Wed Aug 16 21:15:44 EDT 2006

```ajb at spamcop.net wrote:
> G'day Tamas.
>
> Quoting Tamas K Papp <tpapp at Princeton.EDU>:
>
>> f is an a->a function, and there is a stopping rule
>> goOn(a,anext) :: a a -> Bool which determines when to stop.  The
>> algorithm looks like this (in imperative pseudocode):
>>
>> a = ainit
>>
>> while (true) {
>>       anext <- f(a)
>>       if (goOn(a,anext))
>>       	 a <- anext
>>       else
>>          stop and return anext
>> }
>
> Here are a couple more suggestions.
>
> First, this function scans an infinite list and stops when p x1 x2
> is true for two adjacent elements x1 and x2:
>
>     findFixpoint p (x1:xs@(x2:_))
>         | p x1 x2   = x2
>         | otherwise = findFixpoint p xs
>
> Then you just need to pass it [ainit, f ainit, f (f ainit), ...]:
>
>     findFixpoint dontGoOn (iterate f ainit)
>
> Note that the function to pass to findFixpoint here is the condition
> to use to _stop_.

The compiler may not deforest that list, so creating the list may be a small

>
> If you're comfortable with monads, it's possible to directly simulate
> complex imperative control flow.  It's not recommended to do this
> unless the flow is very complex, but here we are for the record:
>
>
>     -- I used a Newton-Raphson square root evaluation for testing,
>     -- but it has the same structure as your algorithm.
>     mysqrt :: Double -> Double
>     mysqrt x
>       = runCont (callCC loop) id
>       where
>         ainit = x * 0.5
>
>         f x = 0.5 * (a + x/a)
>
>         goOn a1 a2 = abs (a1 - a2) > 1e-5
>
>         loop break
>           = loop' ainit
>           where
>             loop' a
>               = do
>                 let anext = f a
>                 if goOn a anext
>                  then loop' anext
>                  else break anext
>
> callCC defines a point outside the loop that you can "break" to.
> You simply call that function (called a "continuation") and the
> loop is broken.
>
> Cheers,
> Andrew Bromage

Note that "f x" should be "f a" above.  But I like it.  My version of the above
looks like

>
> mysqrt :: Double -> Double
> mysqrt x = doWhile goOn f aInit
>   where
>     aInit = x * 0.5
>     f a = 0.5 * (a + x/a)
>     goOn a1 a2 = abs (a1 - a2) > 1e-5
>
> doWhile :: (a -> a -> Bool) -> (a -> a) -> a -> a
> doWhile goOn f x0 = runCont (callCC withBreak) id
>   where withBreak break =
>           let loop x = do let x' = f x
>                           when (not (goOn x x')) (break x')
>                           loop x'
>           in loop x0

```