Re: Re: Number of interval Intersections for a large number of intervals

*To*: mathgroup at smc.vnet.net*Subject*: [mg81767] Re: [mg81718] Re: [mg81679] Number of interval Intersections for a large number of intervals*From*: DrMajorBob <drmajorbob at bigfoot.com>*Date*: Wed, 3 Oct 2007 02:29:23 -0400 (EDT)*References*: <4236415.1191246194428.JavaMail.root@m35> <200710020936.FAA28508@smc.vnet.net> <31150092.1191354562039.JavaMail.root@m35>*Reply-to*: drmajorbob at bigfoot.com

I realized, after sending, that it's only the Piecewise discontinuities that Exclusions detects, not the others caused by InterpolationOrder->0. It appears that Exclusions occur only at min and minHigh, not at max and maxLow, yet that's an illusion, as these plots show: Plot[count2@x, {x, maxLow - 0.1, maxLow + 0.1}] Plot[count2@x, {x, max - 0.1, max + 0.1}] count2 is constant near maxLow and max, but Exclusions causes a spurious gap in the horizontal line at those points. That gap is present but invisible, I suppose, in the broader chart, Plot[count2@x, {x, -1, 1}] Bobby On Tue, 02 Oct 2007 09:41:43 -0500, Brett Champion <brettc at wolfram.com> wrote: > On Oct 2, 2007, at 4:36 AM , DrMajorBob wrote: > >> [WRI: there's a Plot/Piecewise insect (bug?) noted below.] >> >> I'm having to guess a bit, but maybe this is what you mean: >> >> intervals = Sort /@ RandomReal[{-1, 1}, {10^5, 2}]; >> trans = Transpose@intervals; >> {{min, maxLow}, {minHigh, max}} = Through[{Min, Max}@#] & /@ trans; >> {low, high} = >> Interpolation[Transpose@{Sort@#, -1 + Range@Length@#}, >> InterpolationOrder -> 0] & /@ trans; >> >> Clear[count] >> count[x_] /; x < min = 0; >> count[x_] /; x > max = 0; >> count[x_] /; x < minHigh := low[x] >> count[x_] /; x > maxLow := low[maxLow] - high[x] >> count[x_] := low[x] - high[x]; >> First@Timing@Plot[count@x, {x, -1, 1}] >> >> 0.031 >> >> or >> >> Clear[count2] >> count2[x_] = >> Piecewise[{{0, x < min}, {low[x], x < minHigh}, {low[x] - high[x], >> x <= maxLow}, {low[maxLow] - high[x], x <= max}}]; >> First@Timing@Plot[count2@x, {x, -1, 1}] >> >> 0.047 >> >> INSECT report: If I use a small number of Intervals -- 10, say -- the >> two >> plots are different. The Piecewise version doesn't have "returns" at >> some >> of the discontinuities, even though the following plots the zero >> function >> as expected: >> >> Plot[count2@x - count@x, {x, -1, 1}] >> >> Is this an example of Exclusions->Automatic at work? >> >> If so, why aren't all the returns omitted? >> > > Conditional definitions of functions, such as used to define count[], > are not detected by the automatic calculation of exclusions. I don't > think this is a bug, although it is an interesting idea. > > In the case of count2[], there are only going to be exclusions at the > Piecewise boundaries: x==min, x==max, x==minHigh, x==maxLow. The > interpolating function used by low[] and high[] will be a black box to > it. > > Brett Champion > Wolfram Research > -- DrMajorBob at bigfoot.com

**References**:**Re: Number of interval Intersections for a large number of intervals***From:*DrMajorBob <drmajorbob@bigfoot.com>