More /.{I->-1} craziness. Schools
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From: DrMajorBob on 28 Jan 2010 02:44 0 and 1 are not "fuzzballs", so what interval could be >= 1 and also 0.? Bobby On Wed, 27 Jan 2010 00:44:22 -0600, Daniel Lichtblau <danl(a)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(a)yahoo.com
From: Daniel Lichtblau on 28 Jan 2010 02:44 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(a)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
From: DrMajorBob on 28 Jan 2010 02:45 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(a)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(a)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(a)yahoo.com
From: Richard Fateman on 29 Jan 2010 07:45 Daniel Lichtblau wrote: .... > > 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. > If you insist on equality meaning "overlap", then maybe you should give up on trichotomy for fuzz-balls, since its preservation leads to nonsense. That is, if x==0 and x==1 and x==2, on the grounds that the fuzzball for x includes 0,1,2, and x might "possibly be equal" to 0, 1, or 2, then doesn't it follow that x<1 and x>1 are also "possible", and therefore these are true too? In the interval arithmetic world these would be called 'possibly equal', 'possibly greater than' 'possibly less than' etc. As opposed to the more usual 'certainly equal' etc. RJF
From: Andrzej Kozlowski on 29 Jan 2010 07:47 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(a)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(a)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(a)yahoo.com >
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