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Re: Re: Refine, assumptions, domains

• To: mathgroup at smc.vnet.net
• Subject: [mg101753] Re: [mg101715] Re: Refine, assumptions, domains
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Thu, 16 Jul 2009 08:18:11 -0400 (EDT)
• References: <h3hjkc$1ue$1@smc.vnet.net> <h3hqqr$5l3$1@smc.vnet.net> <200907151108.HAA16411@smc.vnet.net> <454715FB-1A82-4D01-ACE9-E700A96FCADD@mimuw.edu.pl> <4A5E684A.7090608@cs.berkeley.edu> <55CFD0EA-0001-4BAA-9414-40A1BF9DCE17@gmail.com>

More precisely, Reduce was greatly expanded in version 5 and could  
give complete solutions to trigonometric equations. Version 7 can do  
much more and find roots in the complex plane of any analytic  
functions. For example:

x == Root[{E^#1 - Sin[#1] & ,
           
-9.4248586537754136567567778432489211323797029251636558910832`20.\
3008606\
       01073058}] || x == Root[{E^#1 - Sin[#1] & ,
           
-6.281314366210795486987123240583963390549132286186244538368`20.\
29036662\
       9310228}] || x == Root[{E^#1 - Sin[#1] & ,
           
-3.1830630119333635919397448876301537357136339778662659227848`20.\
2961913\
       3152797}]

Note that these are exact solutions, the approximate numbers you see  
are only isolating approximations. The solutions themselves can now be  
computed to arbitrary precision:

N[First[%], 100]
x ==  
-9.424858653775413656756771883285883444399179101807799929806950013147702 
\

        
28492314961406843365451610069656121800751803514807455672278388074`100.

This will work also in the complex plane, try:

Reduce[Exp[x] == Sin[x] && Abs[x] < 10, x]


Andrzej Kozlowski

On 15 Jul 2009, at 16:58, Andrzej Kozlowski wrote:

>
> On 15 Jul 2009, at 16:37, Richard Fateman wrote:
>
>> That works, too.
>>
>> Last time I tried using Reduce it worked only for polynomials, so I  
>> stopped using it.
>> Thanks.
>
> Yes, that was the case before version 5, I think.
>
> Andrzej Kozlowski
>
>
>
>
>> Andrzej Kozlowski wrote:
>>> ...
>>> It may be just me but I can't understand what you are trying to do  
>>> at all. If you just want:
>>>
>>>> a way of finding the solutions of any equation,
>>>> say Sin[x]==0 that lie in a particular range
>>>
>>> then what's wrong with:
>>>
>>> x /. {ToRules[Reduce[Sin[x] == 0 && 0 < x < 10, x]]}
>>>
>>> {Pi, 2*Pi, 3*Pi}
>>>
>> FindInstance also works..
>>
>> RJF
>>
>

• References:
  • Re: Refine, assumptions, domains
    • From: Richard Fateman <fateman@cs.berkeley.edu>