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Topic: Difficult Set of Equations
Replies: 10   Last Post: Dec 13, 2007 6:15 PM

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 Bill Daly Posts: 69 Registered: 12/8/04
Re: Difficult Set of Equations
Posted: Dec 12, 2007 7:15 PM
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On Dec 7, 3:03 pm, sonicb11 <williamp...@hotmail.com> wrote:
> Here are the 4 equations:
>
> 1) mu*[a/(a+b)] = sum(Wi)/n
>
> 2) (mu^2+s^2)*[a(a+1) / (a+b)(a+b+1)] = sum(Wi^2)/n
>
> 3) (mu^3+3*mu*s^2)*[a(a+1)(a+2) / (a+b)(a+b+1)(a+b+2)] = sum(Wi^3)/n
>
> 4) (mu^4+6*mu^2*s^2+3s^4)*[a(a+1)(a+2)(a+3)/ (a+b)(a+b+1)(a+b+2)(a+b
> +3)] = sum(Wi^4)n

Well, one approach is to use polynomial reaultants to eliminate
variables. Rewrite your equations as 4 polynomials, using S1..S4 for
the right-hand sides of the original equations:

pol1 = mu*a - S1*(a+b)
pol2 = (mu^2+s^2)*a*(a+1) - S2*(a+b)*(a+b+1)
pol3 = (mu^3+3*mu*s^2)*a*(a+1)*(a+2) - S3*(a+b)*(a+b+1)*(a+b+2)
pol4 = (mu^4+6*mu^2*s^2+3*s^4)*a*(a+1)*(a+2)*(a+3) - S4*(a+b)*(a+b
+1)*(a+b+2)*(a+b+3)

Then, calculate the resultant of pol1 and pol2 using the common
variable mu:

res12 = (S1^2 + (s^2 - S2))*a^4 + ((2*S1^2 - 2*S2)*b + (S1^2 + (s^2 -
S2)))*a^3 + ((S1^2 - S2)*b^2 + (2*S1^2 - S2)*b)*a^2 + S1^2*b^2*a

This eliminates mu as a common variable. Then calculate the resultant
of res12 and pol3 using the common variable a:

res123 = <some big polynomial in the variables b and s>

This eliminates the variable a. Then calculate the resultant of res123
and pol4 using the common variable b:

res1234 = <some humongous polynomial in the variable s>

You'll need a symbolic math package to calculate the resultants for
you. If you don't have one, you could prevail on someone who does to
run it and tell you the answers. The work could be simplified in a
couple of ways:

1) One of your variables, s, only appears with even exponents, so you
can write s2 = s^2, then at the end, take the square root of s2 to get
s.

2) The resultants can usually be simplified, e.g., the above res12
that I calculated is divisible by a, so you can divide it by a to get
a simpler equation (assuming of course that you don't want a=0 as part
of your solution).

Once you know the resultants, you can proceed as follows:

1) Calculate s (or s2) as one of the roots (zeroes) of res1234.

2) Substitute the value of s into res123, then calculate b as one of
the roots of the resulting polynomial.

3) Substitute the values of s and b into res12, then calculate a as
one of the roots of the resulting polynomial.

4) Substitute s, a and b into pol1, then solve this to get mu.

I'd be glad to do this, but I don't have a powerful enough symbolic
math package. Sorry.

Date Subject Author
12/7/07 sonicb11
12/7/07 Maxime
12/7/07 Virgil
12/7/07 sonicb11
12/7/07 sonicb11
12/7/07 RGVickson@shaw.ca
12/7/07 RGVickson@shaw.ca
12/11/07 sonicb11
12/12/07 Ken
12/12/07 Bill Daly
12/13/07 sonicb11

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