Re: [lmi] TAOCP [4.2.2]: to what can zero be "essentially equal"?

[Vadim Zeitlin]

On Thu, 15 Feb 2018 00:45:15 +0000 Greg Chicares <address@hidden> wrote:

GC> Professor--in 'materially_equal.hpp' I claim:
GC> 
GC> /// This is the "essentially equal to" operator defined in Knuth,
GC> /// TAOCP, volume 2, section 4.2.2, equation (34), on page 219 of the
GC> /// second edition.
GC> 
GC>   tolerance = 1.0E-13
GC>   std::fabs(t - u) <= tolerance * std::min(std::fabs(t), std::fabs(u))
GC> 
GC> But is that comment rigorous, even though I seem to have translated
GC> equation (34) faithfully?

 AFAICS it isn't, the condition is indeed sufficient but not required.

GC> (1) Knuth says that satisfying this test implies "essential" equality,
GC> so it's a sufficient condition, and not a (biconditional) definition.
GC> But I don't find anything in the text that's closer to a definition.

 Although it's also given as an implication, rather than definition, I find
it useful to think in terms of the neighbourhoods defined in the text just
before this group of equations. If N(u) is the neighbourhood of u (with
the given implicit choice of fixed epsilon), then

        u ≈ v ⇔ u ∈ N(v)  ∧  v ∈ N(u)

which looks like a nice definition.

GC> (2) Can any nonzero value be considered "essentially" equal to zero?

 According to the definition above, yes.

GC> The above code determines that 1e-100 is not essentially zero for the
GC> given tolerance of 1e-13;

 This is incompatible with the definition above. Whether we want to use
this definition or not is another question.

GC> is that sensible? and would Knuth agree or disagree with that
GC> conclusion--or has he perhaps chosen to leave the answer undefined?

 This is hard to say. All I can provide here is a cop out: I think it's
sensible in Knuth's framework, but I'm not so sure that it is for the lmi
use of it. But I'm not exactly sure how does lmi use it exactly, e.g.
looking at the code in premium_tax::calculate_load() I have really no idea
what is it supposed to do when x or y are 0 (nor even what they actually
are).

 Sorry for lack of any real help,
VZ