Re: Re: Re: Re: Why can't Mathematica find this root?
- To: mathgroup at smc.vnet.net
- Subject: [mg38460] Re: [mg38440] Re: [mg38417] Re: [mg38362] Re: Why can't Mathematica find this root?
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Wed, 18 Dec 2002 01:53:42 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
I realized that the function g I defined below will eliminate very few
cases and it was really the result of a mental confusion. It can't even
identify as equivalent expressions such as Pi/3+ 2Pi n and Pi/3 - 2Pi
n. Here is a definition that works much better:
g[x_, y_] := PolynomialMod[x, 2*Pi] ===
PolynomialMod[y, 2*Pi]
Union[l, SameTest -> (g[x /. #1, x /. #2] & )]
will now work in a lot more cases.
Andrzej
On Sunday, December 15, 2002, at 04:10 PM, Andrzej Kozlowski wrote:
>
> On Saturday, December 14, 2002, at 05:20 PM, Daniel Lichtblau wrote:
>
>> Konrad Den Ende wrote:
>>>
>>>> No, you don't. There are infinitely many roots, of which Mathematica
>>>> gives
>>>> you only three: -Pi/2, 0, and Pi/2.
>>>
>>> Actually, you get five different roots, not three. Nevertheless -
>>> point
>>> taken.
>>> I ment of course that i'd like Mathematica to give me ALL the roots
>>> on form:
>>> angle + (period * n). I'll try to be more exact the next time.
>>>
>>>> Didn't you notice Mathematica's comment that, since inverse
>>>> functions were
>>>> being used, some solutions might not be found? That's the
>>>> explanation.
>>>
>>> Yes, i did. I wonder why it's the ONLY method used by Mathematica. It
>>> feels
>>> pretty bad to know that the program "misses" some simple roots. I
>>> don't
>>> wonder
>>> "what kind of algorithm gave that", rather "why do they use only that
>>> algorithm".
>>>
>>> I appologize if my question caused any trouble by it's "fuzziness".
>>>
>>> --
>>> Vânligen
>>> Konrad
>>> -------------------
>>
>> I did not really understand at first the nature of the question. Now
>> that I do I'll take a stab at it. Under Solve>Further
>> Examples>"Getting
>> infinite solution sets for some equations" in the Help Browser, there
>> is
>> code to do something called "GeneralizeSolve". It is based on finding
>> periodic inverses to various known functions, and takes advantage of
>> the
>> fact that Solve will find solutions in terms of these inverse
>> functions.
>> It does not quite work directly on your problem for the simple reason
>> that e.g. ArcSin[0] will evaluate to 0. So we make it work by solving
>> f'[x]==a and substituting a->0 afterward.
>>
>> Here is the code in question.
>>
>> arctrigs = {ArcSin, ArcCos, ArcCsc, ArcSec, ArcTan, ArcCot, ArcSinh,
>> ArcCosh,
>> ArcCsch, ArcSech, ArcTanh, ArcCoth};
>>
>> periods = {2*Pi, 2*Pi, 2*Pi, 2*Pi, Pi, Pi, 2*I*Pi,
>> 2*I*Pi, 2*I*Pi, 2*I*Pi, I*Pi, I*Pi};
>>
>> (* We use n to denote an arbitrary integer *)
>>
>> Generalize[f_[x_], n_] :=
>> f[x] + nperiods[[Position[arctrigs, f][[1, 1]]]] /;
>> MemberQ[arctrigs,
>> f]
>>
>> Generalize[Log[x_],n_]:=Log[x]+2\[Pi]\[ImaginaryI]n
>>
>> Generalize[ProductLog[x_],n_]:=ProductLog[n,x]
>>
>> Generalize[x___, n_] := x
>>
>> GeneralizedSolve[eqns_, vars_] := Generalize[#, n] & //@ Solve[eqns,
>> vars]
>>
>> In[12]:=
>> GeneralizedSolve[f'[x]==a,x]/. a->0
>>
>> Solve::ifun: Inverse functions are being used by Solve, so some
>> solutions may \
>> not be found.
>>
>> Out[12]=
>> {{x -> -((2*Pi)/3) - 2*n*Pi}, {x -> (2*Pi)/3 + 2*n*Pi},
>> {x -> -(Pi/3) - 2*n*Pi}, {x -> Pi/3 + 2*n*Pi},
>> {x -> -Pi - 2*n*Pi}, {x -> Pi + 2*n*Pi},
>> {x -> -2*n*Pi}, {x -> 2*n*Pi}}
>>
>> A minor inconvenience is that some solutions may be listed more than
>> once e.g. the last two are equivalent if we regard n as ranging over
>> all
>> integers.
>>
>>
>> Daniel Lichtblau
>> Wolfram Research
>>
>>
>>
>
>
> This last "minor inconvenience" can be dealt with as follows:
>
> g[x_, y_] := TrueQ[Simplify[Element[x,
> Integers], Element[y, Integers]] &&
> Simplify[Element[y, Integers], Element[x, Integers]]]
>
>
> l = GeneralizedSolve[Derivative[1][f][x] == a, x] /. a -> 0
>
>
> Solve::ifun:Inverse functions are being used by Solve, so some
> solutions may \
> not be found.
>
>
> {{x -> -((2*Pi)/3) - 2*n*Pi}, {x -> (2*Pi)/3 + 2*n*Pi},
> {x -> -(Pi/3) - 2*n*Pi}, {x -> Pi/3 + 2*n*Pi},
> {x -> -Pi - 2*n*Pi}, {x -> Pi + 2*n*Pi}, {x -> -2*n*Pi},
> {x -> 2*n*Pi}}
>
>
>
> Union[l, SameTest -> (g[x /. #1, x /. #2] & )]
>
>
> {{x -> -2*n*Pi}, {x -> -Pi - 2*n*Pi},
> {x -> -((2*Pi)/3) - 2*n*Pi}, {x -> -(Pi/3) - 2*n*Pi},
> {x -> Pi/3 + 2*n*Pi}, {x -> (2*Pi)/3 + 2*n*Pi},
> {x -> Pi + 2*n*Pi}}
>
>
> (One could of course write a version of GeneralizedSolve that does this
> automatically).
>
>
> Andrzej Kozlowski
> Yokohama, Japan
> http://www.mimuw.edu.pl/~akoz/
> http://platon.c.u-tokyo.ac.jp/andrzej/
>
>
>
>
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/