> From: "Richard O'Keefe" <o...@cs.otago.ac.nz>
> Date: Thu, 29 Mar 2012 16:34:46 +1300
>
> On 29/03/2012, at 3:08 PM, Doug McIlroy wrote:
> > --------- without newtype
> >
> > toSeries f = f : repeat 0 -- coerce scalar to series
> >
> > instance Num a => Num [a] where
> > (f:fs) + (g:gs) = f+g : fs+gs
> > (f:fs') * gs@(g:gs') = f*g : fs'*gs + (toSeries f)*gs'
> >
> > --------- with newtype
> >
> > newtype Num a => PS a = PS [a] deriving (Show, Eq)
> >
> > fromPS (PS fs) = fs -- extract list
> > toPS f = PS (f : repeat 0) -- coerce scalar to series
> >
> > instance Num a => Num (PS a) where
> > (PS (f:fs)) + (PS (g:gs)) = PS (f+g : fs+gs)
> > (PS (f:fs)) * gPS@(PS (g:gs)) =
> > PS $ f*g : fromPS ((PS fs)*gPS + (toPS f)*(PS gs))
>
> Try it again.
>
> newtype PS a = PS [a] deriving (Eq, Show)
>
> u f (PS x) = PS $ map f x
> b f (PS x) (PS y) = PS $ zipWith f x y
> to_ps x = PS (x : repeat 0)
>
> ps_product (f:fs) (g:gs) = whatever
>
> instance Num a => Num (PS a)
> where
> (+) = b (+)
> (-) = b (-)
> (*) = b ps_product
> negate = u negate
> abs = u abs
> signum = u signum
> fromInteger = to_ps . fromInteger
>
> I've avoided defining ps_product because I'm not sure what
> it is supposed to do: the definition doesn't look commutative.
You have given the Hadamard product--a construction with somewhat
esoteric properties. The product I have in mind is the ordinary
mathematical product that one meets in freshman calculus. The
distributive law yields this symmetric formulation
(f:fs) * (g:gs) = f*g : (toSeries f)*gs + fs*(toSeries g) + (0 : fs*gs)
The version I gave is an optimization (which I learned from Jerzy). For more
explanation see www.cs.dartmouth.edu/~doug/powser.html.
I like the lifting functions b and u, but they don't get one very far.
The product is where the PS pox begins to bite badly. I would welcome
a perspicuous formulation of that using newtype.
Incidentally, a more efficient way to write the symmetric product is
(f:fs) * (g:gs) = f*g : zipWith (f*) gs + zipWith fs (*g) + (0 : fs*gs)
toSeries and zipWith appear in the formulas because Haskell overloading
won't let one use the multiplication symbol for both series*series and
scalar*series. One might also invent a distinct operator for the purpose.
Coercion with toSeries strikes me as the least jarring of these approaches.
zipWith suffers from mixing levels of abstraction--it signifies not the
idea of multiplication, but the algorithm. A new operator would suffer
both from unfamiliarity and lack of commutativity.
>
> > The code suffers a pox of PS.
>
> But it doesn't *need* to.
>
>
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