[R] Matrix Multiplication, Floating-Point, etc.

[Moshe Olshansky]

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 <topkatz at msn.com> 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" <cberry at tajo.ucsd.edu>
> >To: Talbot Katz <topkatz at msn.com>
> >CC: r-help at 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 at 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:cberry at tajo.ucsd.edu	            UC San
> Diego
> >http://famprevmed.ucsd.edu/faculty/cberry/  La
> Jolla, San Diego 92093-0901
> >
> >
> 
> ______________________________________________
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> PLEASE do read the posting guide
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> and provide commented, minimal, self-contained,
> reproducible code.
>