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Re: Help needed

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132039] Re: Help needed
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Sat, 23 Nov 2013 02:55:06 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • Delivered-to: l-mathgroup@wolfram.com
  • Delivered-to: mathgroup-outx@smc.vnet.net
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  • References: <l6i3h3$80f$1@smc.vnet.net>

Am 20.11.2013 11:39, schrieb Artur:
> Dear Mathematica Gurus!
> Who have good computer with big RAM memory and will be able to help me
> solve following 13 eq with 13 variables (my computer have 8MB and it is
> not sufficient). If You will be reached some result write to me
> privately because from unknown reason I don't receiving posts now from
> mathgroup):
>
> Solve[{e^3 == e1 e2^2, -1 == 2 c2 e1 e2 + c1 e2^2,
>     3 b e^2 == c2^2 e1 + 2 c1 c2 e2 + 2 b2 e1 e2 + b1 e2^2,
>     0 == c1 c2^2 + 2 b2 c2 e1 + 2 b2 c1 e2 + 2 b1 c2 e2 + 2 a2 e1 e2 +
>       a1 e2^2,
>     3 b^2 e + 3 e^2 f ==
>      2 b2 c1 c2 + b1 c2^2 + b2^2 e1 + 2 a2 c2 e1 + 2 b1 b2 e2 +
>       2 a2 c1 e2 + 2 a1 c2 e2 + e2^2 f1 + 2 e1 e2 f2,
>     0 == b2^2 c1 + 2 b1 b2 c2 + 2 a2 c1 c2 + a1 c2^2 + 2 a2 b2 e1 +
>       2 a2 b1 e2 + 2 a1 b2 e2 + 2 c2 e2 f1 + 2 c2 e1 f2 + 2 c1 e2 f2,
>     b^3 + 6 b e f ==
>      b1 b2^2 + 2 a2 b2 c1 + 2 a2 b1 c2 + 2 a1 b2 c2 + a2^2 e1 +
>       2 a1 a2 e2 + c2^2 f1 + 2 b2 e2 f1 + 2 c1 c2 f2 + 2 b2 e1 f2 +
>       2 b1 e2 f2,
>     0 == 2 a2 b1 b2 + a1 b2^2 + a2^2 c1 + 2 a1 a2 c2 + 2 b2 c2 f1 +
>       2 a2 e2 f1 + 2 b2 c1 f2 + 2 b1 c2 f2 + 2 a2 e1 f2 + 2 a1 e2 f2,
>     3 b^2 f + 3 e f^2 ==
>      a2^2 b1 + 2 a1 a2 b2 + b2^2 f1 + 2 a2 c2 f1 + 2 b1 b2 f2 +
>       2 a2 c1 f2 + 2 a1 c2 f2 + 2 e2 f1 f2 + e1 f2^2,
>     0 == a1 a2^2 + 2 a2 b2 f1 + 2 a2 b1 f2 + 2 a1 b2 f2 + 2 c2 f1 f2 +
>       c1 f2^2, 3 b f^2 == a2^2 f1 + 2 a1 a2 f2 + 2 b2 f1 f2 + b1 f2^2,
>     0 == 2 a2 f1 f2 + a1 f2^2, f^3 == f1 f2^2}, {f, f1, f2, e, e1, e2,
>     a1, a2, b, b1, b2, c1, c2}]

Coupled nonlinear equations have potentially the the algebraic structure 
of a rational function with powers up to product of highest powers. So 
there is no chance solving for a single variable in general.

The growing complexity can be seen by eliminating linear variables:

sol = Solve[
    Join[ eq[[1 ;; 2]], eq[[-4 ;; -1]]], {e1, c1, f1, a1, b1, b2}][[1]]

