[R] Season's Greetings (and great news ... )!

Sun Dec 22 18:35:56 CET 2013

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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>> 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