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Re: Integrate 'learns'?

• To: mathgroup at smc.vnet.net
• Subject: [mg106324] Re: [mg106269] Integrate 'learns'?
• From: "Tony Harker" <a.harker at ucl.ac.uk>
• Date: Fri, 8 Jan 2010 04:16:54 -0500 (EST)
• References: <201001070728.CAA23688@smc.vnet.net> <4B4637A8.6030505@wolfram.com>

Many thanks, Daniel and Bob. I, too, thought this must have come up before
but failed to locate relevant threads. What you say about not seeing my
first-described behaviour is consistent with what Szabolcs Horv=E1t told me:
"I get the first result you cite only when the system is under load (and
also got it when running the computation right after reboot).  I think
people with fast computers won't get it at all."  Sounds as if it's time for
me to upgrade my hardware!

  Tony 

]-> -----Original Message-----
]-> From: Daniel Lichtblau [mailto:danl at wolfram.com]
]-> Sent: 07 January 2010 19:36
]-> To: Tony Harker
]-> Cc: mathgroup at smc.vnet.net
]-> Subject: Re: [mg106269] Integrate 'learns'?
]->
]-> Tony Harker wrote:
]-> >   If I open a clean notebook in Version 7.0 for Microsoft Windows
]-> > (32-bit) and enter
]-> >   Integrate[x/(3 Sin[x]),{x,\[Pi]/4,\[Pi]/2}]
]-> >   the result (after a warning message) is
]-> >   (8*Catalan - I*Pi^2 + Pi*(-Log[1 - (-1)^(1/4)] + Log[1 +
]-> > (-1)^(1/4)]) - (4*I)*(PolyLog[2, -(-1)^(1/4)] -
]-> PolyLog[2, (-1)^(1/4)]))/12
]-> >   and if I then repeat the command I get no error and
]-> >   (5*Catalan)/6 - ((23*I)/288)*Pi^2 +
]-> (Pi*ArcTanh[(-1)^(1/4)])/6 +
]-> > ((2*I)/3)*PolyLog[2, (-1)^(1/4)]
]-> >   which seems to be Mathematica's final answer.
]-> >
]-> >   I am happy that the results are equivalent, but puzzled
]-> about what
]-> > has been saved, and where, to generate this difference. Did
]-> > Mathematica ask the audience or phone a friend?[...]
]->
]-> That's exactly right. Sometimes it also uses the "remove
]-> two wrong answers" option.
]->
]->
]->  > [...] Can anyone enlighten me?
]-> >   Tony Harker
]->
]-> Okay, actually Integrate and functions it uses (e.g.
]-> Simplify and Limit)
]-> can make use of partial results cached along the way, in
]-> beating out
]-> time constraints in later invocations. This is sort of
]-> hinted at but
]-> perhaps not stated outright, in the materials at the URL below.
]->
]-> http://library.wolfram.com/infocenter/Conferences/5832/
]->
]-> The upshot is that results can change with further calls. I
]-> believe this
]-> has come up on this group in past, but my searching
]-> capabilities have
]-> not sufficed to locate relevant prior threads.
]->
]-> I am not seeing the behavior you describe on my machine, by
]-> the way.
]-> Versions 7.0.0, 7.0.1, and development all give me the
]-> second result and
]-> no message, each time I run the example. I may try it on my
]-> machine at
]-> home to see if the platform change makes a difference.
]->
]->
]-> Daniel Lichtblau
]-> Wolfram Research
]->
]->

• References:
  • Integrate 'learns'?
    • From: "Tony Harker" <a.harker@ucl.ac.uk>