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Re: Re: Re: Re: Re: algebraic
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106320] Re: [mg106285] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Fri, 8 Jan 2010 04:16:08 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001070731.CAA23893@smc.vnet.net>
DrMajorBob wrote:
>> But Mathematica does or if you prefer "simulates" a lot of mathematics
>> that only makes sense under the assumption of continuity.
>
> Continuity of a function does NOT depend on completeness in the domain,
> and I suspect that
>
> Resolve[Exists[x, x^2 == 2], Reals]
>
> True
>
> succeeds based on algebra, not topology.
It's based on cylindrical algebraic decomposition (CAD). As the name
indicates, that has an algebraic basis. But it is also intimately
connected to what is called real algebraic geometry, and that has a bit
of a topological flavor to it.
> You or I might (MIGHT) treat it as a topological problem, but I doubt
> Resolve can do so.
>
> A better example might be
>
> Reduce[Exists[x, Exp[x^7 + 3 x - 11] + x - 6/10 == 0], Reals]
>
> True
That might almost be called real analytic (as opposed to algebraic)
geometry. [It has something of a local flavor. In the sense of
geography, not math; it is relatively recent work at WRI. Serious
development is found in
Adam W. Strzebonski: Real root isolation for exp-log functions. ISSAC
2008: 303-314
This received the conference Best Paper award, I might add. It would be
a breach of the MathGroup rules to mention the prize...]
> A human might have great difficulty solving the equation, but he might
> easily establish that the LHS is negative for some value and positive for
> another, hence a solution exists in between.
>
> Yet, since this works:
As of version 7 of Mathematica...
> FindInstance[ Exp[x^7 + 3 x - 11] + x - 6/10 == 0, x]
>
> {{x -> Root[{-3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 &,
> 0.599896128076431511686789719766}]}}
>
> I suspect that algebra and root-search was used in Resolve.
>
> Unless a developer can confirm that Resolve didn't find a solution, merely
> proved that one could be bracketed?
I do not know but I'd imagine it could be done either way. Probably the
latter if at all possible, since that could be faster.
> To do that, Resolve would have to know the LHS is continuous on the real
> line, and haven't we found, frequently, that Mathematica CAN'T identify
> continuous functions?
Reduce, FindInstance, and probably Resolve have a reasonable
understanding of exp-log-quasipolynomial functions.
To answer a possible question from the philosophically-minded, yes, the
software really does understand these things.
> And what does THIS mean?
>
> 0.5998961280764315116867897197655402817356291002252018609367`30. // \
> RootApproximant
>
> Root[1 - #1 + #1^3 - 7 #1^4 + 10 #1^5 - 16 #1^6 + 15 #1^7 - 15 #1^8 +
> 15 #1^9 - 16 #1^10 + 12 #1^11 - 10 #1^12 + 11 #1^13 -
> 2 #1^14 + #1^15 + 8 #1^16 &, 3]
>
> (Note the constant included in the output from FindInstance.)
>
> Did FindInstance (and Resolve) generate and solve a series approximation
> to -3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 & ?
>
> Or is the RootApproximant result a pure accident?
>
> Bobby
An accident, of sorts. It is designed behavior, but it is not obvious
whether the design is without flaw. I say something about this (not
much, but something) in a response to another post from this
thread-of-all-threads.
Daniel Lichtblau
Wolfram Research
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