

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,] 2.664427e17
>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 
2124605430 home
9176565351 cell
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


This is giving you exactly what you are asking for. The operator * does
element by element multiplication. So, .48 + .48 =0, right? Is there
another mathematical possibility you were expecting?
> Original Message
> From: [hidden email]
> [mailto: [hidden email]] On Behalf Of Talbot Katz
> Sent: Friday, July 27, 2007 6:31 PM
> To: [hidden email]
> Subject: [R] Matrix Multiplication, FloatingPoint, etc.
>
> 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,] 2.664427e17
> >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 
> 2124605430 home
> 9176565351 cell
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/rhelp> PLEASE do read the posting guide
> http://www.Rproject.org/postingguide.html> and provide commented, minimal, selfcontained, reproducible code.
>
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


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,] 2.664427e17
>> 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 
> 2124605430 home
> 9176565351 cell
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/rhelp> PLEASE do read the posting guide http://www.Rproject.org/postingguide.html> and provide commented, minimal, selfcontained, reproducible code.
>
Charles C. Berry (858) 5342098
Dept of Family/Preventive Medicine
E mailto: [hidden email] UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 920930901
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


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 
2124605430 home
9176565351 cell
>From: "Charles C. Berry" < [hidden email]>
>To: Talbot Katz < [hidden email]>
>CC: [hidden email]
>Subject: Re: [R] Matrix Multiplication, FloatingPoint, 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,] 2.664427e17
>>>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 
>>2124605430 home
>>9176565351 cell
>>
>>______________________________________________
>> [hidden email] mailing list
>> https://stat.ethz.ch/mailman/listinfo/rhelp>>PLEASE do read the posting guide
>> http://www.Rproject.org/postingguide.html>>and provide commented, minimal, selfcontained, reproducible code.
>>
>
>Charles C. Berry (858) 5342098
> Dept of Family/Preventive
>Medicine
>E mailto: [hidden email] UC San Diego
> http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 920930901
>
>
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


Talbot
The general advice on this list is to read the following
http://docs.sun.com/source/8063568/ncg_goldberg.html
> Original Message
> From: Talbot Katz [mailto: [hidden email]]
> Sent: Monday, July 30, 2007 1:55 PM
> To: [hidden email]
> Cc: [hidden email]; Doran, Harold
> Subject: Re: [R] Matrix Multiplication, FloatingPoint, etc.
>
> 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 
> 2124605430 home
> 9176565351 cell
>
>
>
> >From: "Charles C. Berry" < [hidden email]>
> >To: Talbot Katz < [hidden email]>
> >CC: [hidden email]
> >Subject: Re: [R] Matrix Multiplication, FloatingPoint, 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,] 2.664427e17
> >>>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 
> >>2124605430 home
> >>9176565351 cell
> >>
> >>______________________________________________
> >> [hidden email] mailing list
> >> https://stat.ethz.ch/mailman/listinfo/rhelp> >>PLEASE do read the posting guide
> >> http://www.Rproject.org/postingguide.html> >>and provide commented, minimal, selfcontained, reproducible code.
> >>
> >
> >Charles C. Berry (858) 5342098
> > Dept of
> Family/Preventive
> >Medicine
> >E mailto: [hidden email] UC San Diego
> > http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego
> >920930901
> >
> >
>
>
>
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


One thing to realize is that although it appears that the operations
are the same, the code that is being executed is different in the two
cases. Due to the different sequence of instructions, there may be
roundoff errors that are then introduced
On 7/30/07, Talbot Katz < [hidden email]> 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 
> 2124605430 home
> 9176565351 cell
>
>
>
> >From: "Charles C. Berry" < [hidden email]>
> >To: Talbot Katz < [hidden email]>
> >CC: [hidden email]
> >Subject: Re: [R] Matrix Multiplication, FloatingPoint, 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,] 2.664427e17
> >>>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 
> >>2124605430 home
> >>9176565351 cell
> >>
> >>______________________________________________
> >> [hidden email] mailing list
> >> https://stat.ethz.ch/mailman/listinfo/rhelp> >>PLEASE do read the posting guide
> >> http://www.Rproject.org/postingguide.html> >>and provide commented, minimal, selfcontained, reproducible code.
> >>
> >
> >Charles C. Berry (858) 5342098
> > Dept of Family/Preventive
> >Medicine
> >E mailto: [hidden email] UC San Diego
> > http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 920930901
> >
> >
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/rhelp> PLEASE do read the posting guide http://www.Rproject.org/postingguide.html> and provide commented, minimal, selfcontained, reproducible code.
>

