After multiplication by 10 you get 6*8 = 48 - the result is an exact machine number so there is no roundoff, while 0.6*0.8 = 0.48, where neither of the 3 numbers (0.6, 0.8, 0.48) is an exact machine mumber. However, (-0.6)*0.8 should be equal EXACTLY to -(0.6*0.8), and in fact you get that sum(ev1*ev2) is exactly 0. What is strange is that you are not getting this result from ev1 %*% ev2. This means that either %^% uses some non-straightforward algorithm or it somehow sets the rounding control to something different from "round to nearest". In the later case (-0.6) does not necessarily equal to -(0.6) and the rounding after multiplication is not necessarily symetric.
Regards, Moshe. --- Talbot Katz <[EMAIL PROTECTED]> wrote: > Thank you for responding! > > I realize that floating point operations are often > inexact, and indeed, the > difference between the two answers is within the > all.equal tolerance, as > mentioned in FAQ 7.31 (cited by Charles): > > >(as.numeric(ev1%*%ev2))==(sum(ev1*ev2)) > [1] FALSE > >all.equal((as.numeric(ev1%*%ev2)),(sum(ev1*ev2))) > [1] TRUE > > > > I suppose that's good enough for numerical > computation. But I was still > surprised to see that matrix multiplication > (ev1%*%ev2) doesn't give the > exact right answer, whereas sum(ev1*ev2) does give > the exact answer. I > would've expected them to perform the same two > multiplications and one > addition. But I guess that's not the case. > > However, I did find that if I multiplied the two > vectors by 10, making the > entries integers (although the class was still > "numeric" rather than > "integer"), both computations gave equal answers of > 0: > > >xf1<-10*ev1 > >xf2<-10*ev2 > >(as.numeric(xf1%*%xf2))==(sum(xf1*xf2)) > [1] TRUE > > > > Perhaps the moral of the story is that one should > exercise caution and keep > track of significant digits. > > -- TMK -- > 212-460-5430 home > 917-656-5351 cell > > > > >From: "Charles C. Berry" <[EMAIL PROTECTED]> > >To: Talbot Katz <[EMAIL PROTECTED]> > >CC: r-help@stat.math.ethz.ch > >Subject: Re: [R] Matrix Multiplication, > Floating-Point, etc. > >Date: Mon, 30 Jul 2007 09:27:42 -0700 > > > > > > > >7.31 Why doesn't R think these numbers are equal? > > > >On Fri, 27 Jul 2007, Talbot Katz wrote: > > > >>Hi. > >> > >>I recently tried the following in R 2.5.1 on > Windows XP: > >> > >>>ev2<-c(0.8,-0.6) > >>>ev1<-c(0.6,0.8) > >>>ev1%*%ev2 > >> [,1] > >>[1,] -2.664427e-17 > >>>sum(ev1*ev2) > >>[1] 0 > >>> > >> > >>(I got the same result with R 2.4.1 on a different > Windows XP machine.) > >> > >>I expect this issue is very familiar and probably > has been discussed in > >>this > >>forum before. Can someone please point me to some > documentation or > >>discussion about this? Is there some standard way > to get the "correct" > >>answer from %*%? > >> > >>Thanks! > >> > >>-- TMK -- > >>212-460-5430 home > >>917-656-5351 cell > >> > >>______________________________________________ > >>R-help@stat.math.ethz.ch 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. > >> > > > >Charles C. Berry (858) > 534-2098 > > Dept > of Family/Preventive > >Medicine > >E mailto:[EMAIL PROTECTED] UC San > Diego > >http://famprevmed.ucsd.edu/faculty/cberry/ La > Jolla, San Diego 92093-0901 > > > > > > ______________________________________________ > R-help@stat.math.ethz.ch 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@stat.math.ethz.ch 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.