{e1 -> e^3/e2^2, c1 -> (-2 c2 e^3 - e2)/e2^3, f1 -> f^3/f2^2,
  a1 -> -((2 a2 f^3)/f2^3),
  b1 -> (5 a2^3 e2^3 f^3 + 3 a2 b e2^3 f^2 f2^2 - 2 c2 e2^3 f^3 f2^2 +
    2 c2 e^3 f2^5 + e2 f2^5)/(3 a2 e2^3 f2^4),
  b2 -> (4 a2^3 e2^3 f^3 + 6 a2 b e2^3 f^2 f2^2 + 2 c2 e2^3 f^3 f2^2 -
    2 c2 e^3 f2^5 - e2 f2^5)/(6 a2 e2^3 f^3 f2)}

eq /. sol // Simplify

{True, True, (1/(
   a2 e2 f f2))(-f2^2 (-e2^3 f^3 + e^3 f2^3) (-2 c2 e2^3 f^3 +
        2 c2 e^3 f2^3 + e2 f2^3) +
     a2^3 (5 e2^6 f^6 + 4 e^3 e2^3 f^3 f2^3) +
     3 a2 e2^2 f^2 f2^2 (-c2 (3 c2 e^3 + 2 e2) f f2^2 +
        b e2 (e2 f - e f2)^2 (e2 f + 2 e f2))) == 0, (1/(
   a2 e2 f f2))(3 c2 e^3 e2 f2^8 + e2^2 f2^8 +
     2 a2^3 e2^3 f^3 (5 c2 e2^3 f^3 - 2 c2 e^3 f2^3 - 2 e2 f2^3) +
     a2^2 (-6 e2^7 f^6 f2 + 6 e^3 e2^4 f^3 f2^4) +
     2 c2^2 (-2 e2^6 f^6 f2^2 + e^3 e2^3 f^3 f2^5 + e^6 f2^8) -
     3 a2 e2^2 f^2 f2^2 (c2^2 (2 c2 e^3 + e2) f f2^2 +
        2 b e2 (-c2 e2^3 f^3 + c2 e^3 f2^3 + e2 f2^3))) == 0, (1/(
   a2 e2 f f2))(f2^4 (-4 e2^3 f^3 + e^3 f2^3) (-2 c2 e2^3 f^3 +
        2 c2 e^3 f2^3 + e2 f2^3)^2 +
     16 a2^6 (5 e2^9 f^9 + e^3 e2^6 f^6 f2^3) +
     36 a2^2 e2^5 f^4 f2^3 (b^2 e2 f2 (e2 f - e f2)^2 (2 e2 f + e f2) +
         e2^2 f^2 (e2 f - e f2)^2 (e2 f + 2 e f2) +
        b c2 f (c2 e2^3 f^3 - 4 c2 e^3 f2^3 - 2 e2 f2^3)) +
     12 a2^4 e2^5 f^5 f2 (-4 c2 e2 f f2^3 +
        2 b e2 f2 (7 e2^3 f^3 + 2 e^3 f2^3) +
        c2^2 (5 e2^3 f^4 - 8 e^3 f f2^3)) -
     12 a2 e2^2 f^2 f2^3 (2 c2 e2^3 f^3 - 2 c2 e^3 f2^3 -
        e2 f2^3) (c2 e2 f f2^3 + b e2 f2 (e2^3 f^3 - e^3 f2^3) +
        c2^2 (e2^3 f^4 + 2 e^3 f f2^3)) -
     4 a2^3 e2^3 f^3 f2^2 (19 e2^4 f^3 f2^3 + 2 e^3 e2 f2^6 +
        2 c2 (17 e2^6 f^6 + 8 e^3 e2^3 f^3 f2^3 + 2 e^6 f2^6))) == 0, (
   1/(36 a2^2 e2^9 f^6 f2^5))(-24 a2^5 e2^7 f^6 f2 (e2^3 f^3 +
        2 e^3 f2^3) -
     24 a2^4 b e2^6 f^5 f2^2 (7 c2 e2^3 f^3 - 4 c2 e^3 f2^3 -
        2 e2 f2^3) +
     16 a2^6 e2^6 f^6 (-5 c2 e2^3 f^3 + 2 c2 e^3 f2^3 + e2 f2^3) +
     f2^4 (-2 c2 e2^3 f^3 + 2 c2 e^3 f2^3 +
        e2 f2^3)^2 (4 c2 e2^3 f^3 + 2 c2 e^3 f2^3 + e2 f2^3) +
     12 a2^2 e2^4 f^3 f2^3 (2 c2 (e2^3 f^3 - e^3 f2^3) (e2^3 f^3 -
