[R] Season's Greetings (and great news ... )!
(Ted Harding)
Ted.Harding at wlandres.net
Sun Dec 22 19:37:18 CET 2013
Thanks for the comments, Bert and Mehmet! It is of course a serious
and interesting area to explore (and I'm aware of the chaos context;
I initially got into this areas year ago when I was exploring the
possibilities for chaos in fish population dynamics -- and they're
certainly there)!
But, before anyone takes my posting *too* seriously, let me say that
it was written tongue-in-cheek (or whatever the keyboard analogue of
that may be). I'm certainly not "blaming R".
Have fun anyway!
Ted.
On 22-Dec-2013 17:35:56 Bert Gunter wrote:
> Yes.
>
> See also Feigenbaum's constant and chaos theory for the general context.
>
> Cheers,
> Bert
>
> On Sun, Dec 22, 2013 at 8:54 AM, Suzen, Mehmet <msuzen at gmail.com> wrote:
>> I wouldn't blame R for floating-point arithmetic and our personal
>> feeling of what 'zero' should be.
>>
>>> options(digits=20)
>>> pi
>> [1] 3.141592653589793116
>>> sqrt(pi)^2
>> [1] 3.1415926535897926719
>>> (pi - sqrt(pi)^2) < 1e-15
>> [1] TRUE
>>
>> There was a similar post before, for example see:
>> http://r.789695.n4.nabble.com/Why-does-sin-pi-not-return-0-td4676963.html
>>
>> There is an example by Martin Maechler (author of Rmpfr) on how to use
>> arbitrary precision
>> with your arithmetic.
>>
>> On 22 December 2013 10:59, Ted Harding <Ted.Harding at wlandres.net> wrote:
>>> Greetings All!
>>> With the Festive Season fast approaching, I bring you joy
>>> with the news (which you will surely wish to celebrate)
>>> that R cannot do arithmetic!
>>>
>>> Usually, this is manifest in a trivial way when users report
>>> puzzlement that, for instance,
>>>
>>> sqrt(pi)^2 == pi
>>> # [1] FALSE
>>>
>>> which is the result of a (generally trivial) rounding or
>>> truncation error:
>>>
>>> sqrt(pi)^2 - pi
>>> [1] -4.440892e-16
>>>
>>> But for some very simple calculations R goes off its head.
>>>
>>> I had originally posted this example some years ago, but I
>>> have since generalised it, and the generalisation is even
>>> more entertaining than the original.
>>>
>>> The Original:
>>> Consider a sequence generated by the recurrence relation
>>>
>>> x[n+1] = 2*x[n] if 0 <= x[n] <= 1/2
>>> x[n+1] = 2*(1 - x[n]) if 1/2 < x[n] <= 1
>>>
>>> (for 0 <= x[n] <= 1).
>>>
>>> This has equilibrium points (x[n+1] = x[n]) at x[n] = 0
>>> and at x[n] = 2/3:
>>>
>>> 2/3 -> 2*(1 - 2/3) = 2/3
>>>
>>> It also has periodic points, e.g.
>>>
>>> 2/5 -> 4/5 -> 2/5 (period 2)
>>> 2/9 -> 4/9 -> 8/9 -> 2/9 (period 3)
>>>
>>> The recurrence relation can be implemented as the R function
>>>
>>> nextx <- function(x){
>>> if( (0<=x)&(x<=1/2) ) {x <- 2*x} else {x <- 2*(1 - x)}
>>> }
>>>
>>> Now have a look at what happens when we start at the equilibrium
>>> point x = 2/3:
>>>
>>> N <- 1 ; x <- 2/3
>>> while(x > 0){
>>> cat(sprintf("%i: %.9f\n",N,x))
>>> x <- nextx(x) ; N <- N+1
>>> }
>>> cat(sprintf("%i: %.9f\n",N,x))
>>>
>>> Run that, and you will see that successive values of x collapse
>>> towards zero. Things look fine to start with:
>>>
>>> 1: 0.666666667
>>> 2: 0.666666667
>>> 3: 0.666666667
>>> 4: 0.666666667
>>> 5: 0.666666667
>>> ...
>>>
>>> but, later on,
>>>
>>> 24: 0.666666667
>>> 25: 0.666666666
>>> 26: 0.666666668
>>> 27: 0.666666664
>>> 28: 0.666666672
>>> ...
