Re: Re: Another Bug in Mathematica 3.0.0 definite integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg9311] Re: [mg9254] Re: Another Bug in Mathematica 3.0.0 definite integration*From*: David Withoff <withoff>*Date*: Mon, 27 Oct 1997 02:47:24 -0500*Sender*: owner-wri-mathgroup at wolfram.com

> >> a=Integrate[1/Sqrt[Sin[x]+Cos[x]], {x,0,Pi/2}] > > This is a pattern I have observed in many integrals involving > trigonometric functions that lead to hypergeometric and/or elliptic > functions. The time comparison of 2.2 vs 3.0 is interesting. > > Results from Mathematica 2.2.1 on above test function: > > (Pi^(1/2)*Gamma[1/4])/(2*2^(1/4)*Gamma[3/4]) - > 2*Hypergeometric2F1[1/2, 1, 5/4, -1] + Hypergeometric2F1[3/4, 1, 3/2, > -1] > > 1.397395299268851 > > Time: 0.95 seconds on Mac 8500/120. > > Results from Mathematica 3.0.1, same machine: > > -2*2^(3/4)*HypergeometricPFQ[{1/4, 3/4}, {5/4}, -1] > > -3.012362296717479 > > Time: 10.4 seconds, a factor of 10. > > Both 3.0 and 3.0.1 give the warning about convergence. I have not noticed or seen other reports of a pattern of behavior like this. A slowdown of a factor of 10 is sufficiently atypical that it should be reported to Wolfram Research so that it can be investigated. The only pattern that I have noticed is that Version 3.0 can do more integrals that Version 2.2, and that the integration packages load much more quickly. Just to check that I wasn't making this up, I tried a few integrals just now in Version 2.2 and in Version 3.0 for Windows 95 on my 100 MHz Pentium computer. The first elliptic integral that I tried finished about 18 times faster in Version 3.0 than in Version 2.2, and in Version 3.0 I didn't need to load any separate packages. The first time through, Version 2.2 gave up on my integral after 27.736 seconds. In[1]:= Integrate[1/Sqrt[2 + Cos[x] + Sin[x]], {x, 0, Pi}] //Timing General::intinit: Loading integration packages -- please wait. 1 Out[1]= {27.736 Second, Integrate[-------------------------, {x, 0, Pi}]} Sqrt[2 + Cos[x] + Sin[x]] The second time through it gave up after 1.982 seconds, which suggests that it took about 25 seconds to load the integration packages. After loading the separate Calculus`EllipticIntegrate` package (which took 28.028 seconds) Version 2.2 gave me a correct result in terms of elliptic integrals in 29.507 seconds. In Version 3.0 I got a correct result in 3.9 seconds the first time through, and 1.59 seconds the second time, which suggests that it took about 2.3 seconds (instead of 25+28 seconds) to load all of the necessary packages automatically, and that the integral itself takes 1.59 seconds in Version 3.0, about 18 times faster than the 29.507 seconds that I found in Version 2.2. I also tried some simpler integrals -- just the first trigonometric integrals that popped into my head and that I guessed that Version 2.2 would be able to do. In Version 2.2 these integrals finished in 12.77 seconds: In[5]:= Timing[{ Integrate[1/(1 + Sin[b x]), x], Integrate[Sin[3 a x] + Cos[b x], {x, x1, x2}], Integrate[Sin[x]^3, {x, 0, Pi}], Integrate[1/Sin[c x], {x, a, b}], Integrate[x^3 Sin[x]^2 Cos[x], {x, 1, 2}], Integrate[Sqrt[x] Sin[x], x]}][[1]] Out[5]= 12.77 Second The same integrals took 6.92 seconds in Version 3.0, almost twice as fast as in Version 2.2. A speedup of a factor of 18, as in my example, is unusual, but a slowdown of a factor of 10 is also unusual. Any significant slowdown like this should be reported to Wolfram Research. Dave Withoff Wolfram Research