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




Re: Difficult Set of Equations
Posted:
Dec 12, 2007 7:15 PM


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 righthand 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.



