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.