           3 b^2 e2^2 f f2 - e^3 f2^3) +
        e2 f2^3 (2 e2^3 f^3 + 3 b^2 e2^2 f f2 + e^3 f2^3)) +
     12 a2 b e2^3 f^2 f2^4 (-e2^2 f2^6 +
        2 c2^2 (e2^6 f^6 + e^3 e2^3 f^3 f2^3 - 2 e^6 f2^6) +
        c2 (e2^4 f^3 f2^3 - 4 e^3 e2 f2^6)) +
     4 a2^3 e2^3 f^3 f2^2 (2 c2^2 (8 e2^6 f^6 + 23 e^3 e2^3 f^3 f2^3 -
           4 e^6 f2^6) + c2 (23 e2^4 f^3 f2^3 - 8 e^3 e2 f2^6) -
        2 (-9 b e2^7 f^5 f2 + 9 b e^3 e2^4 f^2 f2^4 + e2^2 f2^6))) ==
   0, (1/(a2 e2 f f2))(80 a2^9 e2^9 f^9 + 288 a2^7 b e2^9 f^8 f2^2 +
     36 a2^5 e2^7 f^6 f2^3 (2 e2^3 f^3 + 9 b^2 e2^2 f f2 +
        7 e^3 f2^3) +
     24 a2^6 e2^6 f^6 f2^2 (5 c2 e2^3 f^3 - 14 c2 e^3 f2^3 -
        7 e2 f2^3) -
     36 a2^2 e2^4 f^3 f2^5 (e2^3 f^3 - e^3 f2^3) (2 c2 e2^3 f^3 -
        2 c2 e^3 f2^3 - e2 f2^3) -
     9 a2 b e2^3 f^2 f2^6 (-2 c2 e2^3 f^3 + 2 c2 e^3 f2^3 +
        e2 f2^3)^2 +
     f2^6 (-2 c2 e2^3 f^3 + 2 c2 e^3 f2^3 + e2 f2^3)^3 -
     36 a2^4 b e2^6 f^5 f2^4 (4 c2 e2^3 f^3 + 14 c2 e^3 f2^3 +
        7 e2 f2^3) -
     3 a2^3 e2^3 f^3 f2^4 (c2^2 (64 e2^6 f^6 + 88 e^3 e2^3 f^3 f2^3 -
           44 e^6 f2^6) + 44 c2 (e2^4 f^3 f2^3 - e^3 e2 f2^6) -
        e2^2 f2 (11 f2^5 +
           72 b e2^2 f^2 (e2 f - e f2)^2 (2 e2 f + e f2)))) == 0, (1/(
   a2 e2 f f2))(8 a2^6 e2^6 f^6 + 12 a2^4 b e2^6 f^5 f2^2 -
     12 a2^3 e2^3 (2 c2 e^3 + e2) f^3 f2^5 +
     6 a2 b e2^3 f^2 f2^4 (2 c2 e2^3 f^3 - 2 c2 e^3 f2^3 - e2 f2^3) -
     12 a2^2 (e2^7 f^6 f2^3 - e^3 e2^4 f^3 f2^6) +
     f2^4 (e2^2 f2^6 +
        4 c2^2 (-2 e2^6 f^6 + e^3 e2^3 f^3 f2^3 + e^6 f2^6) +
        2 c2 (e2^4 f^3 f2^3 + 2 e^3 e2 f2^6))) == 0, (1/(
   a2 e2 f f2))(20 a2^6 e2^6 f^6 + 36 a2^4 b e2^6 f^5 f2^2 +
     12 a2^2 e2^4 f^3 f2^3 (e2 f - e f2)^2 (2 e2 f + e f2) -
     f2^4 (-2 c2 e2^3 f^3 + 2 c2 e^3 f2^3 + e2 f2^3)^2 -
     8 a2^3 e2^3 f^3 f2^2 (5 c2 e2^3 f^3 + 4 c2 e^3 f2^3 +
        2 e2 f2^3)) == 0, True, True, True, True}


I don't see much further improvement. Numerical methods apply if the 
solution is unique and not a variety.

-- 

Roland Franzius









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