>>>
>>> 46: 0.667968750
>>> 47: 0.664062500
>>> 48: 0.671875000
>>> 49: 0.656250000
>>> 50: 0.687500000
>>> 51: 0.625000000
>>> 52: 0.750000000
>>> 53: 0.500000000
>>> 54: 1.000000000
>>> 55: 0.000000000
>>>
>>> What is happening is that, each time R multiplies by 2, the binary
>>> representation is shifted up by one and a zero bit is introduced
>>> at the bottom end. To illustrate this, do the calculation in
>>> 7-bit arithmetic where 2/3 = 0.1010101, so:
>>>
>>> 0.1010101 x[1], >1/2 so subtract from 1 = 1.0000000 -> 0.0101011,
>>> and then multiply by 2 to get x[2] = 0.1010110. Hence
>>>
>>> 0.1010101 x[1] -> 2*(1 - 0.1010101) = 2*0.0101011 ->
>>> 0.1010110 x[2] -> 2*(1 - 0.1010110) = 2*0.0101010 ->
>>> 0.1010100 x[3] -> 2*(1 - 0.1010100) = 2*0.0101100 ->
>>> 0.1011000 x[4] -> 2*(1 - 0.1011000) = 2*0.0101000 ->
>>> 0.1010000 x[5] -> 2*(1 - 0.1010000) = 2*0.0110000 ->
>>> 0.1100000 x[6] -> 2*(1 - 0.1100000) = 2*0.0100000 ->
>>> 0.1000000 x[7] -> 2*0.1000000 = 1.0000000 ->
>>> 1.0000000 x[8] -> 2*(1 - 1.0000000) = 2*0 ->
>>> 0.0000000 x[9] and the end of the line.
>>>
>>> The final index of x[i] is i=9, 2 more than the number of binary
>>> places (7) in this arithmetic, since 8 successive zeros have to
>>> be introduced. It is the same with the real R calculation since
>>> this is working to .Machine$double.digits = 53 binary places;
>>> it just takes longer (we reach 0 at x[55])! The above collapse
>>> to 0 occurs for any starting value in this simple example (except
>>> for multiples of 1/(2^k), when it works properly).
>>>
>>> Generalisation:
>>> This is basically the same, except that everything is multiplied
>>> by a scale factor S, so instead of being on the interval [0,1].
>>> it is on [0,S], and
>>>
>>> x[n+1] = 2*x[n] if 0 <= x[n] <= S/2
>>> x[n+1] = 2*(S - x[n]) if S/2 < x[n] <= S
>>> (for 0 <= x[n] <= S).
>>>
>>> Again, x[n] = 2*S/3 is an equilibrium point. 2*S/3 > S/2, so
>>>
>>> x[n] -> 2*(S - 2*S/3) = 2*(S/3) = 2*S/3
>>>
>>> Functions to implement this:
>>>
>>> nxtS <- function(x,S){
>>> if((x >= 0)&(x <= S/2)){ x<- 2*x } else {x <- 2*(S-x)}
>>> }
>>>
>>> S <- 6 ## Or some other value of S
>>> Nits <- 100
>>> x <- 2*S/3
>>> N <- 1 ; print(c(N,x))
>>> while(x>0){
>>> if(N > Nits) break ### to stop infinite looping
>>> N <- (N+1) ; x <- nxtS(x,S)
>>> print(c(N,x))
>>> }
>>>
>>> The behaviour of the sequence now depends on the value of S.
>>>
>>> If S is a multiple of 3, then with x[1] = 2*S/3 the equilibrium
>>> is immediately attained and x[n] = 2*S/3 forever after, since
>>> R is now calculating with integers. E.g. try the above with S<-6
>>> That is what arithmetic ought to be like! But for S not a multiple
>>> of 3 one can get the impression that R is on some sort of drug!
>>>
>>> For other values of S (but not all) we observe the same collapse
>>> to x=0 as before, and again it takes 54 steps (ending with x[55]).
>>> Try e.g. S <- 16
>>>
>>> For some values of S, however, the iteration ends up in a periodic loop.
>>>
>>> For example, with S<-7, at x[52] we get x[52]=4, x[53]=6, x[54]=2,
>>> and then 4 6 2 4 6 2 4 6 2 ... forever (or until Nits cuts in),
>>> so period = 3.
>>>
>>> For S<-11, x[52]=8 then 6 then 10 then 2 then 4 then 8 6 10 2 4 ...
>>> so period = 5.
>>>
>>> For S<-13, x[51]=4 then 8 10 6 12 2 4 8 10 6 12 2 4 8 ...
>>> so period = 6.
>>>
>>> For S<-19, x[51]=12 then 14 10 18 2 4 8 16 6 12 ...
>>> so period = 9.
>>>
>>> And so on ...
>>>
>>> So, one sniff of something like S<-19, and R is off its head!
>>>
>>> All it has to do is multiply by 2 -- and it gets it cumulatively wrong!
>>> R just doesn't add up ...
>>>
>>> Season's Greetings to all -- and may your calculations always
>>> be accurate -- to within machine precision ...
>>>
>>> Ted.
>>>
>>> -------------------------------------------------
>>> E-Mail: (Ted Harding) <Ted.Harding at wlandres.net>
>>> Date: 22-Dec-2013 Time: 09:59:00
>>> This message was sent by XFMail
>>>
>>> ______________________________________________
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>>> PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
>
>
> --
>
> Bert Gunter
> Genentech Nonclinical Biostatistics
>
> (650) 467-7374
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
-------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at wlandres.net>
Date: 22-Dec-2013 Time: 18:37:15
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