Jim Holtman
Cincinnati, OH
+1 513 646 9390
What is the problem you are trying to solve?
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


> Original Message
> From: [hidden email] [mailto: [hidden email]]
> On Behalf Of Talbot Katz
> Sent: Monday, July 30, 2007 10:55 AM
> To: [hidden email]
> Cc: [hidden email]; [hidden email]
> Subject: Re: [R] Matrix Multiplication, FloatingPoint, etc.
>
> 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 
> 2124605430 home
> 9176565351 cell
>
There may other issues involved here besides R version, floating point precision, and OS version. On my WinXP system running R2.5.1 binary from CRAN, I get what you expected:
> ev2<c(0.8,0.6)
> ev1<c(0.6,0.8)
> ev1%*%ev2
[,1]
[1,] 0
>
There could be differences in OS release, service packs installed, cpu, etc. But the moral you draw is probably a reasonable one.
Dan
Daniel Nordlund
Bothell, WA
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


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 nonstraightforward 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 < [hidden email]> 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 
> 2124605430 home
> 9176565351 cell
>
>
>
> >From: "Charles C. Berry" < [hidden email]>
> >To: Talbot Katz < [hidden email]>
> >CC: [hidden email]
> >Subject: Re: [R] Matrix Multiplication,
> FloatingPoint, 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,] 2.664427e17
> >>>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 
> >>2124605430 home
> >>9176565351 cell
> >>
> >>______________________________________________
> >> [hidden email] mailing list
> >> https://stat.ethz.ch/mailman/listinfo/rhelp> >>PLEASE do read the posting guide
> >> http://www.Rproject.org/postingguide.html> >>and provide commented, minimal, selfcontained,
> reproducible code.
> >>
> >
> >Charles C. Berry (858)
> 5342098
> > Dept
> of Family/Preventive
> >Medicine
> >E mailto: [hidden email] UC San
> Diego
> > http://famprevmed.ucsd.edu/faculty/cberry/ La
> Jolla, San Diego 920930901
> >
> >
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/rhelp> PLEASE do read the posting guide
> http://www.Rproject.org/postingguide.html> and provide commented, minimal, selfcontained,
> reproducible code.
>
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.


On Mon, 30 Jul 2007, Moshe Olshansky wrote:
> 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 nonstraightforward 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.
Mr Olshansky seems unaware of the effects of extendedprecision
intermediate arithmetic on ix86 CPUs.
sum() does use a higherprecision accumulator (where available, including
on Windows), but ev1*ev2 is done in R and so stored to basic precision.
OTOH, %*% (sic) calls the BLAS routine dgemm and hence may accumulate in
80bit floatingpoint registers. What result you get will depend on what
compiler, compiler flags and BLAS is in use, but with the default
reference BLAS it is very likely that some of the intermediate results are
stored in FP registers to extended precision.
It is a simple experiment to confirm this: recompile the BLAS with
fforcestore and you do get 0 (at least on my Windows build system).
Let's see less speculation and more homework in future.
>
> Regards,
>
> Moshe.
>
>  Talbot Katz < [hidden email]> 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 
>> 2124605430 home
>> 9176565351 cell
>>
>>
>>
>>> From: "Charles C. Berry" < [hidden email]>
>>> To: Talbot Katz < [hidden email]>
>>> CC: [hidden email]
>>> Subject: Re: [R] Matrix Multiplication,
>> FloatingPoint, 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,] 2.664427e17
>>>>> 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 
>>>> 2124605430 home
>>>> 9176565351 cell

Brian D. Ripley, [hidden email]
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272866 (PA)
Oxford OX1 3TG, UK Fax: +44 1865 272595
______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/rhelpPLEASE do read the posting guide http://www.Rproject.org/postingguide.htmland provide commented, minimal, selfcontained, reproducible code.

