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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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76 changes: 57 additions & 19 deletions src/arithmetic.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -208,28 +208,66 @@ 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
# Some definitions related to the order of some polynomials
orderTS = get_order() # maximum order, fixed by `TaylorSeries._params_TaylorN_`
order_a = get_order(a)
order_b = get_order(b)
order_prod = 2 * max(order_a, order_b)

# The returned polynomial has no terms beyond `orderTS`
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

# The resulting product has a part whose order is larger than `orderTS`;
# a bound for this polynomial has to be included
apol = TaylorN(a.pol.coeffs, orderTS)
bpol = TaylorN(b.pol.coeffs, orderTS)
res = apol * bpol # Trunctated polynomial to order `orderTS`
Δa = apol(aux)
Δb = bpol(aux)
Δ = Δb * a.rem + Δa * b.rem + a.rem * b.rem
N = get_numvars()

# We evaluate *explicitely* each term of the product of coefficients at `aux`
Δnegl = zero(Δ)
for order = orderTS+1:order_prod
orderini = order - orderTS
for inda = orderini:order_a
indb = order - inda
# Δnegl += evaluate(apol[inda], aux) * evaluate(bpol[indb], aux)
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@dpsanders dpsanders Feb 11, 2019

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You could try evaluating all of these bounds once before the loop (and storing them in new arrays), and then just indexing into the arrays inside the for loops. This should be much faster.

What I had done previously was store these bounds in the TaylorModel object itself when it is created (so that they are calculated exactly once in the history of the object).


# Coefficient tables of corresponding `HomogeneousPolynomial`s
@inbounds cta = TaylorSeries.coeff_table[inda+1]
@inbounds ctb = TaylorSeries.coeff_table[indb+1]

for i in eachindex(cta)
@inbounds a_coeff = apol.coeffs[inda+1][i]
iszero(a_coeff) && continue
for j in eachindex(ctb)
@inbounds b_coeff = bpol.coeffs[indb+1][j]
iszero(b_coeff) && continue

# Evaluation of monomials
tmp = one(Δ)
for n in 1:N
@inbounds tmp *= aux[n]^(cta[i][n]+ctb[j][n])
end
Δnegl += a_coeff * b_coeff * tmp
end
end
end
end

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


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8 changes: 3 additions & 5 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 @@ -100,6 +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 *(ib1[2]-b1[2]) * (2+(ib1[2]-b1[2]))
remT = 5 * (ib1[1]-b1[1])^2 * (2*(ib1[2]-b1[2])+(ib1[2]-b1[2])^2)
@test rpa(x->5*x^2, xm*ym) == TaylorModelN( 5*xT^2, remT, b1, ib1)

Expand Down