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Improve product of TaylorModelNs #31

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Improve product of TaylorModelNs #31

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1d51398
Avoid duplicating order in product of `TaylorModelN`s
lbenet Feb 10, 2019
800c016
Explicit evaluation of monomials, and revert a modified test
lbenet Feb 10, 2019
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57 changes: 37 additions & 20 deletions src/arithmetic.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -208,30 +208,47 @@ end
# Multiplication
function *(a::TaylorModelN, b::TaylorModelN)
@assert a.x0 == b.x0 && a.I == b.I

# Polynomial product with extended order
order = max(get_order(a), get_order(b))
@assert 2*order ≤ get_order()
aext = TaylorN(copy(a.pol.coeffs), 2*order)
bext = TaylorN(copy(b.pol.coeffs), 2*order)
res = aext * bext

# Returned polynomial
bext = TaylorN( copy(res.coeffs[1:order+1]) )

# Bound for the neglected part of the product of polynomials
res[0:order] .= zero(eltype(res))
aux = a.I - a.x0
Δnegl = res(aux)

# Remainder of the product
Δa = a.pol(aux)
Δb = b.pol(aux)
Δ = Δnegl + Δb * a.rem + Δa * b.rem + a.rem * b.rem
# Polynomial product with largest order from TaylorSeries._params_TaylorN_
order_a = get_order(a)
order_b = get_order(b)
order_max = max(order_a, order_b)
order_prod = order_a + order_b
orderTS = get_order()

# The returned polynomial has no neglected part
if order_prod ≤ orderTS
apol = TaylorN(a.pol.coeffs, order_prod)
bpol = TaylorN(b.pol.coeffs, order_prod)
res = apol * bpol
Δa = a.pol(aux)
Δb = b.pol(aux)
Δ = Δb * a.rem + Δa * b.rem + a.rem * b.rem
return TaylorModelN(res, Δ, a.x0, a.I)
end

return TaylorModelN(bext, Δ, a.x0, a.I)
end
# The returned polynomial has a neglected part
# Bound the neglected part of the product of polynomials
apol = TaylorN(a.pol.coeffs, orderTS)
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bpol = TaylorN(b.pol.coeffs, orderTS)
res = apol * bpol
Δa = apol(aux)
Δb = bpol(aux)
Δ = Δb * a.rem + Δa * b.rem + a.rem * b.rem

# We evaluate each term of the product of coefficients at `aux`
Δnegl = zero(Δ)
for order = order_max+1:order_prod
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Why does this start at order_max?

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I am correcting that, it should be orderTS. The idea, as I say above, is to compute the resulting polynomial up to orderTS, and from there bound whatever is not included in the polynomial part.

orderini = order - order_max
for inda = orderini:order_a
indb = order - inda
Δnegl += evaluate(apol[inda], aux) * evaluate(bpol[indb], aux)
end
end

return TaylorModelN(res, Δ+Δnegl, a.x0, a.I)
end

# Multiplication by numbers
function *(b::T, a::TaylorModelN) where {T<:NumberNotSeries}
Expand Down
9 changes: 3 additions & 6 deletions test/TMN.jl
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Expand Up @@ -20,8 +20,7 @@ function check_containment(ftest, xx::TaylorModelN{N,T,S}, tma::TaylorModelN{N,T
end

const _order = 2
const _order_max = 2*(_order+1)
set_variables(Interval{Float64}, [:x, :y], order=_order_max)
set_variables(Interval{Float64}, [:x, :y], order=_order)

@testset "Tests for TaylorModelN " begin
b0 = Interval(0.0) × Interval(0.0)
Expand Down Expand Up @@ -53,7 +52,7 @@ set_variables(Interval{Float64}, [:x, :y], order=_order_max)
@test_throws BoundsError TaylorModelN(5, _order, b0, ib0) # wrong variable number

# Tests for get_order and remainder
@test get_order() == 6
@test get_order() == 2
@test get_order(xm) == 2
@test remainder(ym) == zi
end
Expand All @@ -79,8 +78,6 @@ set_variables(Interval{Float64}, [:x, :y], order=_order_max)
b = b1[2] * a
@test b == TaylorModelN( b1[2]*a.pol, Δ*b1[2], b1, ib1 )
@test b / b1[2] == a
@test_throws AssertionError TaylorModelN(TaylorN(1, order=_order_max), zi, b1, ib1) *
TaylorModelN(TaylorN(2, order=_order_max), zi, b1, ib1)

remt = remainder(1/(1-TaylorModel1(_order, b1[1], ib1[1])))
@test remainder(1 / (1-xm)) == remt
Expand All @@ -100,7 +97,7 @@ set_variables(Interval{Float64}, [:x, :y], order=_order_max)
@test rpa(x->5*x, ym) == 5*ym
remT = remainder(5*TaylorModel1(2, b1[1], ib1[1])^4)
@test rpa(x->5*x^4, xm) == TaylorModelN(zero(xT), remT, b1, ib1)
remT = 5 * (ib1[1]-b1[1])^2 * (2*(ib1[2]-b1[2])+(ib1[2]-b1[2])^2)
remT = 5 * (ib1[1]-b1[1])^2 *(ib1[2]-b1[2]) * (2+(ib1[2]-b1[2]))
@test rpa(x->5*x^2, xm*ym) == TaylorModelN( 5*xT^2, remT, b1, ib1)

# Testing remainders of an RPA
Expand Down