Re: Re: Re: Re: More /.{I->-1} craziness. Schools

*To*: mathgroup at smc.vnet.net*Subject*: [mg106955] Re: [mg106925] Re: [mg106656] Re: [mg106882] Re: More /.{I->-1} craziness. Schools*From*: Andrzej Kozlowski <akozlowski at gmail.com>*Date*: Fri, 29 Jan 2010 07:47:56 -0500 (EST)*References*: <hjbvc0$2tp$1@smc.vnet.net> <hjeqh1$g3c$1@smc.vnet.net> <201001280745.CAA23668@smc.vnet.net>

The definition of precision in Mathematica is this. Suppose x is a number known up to an error of epsilon, that is it can be viewed as lying in the interval (x-epsilon/2,x+epsilon/2). Then its precision is -Log[10,epsilon/x]. Its accuracy is -Log[10,epsilon]. The two are related by the equation: Precision[x] - Accuracy[x] == RealExponent[x] The interpretation in terms of digits is only approximate. Both accuracy and precision can be negative - this depends on the scale of the number i.e. RealExponent. A number will have negative accuracy if its absolute error is large. It is easy to produce such numbers by cancellation With[{x = N[10^100, 50] - N[10^100, 50]}, Accuracy[x]] -50.301 On the other hand, since $MinPrecision 0 You won't normally in Mathematica see numbers with negative Precision. Precision is the main concept, Accuracy is only used because Precision is singular at 0 (remember - its relative error). It's all perfectly documented so this tired scape goat is not available this time. Andrzej Kozlowski On 28 Jan 2010, at 08:45, DrMajorBob wrote: > OK... so numbers are allowed to have NEGATIVE precision? > > LESS than zero digits of accuracy? Really? > > Whatever for? > > Bobby > > On Wed, 27 Jan 2010 18:23:32 -0600, Daniel Lichtblau <danl at wolfram.com> > wrote: > >> DrMajorBob wrote: >>> 0 and 1 are not "fuzzballs", so what interval could be >= 1 and also 0.? >>> Bobby >> >> I had in mind the spoiler answer Richard Fateman provided in his first >> post mentioning this particular tangent, err, example. >> >> http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00638.html >> >> At the bottom we find: >> --- >> I would especially avoid .nb objects, and most especially on topics of >> numerical analysis, where the design flaws are, in my opinion, so >> fundamental. Example (mathematica 7.0): >> {x >= 1, x > 1, x > 0, x} >> evaluates to >> {True, False, False, 0.} >> >> can you construct x? >> >> RJF >> >> One possible answer, below.... >> >> x=0``-.5 >> --- >> >> The point is that with Mathematica's version of significance arithmetic, >> equality, I believe, is effectively treated as having a nontrivial an >> intersection (of the implicit intervals defining two numbers). If >> neither has any fuzz (i.e. both are exact), then Equal allows for no >> fuzz, so this is only a subtlety if at least one of the values is >> approximate. >> >> One implication is that a "zero" of sufficiently low (as in bad) >> accuracy can be regarded as 1, or -1, or Pi, if those values happen to >> fall within the accuracy (which I refer to as fuzz). >> >> The other inequalities follow from the preservation of trichotomy. For >> explicitly real values we regard that as important. mathematica makes no >> pretense that Equal is transitive and I do not see any way to do that >> and also have useful approximate arithmetic. >> >> There has been some amount of communication off-line on this topic, >> which is why some of us (well, me, at least) sometimes forget the >> examples are not universally obvious to those who have not memorized the >> enitre thread. >> >> Daniel >> >> >>> On Wed, 27 Jan 2010 00:44:22 -0600, Daniel Lichtblau <danl at wolfram.com> >>> wrote: >>> >>>> Richard Fateman wrote: >>>>> [...] >>>>> If all of Mathematica functionality were available in the free player >>>>> version, WRI would need to drastically change its business model. And >>>>> even it it were free, we still have behavior like this: (..for some >>>>> values of zero) >>>>> >>>>> {x >== 1, x > 0, x} evaluates to {True, False, 0.} >>>>> >>>>> RJF >>>> >>>> Let's take simple intervals, that is, intervals that are segments. >>>> Define less and greater in the obvious ways, that is, one segment lies >>>> strictly below the other (right endpoint of lesser is less than left >>>> endpoint of larger). Let us further define two intervals to be equal >>>> whenever they have nonempty intersection. >>>> >>>> With these definitions, which I think are sensible, the behavior you >>>> describe above is consistent with arithmetic on intervals. As the >>>> numbers involved, at least some of them, are fuzzballs, this strikes me >>>> as an appropriate behavior. >>>> >>>> Daniel Lichtblau >>>> Wolfram Research >> > > > -- > DrMajorBob at yahoo.com >

**References**:**Re: Re: Re: More /.{I->-1} craziness. Schools***From:*DrMajorBob <btreat1@austin.rr.com>