How to speed up (or memoize) a series of mutually recursive functions

Question

I have a program which produces a series of functions f and g which looks like the following:

step (f,g) = (newF f g, newG f g)

newF f g x = r (f x) (g x)
newG f g x = s (f x) (g x)

foo = iterate step (f0,g0)

Where r and s are some uninteresting functions of f x and g x. I naively hoped that having foo be a list would mean that when I call the n'th f it will not recompute the (n-1)th f if it has already computed it (as would have happened if f and g weren't functions). Is there any way to memoize this without ripping the whole program apart (e.g. evaluating f0 and g0 on all relevant arguments and then working upward)?

Answer 1

You may find Data.MemoCombinators useful (in the data-memocombinators package).

You don't say what argument types your f and g take --- if they both takes integral values then you would use it like this:

import qualified Data.MemoCombinators as Memo

foo = iterate step (Memo.integral f0, Memo.integral g0)

If required, you could memoise the output of each step as well

step (f,g) = (Memo.integral (newF f g), Memo.integral (newG f g))

I hope you don't see this as ripping the whole program apart.

---

In reply to your comment:

This is the best I can come up with. It's untested, but should be working along the right lines.

I worry that converting between Double and Rational is needlessly inefficient --- if there was a Bits instance for Double we could use Memo.bits instead. So this might not ultimately be of any practical use to you.

import Control.Arrow ((&&&))
import Data.Ratio (numerator, denominator, (%))

memoV :: Memo.Memo a -> Memo.Memo (V a)
memoV m f = \(V x y z) -> table x y z
  where g x y z = f (V x y z)
        table = Memo.memo3 m m m g

memoRealFrac :: RealFrac a => Memo.Memo a
memoRealFrac f = Memo.wrap (fromRational . uncurry (%))
                           ((numerator &&& denominator) . toRational)
                           Memo.integral

---

A different approach.

You have

step :: (V Double -> V Double, V Double -> V Double)
     -> (V Double -> V Double, V Double -> V Double)

How about you change that to

step :: (V Double -> (V Double, V Double))
     -> (V Double -> (V Double, V Double))
step h x = (r fx gx, s fx gx)
  where (fx, gx) = h x

And also change

foo = (fst . bar, snd . bar)
  where bar = iterate step (f0 &&& g0)

Hopefully the shared fx and gx should result in a bit of a speed-up.

Answer 2

Is there any way to memoize this without ripping the whole program apart (e.g. evaluating f0 and g0 on all relevant arguments and then working upward)?

This may be what you mean by "ripping the whole program apart", but here is a solution in which (I believe but can't test ATM) fooX can be shared.

nthFooOnX :: Integer -> Int -> (Integer, Integer)
nthFooOnX x = 
    let fooX = iterate step' (f0 x, g0 x)
     in \n-> fooX !! n

step' (fx,gx) = (r fx gx, s fx gx)

-- testing definitions:
r = (+)
s = (*)
f0 = (+1)
g0 = (+1)

I don't know if that preserves the spirit of your original implementation.