>>>>> Ashim Kapoor <ashimkap...@gmail.com>
>>>>> on Tue, 25 Apr 2017 14:02:18 +0530 writes:
> Dear all,
> I am not able to understand the interplay of absolute vs relative and
> tolerance in the use of all.equal
> If I want to find out if absolute differences between 2 numbers/vectors
are
> bigger than a given tolerance I would do:
> all.equal(1,1.1,scale=1,tol= .1)
> If I want to find out if relative differences between 2 numbers/vectors
are
> bigger than a given tolerance I would do :
> all.equal(1,1.1,tol=.1)
>
##################################################################################################################################
> I can also do :
> all.equal(1,3,tol=1)
> to find out if the absolute difference is bigger than 1.But here I won't
be
> able to detect absolute differences smaller than 1 in this case,so I don't
> think that this is a good way.
> My query is: what is the reasoning behind all.equal returning the absolute
> difference if the tolerance >= target and relative difference if tolerance
> < target?
(above, it is tol >/<= |target| ie. absolute value)
The following are desiderata / restrictions :
1) Relative tolerance is needed to keep things scale-invariant
i.e., all.equal(x, y) and all.equal(1000 * x, 1000 * y)
should typically be identical for (almost) all (x,y).
==> "the typical behavior should use relative error tolerance"
2) when x or y (and typically both!) are very close to zero it
is typically undesirable to keep relative tolerances (in the
boundary case, they _are_ zero exactly, and "relative error" is undefined).
E.g., for most purposes, 3.45e-15 and 1.23e-17 should be counted as
equal to zero and hence to themselves.
1) and 2) are typically reconciled by switching from relative to absolute
when the arguments are close to zero (*).
The exact cutoff at which to switch from relative to absolute
(or a combination of the two) is somewhat arbitrary(*2) and for
all.equal() has been made in the 1980's (or even slightly
earlier?) when all.equal() was introduced into the S language at
Bell labs AFAIK. Maybe John Chambers (or Rick Becker or ...,
but they may not read R-help) knows more.
*2) Then, the choice for all.equal() is in some way "least arbitrary",
using c = 1 in the more general tolerance >= c*|target| framework.
*) There have been alternatives in "the (applied numerical
analysis / algorithm) literature" seen in published algorithms,
but I don't have any example ready.
Notably some of these alternatives are _symmetric_ in (x,y)
where all.equal() was designed to be asymmetric using names
'target' and 'current'.
The alternative idea is along the following thoughts:
Assume that for "equality" we want _both_ relative and
absolute (e := tolerance) "equality"
|x - y| < e (|x|+|y|)/2 (where you could use |y| or |x|
instead of their mean; all.equal()
uses |target|)
|x - y| < e * e1 (where e1 = 1, or e1 = 10^-7..)
If you add the two inequalities you get
|x - y| < e (e1 + |x+y|/2)
as check which is a "mixture" of relative and absolute tolerance.
With a somewhat long history, my gut feeling would nowadays
actually prefer this (I think with a default of e1 = e) - which
does treat x and y symmetrically.
Note that convergence checks in good algorithms typically check
for _both_ relative and absolute difference (each with its
tolerance providable by the user), and the really good ones for
minimization do check for (approximate) gradients also being
close to zero - as old timers among us should have learned from
Doug Bates ... but now I'm really diverging.
Last but not least some R code at the end, showing that the *asymmetric*
nature of all.equal() may lead to somewhat astonishing (but very
logical and as documented!) behavior.
Martin
> Best Regards,
> Ashim
> ## The "data" to use:
> epsQ <- lapply(seq(12,18,by=1/2), function(P) bquote(10^-.(P))); names(epsQ)
> <- sapply(epsQ, deparse); str(epsQ)
List of 13
$ 10^-12 : language 10^-12
$ 10^-12.5: language 10^-12.5
$ 10^-13 : language 10^-13
$ 10^-13.5: language 10^-13.5
$ 10^-14 : language 10^-14
$ 10^-14.5: language 10^-14.5
$ 10^-15 : language 10^-15
$ 10^-15.5: language 10^-15.5
$ 10^-16 : language 10^-16
$ 10^-16.5: language 10^-16.5
$ 10^-17 : language 10^-17
$ 10^-17.5: language 10^-17.5
$ 10^-18 : language 10^-18
> str(lapply(epsQ, function(tl) all.equal(3.45e-15, 1.23e-17, tol = eval(tl))))
List of 13
$ 10^-12 : logi TRUE
$ 10^-12.5: logi TRUE
$ 10^-13 : logi TRUE
$ 10^-13.5: logi TRUE
$ 10^-14 : logi TRUE
$ 10^-14.5: chr "Mean relative difference: 0.9964348"
$ 10^-15 : chr "Mean relative difference: 0.9964348"
$ 10^-15.5: chr "Mean relative difference: 0.9964348"
$ 10^-16 : chr "Mean relative difference: 0.9964348"
$ 10^-16.5: chr "Mean relative difference: 0.9964348"
$ 10^-17 : chr "Mean relative difference: 0.9964348"
$ 10^-17.5: chr "Mean relative difference: 0.9964348"
$ 10^-18 : chr "Mean relative difference: 0.9964348"
> ## Now swap `target` and `current` :
> str(lapply(epsQ, function(tl) all.equal(1.23e-17, 3.45e-15, tol = eval(tl))))
List of 13
$ 10^-12 : logi TRUE
$ 10^-12.5: logi TRUE
$ 10^-13 : logi TRUE
$ 10^-13.5: logi TRUE
$ 10^-14 : logi TRUE
$ 10^-14.5: chr "Mean absolute difference: 3.4377e-15"
$ 10^-15 : chr "Mean absolute difference: 3.4377e-15"
$ 10^-15.5: chr "Mean absolute difference: 3.4377e-15"
$ 10^-16 : chr "Mean absolute difference: 3.4377e-15"
$ 10^-16.5: chr "Mean absolute difference: 3.4377e-15"
$ 10^-17 : chr "Mean relative difference: 279.4878"
$ 10^-17.5: chr "Mean relative difference: 279.4878"
$ 10^-18 : chr "Mean relative difference: 279.4878"
>
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