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Topic: How to solve this equation?
Replies: 6   Last Post: Apr 18, 2012 4:56 PM

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quasi

Posts: 12,067
Registered: 7/15/05
Re: How to solve this equation?
Posted: Apr 18, 2012 4:56 PM
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On Wed, 18 Apr 2012 12:30:39 -0700 (PDT), Lucy <comtech.usa@gmail.com>
wrote:

>On Apr 18, 2:27 pm, Lucy <comtech....@gmail.com> wrote:
>> On Apr 18, 1:00 pm, Lucy <comtech....@gmail.com> wrote:
>>

>> > On Apr 18, 12:37 pm, James Waldby <n...@valid.invalid> wrote:
>>
>> > > On Wed, 18 Apr 2012 10:09:02 -0700, Michael wrote:
>> > > > I have sum(w_i, i from 1 to n)=1 given,
>>
>> > > > and also I am given a set of numbers e_i, i from 1 to n.
>>
>> > > > Now I need to find a number u, such that
>>
>> > > > sum(w_i/(u - e_i)^2, i from 1 to n) ...
>>
>> > > > My questions are: are there systematic way of finding all possible
>> > > > solutions u?

>>
>> > > You didn't state any condition on u (except perhaps implying it's not
>> > > equal to any e_i).  State an equation before asking for solutions.  Also
>> > > clarify if w_i are given (besides e_i) and if u is the only unknown.

>>
>> > > > And is the number of solution related to n?
>>
>> > > > I am thinking of maybe for n=2, the number of solutions u is 1?
>>
>> > > > And for general n, the number of solutions u is n-1?
>>
>> > > --
>> > > jiw

>>
>> > yes,u is unconstrained... the only unknown...- Hide quoted text -
>>
>> > - Show quoted text -
>>
>> u is unconstrained... all the rest are given...
>>
>> and yes, w_i >=0 for all i...
>>
>> Thank yuo!- Hide quoted text -
>>
>> - Show quoted text -

>
>[Edit]
>
>Now I need to find a number u, such that
>
>sum(w_i/(u - e_i)^2, i from 1 to n) = 1.
>
>And I am looking for real numbers u...


Not much chance for a closed form,

If you clear denominators, you have a polynomial equation of
degree 2^n.

If n is not too large (say n <= 10), just use a CAS (such as
Maple or Mathematica) or some other other program that can
find polynomial roots numerically.

On the other hand, if n is large, getting the associated
polynomial equation would be a disaster. Instead, find the
approximate solution numerically leaving the equation in its
original form. The bisection method (or possibly Newton's
method) can be applied on intervals which avoid the points
e_i. Use a CAS with high precision to avoid errors due to
rounding.

quasi



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