RE: Trivial integral freezes 3.
- To: mathgroup at smc.vnet.net
- Subject: [mg9800] RE: [mg9755] Trivial integral freezes 3.
- From: Ersek_Ted%PAX1A at mr.nawcad.navy.mil
- Date: Fri, 28 Nov 1997 05:35:13 -0500
- Sender: owner-wri-mathgroup at wolfram.com
Carlos wrote: -------------------------------------------------------- |The following trivial Integrate statement (from a class example) appears |to freeze Mathematica 3.0 running under Mac OS 7.5.5. | |ClearAll[alpha,phi]; alpha=Pi/3; |q=(Cos[alpha]-Cos[alpha+phi])/(-alpha+Pi+Cos[alpha]*Sin[alpha]); |Print["q=",q//InputForm]; |r1=Integrate[q,{phi,0,2*Pi-2*alpha}]; Print["r1=",r1//InputForm]; | |Same freeze happens for other values of alpha, e.g alpha=Pi/6. |Mathematica 2.2 has no problems. | |I am curious as to whether the freeze happens on non-Mac versions. | Below is the result of what happens doing this on my 90 Mhz Pentium. The Pentium was still working on it after 30 minutes, so I Aborted. It seems the algorithm ends up trying to evaluate (2-2(-1)^(1/3)+2(-1)^(2/3)) using numerical approximations, and can never get enough Precision because the value is actually zero. I don't know why numerical approximations are needed in this problem. Anyway, once it became apparent numerical approximations are giving us problems, I think the algorithm should try FullSimplify[2-2(-1)^(1/3)+2(-1)^(2/3)] which evaluates to zero. Once this hurdle is overcome, the result might come out OK. ------------------------------------- Also notice the following gives a result that has a tiny imaginary part, even though the integrand is real along the integrating interval. In[1]:= Integrate[(1/2.0 - Cos[phi + Pi/3.0])/(Sqrt[3.0]/4 + (2.0*Pi)/3), {phi, 0, 5.0 Pi/3}] Out[1]= 1.3785\[InvisibleSpace]+0. I ------------------------------------- I think we will hear from Wolfram Research on this one. Ted Ersek ------------------------------------ In[2]:= Integrate[(1/2 - Cos[phi + Pi/3])/(Sqrt[3]/4 + (2*Pi)/3), {phi, 0, 5 Pi/3}]//Timing \!\($MaxExtraPrecision::"meprec" \( : \ \) "In increasing internal precision while attempting to evaluate \!\(2 - \ \(2\\ \((-1)\)\^\(1/3\)\) + \(2\\ \((-1)\)\^\(2/3\)\)\), the limit \ $MaxExtraPrecision = \!\(49.9999999999999911`\) was reached. Increasing the \ value of $MaxExtraPrecision may help resolve the uncertainty."\) \!\(\* RowBox[{ \(\[Infinity]::"indet"\), \( : \ \), "\<"Indeterminate expression \!\(\((\(-0``16.7608\) + \(0``16.7441\\ I\))\ \)\\ \*InterpretationBox[\"ComplexInfinity\", DirectedInfinity[]]\) \ encountered."\>"}]\) \!\($MaxExtraPrecision::"meprec" \( : \ \) "In increasing internal precision while attempting to evaluate \!\(2 - \ \(2\\ \((-1)\)\^\(1/3\)\) + \(2\\ \((-1)\)\^\(2/3\)\)\), the limit \ $MaxExtraPrecision = \!\(49.9999999999999911`\) was reached. Increasing the \ value of $MaxExtraPrecision may help resolve the uncertainty."\) \!\(\* RowBox[{ \(\[Infinity]::"indet"\), \( : \ \), "\<"Indeterminate expression \!\(\((\(-0``16.7608\) + \(0``16.7441\\ I\))\ \)\\ \*InterpretationBox[\"ComplexInfinity\", DirectedInfinity[]]\) \ encountered."\>"}]\) \!\($MaxExtraPrecision::"meprec" \( : \ \) "In increasing internal precision while attempting to evaluate \!\(2 - \ \(2\\ \((-1)\)\^\(1/3\)\) + \(2\\ \((-1)\)\^\(2/3\)\)\), the limit \ $MaxExtraPrecision = \!\(49.9999999999999911`\) was reached. Increasing the \ value of $MaxExtraPrecision may help resolve the uncertainty."\) General::"stop": "Further output of \!\($MaxExtraPrecision :: \"meprec\"\) will be \ suppressed during this calculation." \!\(\* RowBox[{ \(\[Infinity]::"indet"\), \( : \ \), "\<"Indeterminate expression \!\(\((\(-0``16.7608\) + \(0``16.7441\\ I\))\ \)\\ \*InterpretationBox[\"ComplexInfinity\", DirectedInfinity[]]\) \ encountered."\>"}]\) General::"stop": "Further output of \!\(\\[Infinity] :: \"indet\"\) will be suppressed \ during this calculation." Out[2]= $Aborted