Question about multiplication

[folivetti]

levitate/src/Affine.hs

Lines 226 to 244 in 8dfcf39

	(*@) :: Affine -> Affine -> AffineMonad Affine
	(*@) Empty _ = return Empty
	(*@) _ Empty = return Empty
	(*@) RealLine _ = return RealLine
	(*@) _ RealLine = return RealLine
	(*@) x@(Affine cx _) y@(Affine cy _) = do
	let rx = radius x
	let ry = radius y
	let δ = rx *↑ ry
	let p = (Interval.make cx cx) *! (Interval.make cy cy)
	let δ' = δ +↑ (Interval.radius p)
	aζ <- makeFreshAffine (-(Interval.midpoint p)) []
	aδ <- makeFreshAffine δ' []
	a1 <- cy ·@ x
	a2 <- cx ·@ y
	a3 <- a1 +@ a2
	a4 <- a3 +@ aζ
	a5 <- a4 +@ aδ
	return a5

Hi,

as I understand from this piece of code, for two affines

x = x0 + x1 . e1
y = y0 + y1 . e2

their multiplication are calculated as:

z = x0y0 + y0x0 - x0y0 + y0x1e1 + x0y1e2 + |x1||y1|
  = (x0y0 + |x1||y1|) + y0x1e1 + x0y1e2

But by reading Stolffi reference, it should be:

z = x0y0  + y0x1e1 + x0y1e2 + |x1||y1|e3

In my tests your implementation seems to lead to more accurate results. Do you have any reference for that?

[kevinclancy]

Hi. I'm a little rusty on affine arithmetic at the moment. I was hoping to do some review and get back to you with a fully satisfying answer, but I have been pretty busy lately. My implementation does seem different than the one described by Stolfi, but I recall thinking of this project as an implementation of parts of Stolfi's tutorial. I don't recall drawing from any other sources; perhaps I just made a mistake.

I've been hoping to review affine arithmetic in order to examine it from a category theoretic perspective, but I can't say when that will happen.