[Haskell-cafe] Fwd: Re: Period of a sequence

[Steffen Schuldenzucker]

Michael,

On 06/27/2011 01:51 PM, Steffen Schuldenzucker wrote:
 >
 > Forwarding to -cafe
 >
 > -------- Original Message --------
 > Subject: Re: [Haskell-cafe] Period of a sequence
 > Date: Mon, 27 Jun 2011 04:46:10 -0700 (PDT)
 > From: michael rice <nowgate at yahoo.com>
 > To: Steffen Schuldenzucker <sschuldenzucker at uni-bonn.de>
 >
 > Hi Steffen,
 >
 > Repeating decimals.
 >
 > 5/7 == 0.714285 714285 7142857 ... Period = 6
 >
 > It does seem like a difficult problem.
 >
 > This one is eventually repeating, with Period = 3
 >
 > 3227/555 = 5.8144 144 144…

why not use the well-known division algorithm: (I hope this is readable)

3227 / 555
= 3227 `div` 555 + 3227 `mod` 555 / 555
= 5 + 452 / 555
= 5 + 0.1 * 4520 / 555
= 5 + 0.1 * (4520 `div` 555 + (4520 `mod` 555) / 555)
= 5 + 0.1 * (8 + 80 / 555)
= 5 + 0.1 * (8 + 0.1 * (800 / 555))
= 5 + 0.1 * (8 + 0.1 * (800 `div` 555 + (800 `mod` 555) / 555))
= 5 + 0.1 * (8 + 0.1 * (1 + 245 / 555))
= 5 + 0.1 * (8 + 0.1 * (1 + 0.1 * 2450 / 555))
= 5 + 0.1 * (8 + 0.1 * (1 + 0.1 * (4 + 230 / 555)))
= 5 + 0.1 * (8 + 0.1 * (1 + 0.1 * (4 + 0.1 * 2300 / 555)))
= 5 + 0.1 * (8 + 0.1 * (1 + 0.1 * (4 + 0.1 * (4 + 80 / 555))))
*whoops*, saw 80 already, namely in line 6. Would go on like that 
forever if I continued like this, so the final result has to be:

vvv Part before the place where I saw the '80' first
5.8 144 144 144 ...
     ^^^ Part after I saw the '80'

So you could write a recursive function that takes as an accumulating 
parameter containing the list of numbers already seen:

-- periodOf n m gives the periodic part of n/m as a decimal fraction.
-- (or an empty list if that number has finitely many decimal places)
 > periodOf :: (Integral a) => a -> a -> [a]
 > periodOf = periodOfWorker []
 >   where
 >     periodOfWorker seen n m
 >         | n `mod` m == 0 = ...
 >         | (n `mod` m) `elem` seen = ...
 >         | otherwise = ...

> --- On *Mon, 6/27/11, Steffen Schuldenzucker
> /<sschuldenzucker at uni-bonn.de>/*wrote:
>
>
> From: Steffen Schuldenzucker <sschuldenzucker at uni-bonn.de>
> Subject: Re: [Haskell-cafe] Period of a sequence
> To: "michael rice" <nowgate at yahoo.com>
> Cc: haskell-cafe at haskell.org
> Date: Monday, June 27, 2011, 4:32 AM
>
>
>
> On 06/26/2011 04:16 PM, michael rice wrote:
>  > MathWorks has the function seqperiod(x) to return the period of
> sequence
>  > x. Is there an equivalent function in Haskell?
>
> Could you specify what exactly the function is supposed to do? I am
> pretty sure that a function like
>
> seqPeriod :: (Eq a) => [a] -> Maybe Integer -- Nothing iff non-periodic
>
> cannot be written. If "sequences" are represented by the terms that
> define them (or this information is at least accessible), chances
> might be better, but I would still be interested how such a function
> works. The problem seems undecidable to me in general.
>
> On finite lists (which may be produced from infinite ones via
> 'take'), a naive implementation could be this:
>
>  >
>  > import Data.List (inits, cycle, isPrefixOf)
>  > import Debug.Trace
>  >
>  > -- Given a finite list, calculate its period.
>  > -- The first parameter controls what is accepted as a generator.
> See below.
>  > -- Set it to False when looking at chunks from an infinite sequence.
>  > listPeriod :: (Eq a) => Bool -> [a] -> Int
>  > listPeriod precisely xs = case filter (generates precisely xs)
> (inits xs) of
>  > -- as (last $ init xs) == xs, this will always suffice.
>  > (g:_) -> length g -- length of the *shortest* generator
>  >
>  > -- @generates prec xs g@ iff @g@ generates @xs@ by repitition. If
> @prec@, the
>  > -- lengths have to match, too. Consider
>  > --
>  > -- >>> generates True [1,2,3,1,2,1,2] [1,2,3,1,2]
>  > -- False
>  > --
>  > -- >>> generates False [1,2,3,1,2,1,2] [1,2,3,1,2]
>  > -- True
>  > generates :: (Eq a) => Bool -> [a] -> [a] -> Bool
>  > generates precisely xs g = if null g
>  > then null xs
>  > else (not precisely || length xs `mod` length g == 0)
>  > && xs `isPrefixOf` cycle g
>  >
>
> -- Steffen
>
>
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