), f, polf, polfI, x0, I)
end


Expand All @@ -77,22 +50,98 @@ function bound_remainder(::Type{RTaylorModel1}, f::Function, polf::Taylor1, polf
fTIend = polfI[_order+1]
a = Interval(inf(I))
b = Interval(sup(I))
if (sup(fTIend) < 0 || inf(fTIend) > 0) && isempty(a ∩ x0) && isempty(b ∩ x0)
# Error is monotonic
denom_lo = (a-x0)^_order
Δlo = f(a) - polf(a-x0)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δlo = Δlo / denom_lo
denom_hi = (b-x0)^_order
Δhi = f(b) - polf(b-x0)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δhi = Δhi / denom_hi
Δ = hull(Δlo, Δhi)
else
# Lagrange coefficient
Δ = polfI[_order]
end
return Δ
bb = (sup(fTIend) < 0 || inf(fTIend) > 0) && isempty(a ∩ x0) && isempty(b ∩ x0)
return _monot_bound_remainder(RTaylorModel1, Val(bb), f, polf, polfI, x0, I)
end


"""
_monot_bound_remainder(::Type{TaylorModel1}, ::Val{true}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)
Computes the remainder exploiting monotonicity; see Prop 2.2.1 in Mioara Joldes' PhD thesis (pp 52).
"""
@inline function _monot_bound_remainder(::Type{TaylorModel1}, ::Val{true}, f::Function,
polf::Taylor1, polfI::Taylor1, x0, I::Interval)
# Absolute remainder is monotonic
a = Interval(inf(I))
b = Interval(sup(I))
Δlo = f(a) - polf(a-x0)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δhi = f(b) - polf(b-x0)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δx0 = f(x0) - polf[0]
return hull(Δlo, Δx0, Δhi)
end
@inline function _monot_bound_remainder(::Type{TaylorModel1}, ::Val{true}, f::Function,
polf::Taylor1{TaylorN{T}}, polfI::Taylor1, x0, I::Interval) where {T}
# Absolute remainder is monotonic
a = Interval(inf(I))
b = Interval(sup(I))
symIbox = IntervalBox(-1 .. 1, get_numvars())
Δlo = (f(a) - polf(a-x0))(symIbox)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δhi = (f(b) - polf(b-x0))(symIbox)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δx0 = (f(x0) - polf[0])(symIbox)
return hull(Δlo, Δx0, Δhi)
end
"""
_monot_bound_remainder(::Type{TaylorModel1}, ::Val{false}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)
Computes the remainder using Lagrange bound
"""
@inline function _monot_bound_remainder(::Type{TaylorModel1}, ::Val{false}, f::Function,
polf, polfI, x0, I)
_order = get_order(polf) + 1
fTIend = polfI[_order]
# Lagrange bound
return fTIend * (I-x0)^_order
end

"""
_monot_bound_remainder(::Type{RTaylorModel1}, ::Val{true}, f::Function, polf::Taylor1, polfI::Taylor1, x0, I::Interval)
Computes the remainder exploiting monotonicity; see Prop 2.3.7 in Mioara Joldes' PhD thesis (pp 67).
"""
@inline function _monot_bound_remainder(::Type{RTaylorModel1}, ::Val{true}, f::Function,
polf::Taylor1, polfI::Taylor1, x0, I::Interval)

_order = get_order(polf) + 1
a = Interval(inf(I))
b = Interval(sup(I))
# Error is monotonic
denom_lo = (a-x0)^_order
Δlo = f(a) - polf(a-x0)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δlo = Δlo / denom_lo
denom_hi = (b-x0)^_order
Δhi = f(b) - polf(b-x0)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δhi = Δhi / denom_hi
return hull(Δlo, Δhi)
end
@inline function _monot_bound_remainder(::Type{RTaylorModel1}, ::Val{true}, f::Function,
polf::Taylor1{TaylorN{T}}, polfI::Taylor1, x0, I::Interval) where {T}

_order = get_order(polf) + 1
a = Interval(inf(I))
b = Interval(sup(I))
symIbox = IntervalBox(-1 .. 1, get_numvars())
# Error is monotonic
denom_lo = (a-x0)^_order
Δlo = (f(a) - polf(a-x0))(symIbox)
# Δlo = f(a) - bound_taylor1(polf, a-x0)
Δlo = Δlo / denom_lo
denom_hi = (b-x0)^_order
Δhi = (f(b) - polf(b-x0))(symIbox)
# Δhi = f(b) - bound_taylor1(polf, b-x0)
Δhi = Δhi / denom_hi
return hull(Δlo, Δhi)
end
@inline function _monot_bound_remainder(::Type{RTaylorModel1}, ::Val{false}, f::Function,
polf::Taylor1, polfI::Taylor1, x0, I::Interval)
_order = get_order(polf) + 1
return polfI[_order]
end


Expand Down
8 changes: 6 additions & 2 deletions src/constructors.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -53,6 +53,8 @@ for TM in tupleTMs
@inline polynomial(tm::$TM) = tm.pol
@inline domain(tm::$TM) = tm.dom
@inline expansion_point(tm::$TM) = tm.x0
# Centered domain
@inline centered_dom(tm::$TM) = domain(tm) - expansion_point(tm)
end
end

Expand Down Expand Up @@ -147,6 +149,8 @@ TaylorModelN(a::T, ord::Integer, x0::IntervalBox{N,T}, dom::IntervalBox{N,T}) wh
@inline domain(tm::TaylorModelN) = tm.dom
@inline expansion_point(tm::TaylorModelN) = tm.x0
@inline get_numvars(::TaylorModelN{N,T,S}) where {N,T,S} = N
# Centered domain
@inline centered_dom(tm::TaylorModelN) = domain(tm) .- expansion_point(tm)

"""
TMSol{N,T,V1,V2,M}
Expand All @@ -158,7 +162,7 @@ Structure containing the solution of a validated integration.
`fp :: AbstractVector{IntervalBox{N,T}}` IntervalBox vector representing the flowpipe
`xTMv :: AbstractMatrix{TaylorModel1{TaylorN{T},T}}` Matrix whose entry `xTMv[i,t]` represents
`xTMv :: AbstractMatrix{TaylorModel1{TaylorN{T},T}}` Matrix whose entry `xTMv[i,t]` represents
the `TaylorModel1` of the i-th dependent variable, obtained at time time[t].
"""
struct TMSol{N,T<:Real,V1<:AbstractVector{T},V2<:AbstractVector{IntervalBox{N,T}},
Expand All @@ -167,7 +171,7 @@ struct TMSol{N,T<:Real,V1<:AbstractVector{T},V2<:AbstractVector{IntervalBox{N,T}
fp :: V2
xTM :: M

function TMSol(time::V1, fp::V2, xTM::M) where
function TMSol(time::V1, fp::V2, xTM::M) where
{N,T<:Real,V1<:AbstractVector{T},V2<:AbstractVector{IntervalBox{N,T}},
M<:AbstractMatrix{TaylorModel1{TaylorN{T},T}}}
@assert length(time) == length(fp) == size(xTM,2) && N == size(xTM,1)
Expand Down
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@lbenet lbenet commented on 9789bba Dec 17, 2021

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Registration pull request created: JuliaRegistries/General/50739

After the above pull request is merged, it is recommended that a tag is created on this repository for the registered package version.

This will be done automatically if the Julia TagBot GitHub Action is installed, or can be done manually through the github interface, or via:

git tag -a v0.5.2 -m "<description of version>" 9789bba2d696278970c6a1e03d83fd74881cf2a7
git push origin v0.5.2

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