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Topic: deterministic integration
Replies: 4   Last Post: Mar 3, 2013 10:57 PM

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Alex Krasnov

Posts: 15
Registered: 10/3/12
deterministic integration
Posted: Feb 27, 2013 3:03 AM
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This issue has already been discussed on this list, but the previous
solutions are unsatisfactory. Integrate returns non-deterministic results
depending on machine speed and kernel cache state. Example (Mathematica
8.0.4):

In: Assuming[Element[z, Reals], Integrate[1/Sqrt[x^2+y^2+z^2], {x, -1, 1}, {y, -1, 1}]]
Out: -4*(z*ArcCot[z*Sqrt[2 + z^2]] + Log[1 - I*z] + Log[(1 + I*z)/(3 + z^2 + 2*Sqrt[2 + z^2])])

In: Assuming[Element[z, Reals], Integrate[1/Sqrt[x^2+y^2+z^2], {x, -1, 1}, {y, -1, 1}]]
Out: -4*(z*ArcCot[z*Sqrt[2 + z^2]] + Log[(1 + z^2)/(3 + z^2 + 2*Sqrt[2 + z^2])])

The previous solutions involve adjusting the machine speed and clearing
the kernel cache before each evaluation. This strategy attempts to achieve
the least transformed result, though this is presumably not guaranteed.
Instead, how can one achieve the most transformed result? The behavior of
Integrate appears to be similar to that of Refine, Simplify, FullSimplify.
However, unlike the latter, Integrate does not expose a TimeConstraint
option. How can one achieve the effect of TimeConstraint -> Infinity? Is
the absence of this option related to the undecidability of Risch's
algorithm or is the non-determinism entirely in a subsequent
simplification phase?

Alex




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