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 <msu...@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.hard...@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.hard...@wlandres.net> >> Date: 22-Dec-2013 Time: 09:59:00 >> This message was sent by XFMail >> >> ______________________________________________ >> R-help@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. > > ______________________________________________ > R-help@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@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.