Re: Integrate in version 5.1

*To*: mathgroup at smc.vnet.net*Subject*: [mg52910] Re: [mg52880] Integrate in version 5.1*From*: DrBob <drbob at bigfoot.com>*Date*: Thu, 16 Dec 2004 03:41:23 -0500 (EST)*References*: <200412150927.EAA10723@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

>> Also there still exist some examples where reevaluating the same >> expression leads to a different outcome: I get the same answer for both integrals with 5.1 -- (2*Pi*r^3)/3 -- with either ClearCache OR Quit beforehand. Bobby On Wed, 15 Dec 2004 04:27:40 -0500 (EST), Maxim <ab_def at prontomail.com> wrote: > In version 5.1: > > In[1]:= > Integrate[1/z, {z, z1, z2}] > > Out[1]= > (-z1 + z2)*If[Re[z1/(z1 - z2)] >= 1 || Re[z1/(-z1 + z2)] >= 0 || > Im[z1/(-z1 + z2)] != 0, > (Log[z1] - Log[z2])/(z1 - z2), <<1>>] > > In a way this is a step back: Mathematica 5.0 generated a condition > that described the positions of the points z1, z2 such that the > segment [z1, z2] and the branch cut of Log[z] do not intersect > (although there was an error in the case of one of the endpoints z1, > z2 lying on the branch cut). Now the condition in Out[1] just > specifies that [z1, z2] doesn't go through the origin; thus Out[1] is > incorrect for any values of z1, z2 such that [z1, z2] intersects the > ray (-Infinity, 0), e.g. z1=-1-I and z2=-1+I. But then I fail to see > any point in generating a condition at all, if this condition is not > generically valid. > > Many of the typical integration bugs of version 5.0 are still present > in 5.1: > > In[2]:= > Integrate[DiracDelta''[x^2 - 1]*x^2, {x, -Infinity, Infinity}] > > Out[2]= > -3/4 > > The correct value is -1/4. This is not a single bug; replacing the > integrand with DiracDelta''[x^2 - 1]*phi[x] shows that the result > doesn't contain the terms with phi'', and pretty much all the > integrals with higher derivatives of DiracDelta will come out wrong. > Applying FunctionExpand to the integrand fixes the problem. > > It is strange that even some kernel crash issues haven't been fixed, > as with > > NIntegrate[PolyLog[2, E^(-I*y - 1)], {y, 0, 2*Pi}], > > which crashes the kernel in versions 5.0 and 5.1. > > Also there still exist some examples where reevaluating the same > expression leads to a different outcome: > > In[3]:= > Developer`ClearCache[] > Assuming[r > 0, > Integrate[Sqrt[r^2 - x^2 - y^2], > {x, -r, r}, {y, -Sqrt[r^2 - x^2], Sqrt[r^2 - x^2]}]] > Assuming[r > 0, > Integrate[Sqrt[r^2 - x^2 - y^2], > {x, -r, r}, {y, -Sqrt[r^2 - x^2], Sqrt[r^2 - x^2]}]] > > Out[4]= > (-2*Pi*r^3)/3 > > Out[5]= > (2*Pi*r^3)/3 > > This is a very regrettable bug, because it will affect many standard > integrals over circular regions. What makes this example so peculiar > is that the outcome depends on the hardware. I've tried this example > on two different systems, both running Mathematica 5.1 for Windows: on > Pentium II 450 this glitch is always reproducible, but on Athlon 64 > 3800 I always get the correct result. The explanation may be quite > simple: if somewhere in the integration code there is a line like > Simplify[expr, TimeConstraint -> 5], then a slow machine doesn't have > enough time to simplify the expression and the integration (or taking > the limit or some other complex operation) proceeds with the > unsimplified integrand for which the integration gives a wrong result. > On a fast machine the integrand is reduced to a simple form, and then > the integration works fine. This also explains why the second try on a > slow machine works correctly: it uses cached results, thus saving time > to advance the simplification further. > > Sometimes the changes in the new version break previously working > code: > > Integrate[ > (-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)* > UnitStep[-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3], > {x1, 0, 1}, {x2, 0, x1}, {x3, 0, x2}, > {y1, 0, 1}, {y2, 0, 1}, {y3, 0, 1}] > > The calculus`integration package could handle this integral (it took > approximately 45 seconds on a slow PC), but Mathematica 5.1 locks up > on it -- it generates $RecursionLimit::reclim messages and is still > running after 20 minutes. The irony is that calculus`integration is no > longer included in version 5.1. > > Next, working with a similar integrand, > > NIntegrate[ > Abs[-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3], > {x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}, > {y1, 0, 1}, {y2, 0, 1}, {y3, 0, 1}, > MaxPoints -> 10^6, PrecisionGoal -> 3] > > took about 30 seconds in version 5.0 (with or without > calculus`integration). Mathematica 5.1 again is unable to evaluate > this integral in 20 minutes. The reason is that Mathematica 5.0 > assumed that specifying the option MaxPoints always implied the use of > the method QuasiMonteCarlo for numerical integration. Now, version 5.1 > has two new methods, EvenOddSubdivision and > SymbolicPiecewiseSubdivision, for numerical integration of piecewise > functions (Abs in this case) and Mathematica seems to apply one of > them here. > > Two possible workarounds are either to specify Method->QuasiMonteCarlo > explicitly or to rewrite Abs[z] as Sqrt[z^2], avoiding the piecewise > handling routine. > > The documentation, section A.9.4, still claims that "if an explicit > setting for MaxPoints is given, NIntegrate by default uses > Method->QuasiMonteCarlo." This appears to be incorrect for version > 5.1. The reference for NIntegrate still enumerates only the methods > that existed in version 5.0, and the only way to learn about the two > new methods is by stumbling on the NIntegrate::bdmtd message. > > Besides, I just do not see any reason to change the behaviour of > NIntegrate in this particular case: MaxPoints meant that I wanted to > use stochastic methods, why was it necessary to change this? > > Maxim Rytin > m.r at inbox.ru > > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**Follow-Ups**:**Re: Re: Integrate in version 5.1***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Integrate in version 5.1***From:*ab_def@prontomail.com (Maxim)

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