New user question
- Subject: New user question
- From: Wolfgang Jenkner
- Date: 03 Mar 2003 06:52:22 +0100
"Stavros Macrakis" <stavros.macrakis@verizon.net> writes:
> > f1: (Rb/Bmin) * B * (B - Bmin) * (1 - B/Bmax) - A * B * F = 0;
> > f2: (Rf/Fmin) * F * (F - Fmin) * (1 - F/Fmax) - A * B * F = 0;
> > res: solve ([f1, f2], [B, F]);
> >
> > Now if I input just that, and try to solve it symbolically,
> > Maxima hangs.
>
> Your system reduces to something of the form:
>
> q1: b2 * B^2 + b1 * B + b0 + bf * F = 0
>
> q2: f2 * F^2 + f1 * F + f0 + fb * B = 0
Actually, the original system can be further reduced to a system of
the form
x^2+y=a
x+y^2=b
See below for the corresponding commented Maxima transcript. Clearly,
this is equivalent to the system
y=a-x^2
x^4-2*a*x^2+x+a^2-b=0
(C1) f1: (Rb/Bmin) * B * (B - Bmin) * (1 - B/Bmax) - A * B * F = 0$
(C2) f2: (Rf/Fmin) * F * (F - Fmin) * (1 - F/Fmax) - A * B * F = 0$
(C3) eqs:expand([f1/(A*B),f2/(A*F)]);
2
B Rb B Rb B Rb Rb
(D3) [- ----------- + ------ + ------ - -- - F = 0,
A Bmax Bmin A Bmin A Bmax A
2
F Rf F Rf F Rf Rf
- ----------- + ------ + ------ - -- - B = 0]
A Fmax Fmin A Fmin A Fmax A
/* Complete the squares */
(C4) eqs:expand(sublis([B=B1+(Bmin+Bmax)/2,F=F1+(Fmin+Fmax)/2],%));
2
Bmin Rb Bmax Rb B1 Rb Rb Fmin Fmax
(D4) [-------- + -------- - ----------- - --- - ---- - ---- - F1 = 0,
4 A Bmax 4 A Bmin A Bmax Bmin 2 A 2 2
2
Fmin Rf Fmax Rf F1 Rf Rf Bmin Bmax
-------- + -------- - ----------- - --- - ---- - ---- - B1 = 0]
4 A Fmax 4 A Fmin A Fmax Fmin 2 A 2 2
/* Perform another substitution... */
(C5) eqs:sublis([B1=B2*p,F1=F2*q],%);
2 2
B2 p Rb Bmin Rb Bmax Rb Rb Fmin Fmax
(D5) [- ----------- + -------- + -------- - --- - F2 q - ---- - ---- = 0,
A Bmax Bmin 4 A Bmax 4 A Bmin 2 A 2 2
2 2
F2 q Rf Fmin Rf Fmax Rf Rf Bmin Bmax
- ----------- + -------- + -------- - --- - B2 p - ---- - ---- = 0]
A Fmax Fmin 4 A Fmax 4 A Fmin 2 A 2 2
(C6) eqs:expand([first(%)/q,second(%)/p]);
2 2
B2 p Rb Bmin Rb Bmax Rb Rb Fmin Fmax
(D6) [- ------------- + ---------- + ---------- - ----- - ---- - ---- - F2 =
A Bmax Bmin q 4 A Bmax q 4 A Bmin q 2 A q 2 q 2 q
2 2
F2 q Rf Fmin Rf Fmax Rf Rf Bmin Bmax
0, - ------------- + ---------- + ---------- - ----- - ---- - ---- - B2 = 0]
A Fmax Fmin p 4 A Fmax p 4 A Fmin p 2 A p 2 p 2 p
/* ... and determine p and q such that the coefficients of B2^2 and
F2^2 become -1 */
(C7) solve([ratcoef(lhs(first(%)),B2,2)=-1,ratcoef(lhs(second(%)),F2,2)=-1],[p,q]);
2/3 2/3 1/3 1/3
(SQRT(3) %I - 1) A Bmax Bmin Fmax Fmin
(D7) [[p = --------------------------------------------------,
2/3 1/3
2 Rb Rf
1/3 1/3 2/3 2/3
(SQRT(3) %I + 1) A Bmax Bmin Fmax Fmin
q = - --------------------------------------------------],
1/3 2/3
2 Rb Rf
2/3 2/3 1/3 1/3
(SQRT(3) %I + 1) A Bmax Bmin Fmax Fmin
[p = - --------------------------------------------------,
2/3 1/3
2 Rb Rf
1/3 1/3 2/3 2/3
(SQRT(3) %I - 1) A Bmax Bmin Fmax Fmin
q = --------------------------------------------------],
1/3 2/3
2 Rb Rf
2/3 2/3 1/3 1/3
A Bmax Bmin Fmax Fmin
[p = ---------------------------------,
2/3 1/3
Rb Rf
1/3 1/3 2/3 2/3
A Bmax Bmin Fmax Fmin
q = ---------------------------------], [p = 0, q = 0]]
1/3 2/3
Rb Rf
/* Choose the non-trivial real solution */
(C8) sol:third(%);
2/3 2/3 1/3 1/3
A Bmax Bmin Fmax Fmin
(D8) [p = ---------------------------------,
2/3 1/3
Rb Rf
1/3 1/3 2/3 2/3
A Bmax Bmin Fmax Fmin
q = ---------------------------------]
1/3 2/3
Rb Rf
/* Substitute back */
(C9) eqs:expand(sublis(%,eqs));
2/3 4/3 2/3 4/3 2/3
Bmin Rb Rf Rb Rf
(D9) [---------------------------- - ------------------------------------
2 4/3 2/3 2/3 2 1/3 1/3 2/3 2/3
4 A Bmax Fmax Fmin 2 A Bmax Bmin Fmax Fmin
2/3 4/3 2/3 1/3 1/3 2/3
Bmax Rb Rf Fmin Rb Rf
+ ---------------------------- - ---------------------------
2 4/3 2/3 2/3 1/3 1/3 2/3
4 A Bmin Fmax Fmin 2 A Bmax Bmin Fmax
1/3 1/3 2/3
Fmax Rb Rf 2
- --------------------------- - F2 - B2 = 0,
1/3 1/3 2/3
2 A Bmax Bmin Fmin
2/3 2/3 4/3 2/3 4/3
Fmin Rb Rf Rb Rf
---------------------------- - ------------------------------------
2 2/3 2/3 4/3 2 2/3 2/3 1/3 1/3
4 A Bmax Bmin Fmax 2 A Bmax Bmin Fmax Fmin
2/3 2/3 4/3 1/3 2/3 1/3
Fmax Rb Rf Bmin Rb Rf
+ ---------------------------- - ---------------------------
2 2/3 2/3 4/3 2/3 1/3 1/3
4 A Bmax Bmin Fmin 2 A Bmax Fmax Fmin
1/3 2/3 1/3
Bmax Rb Rf 2
- --------------------------- - F2 - B2 = 0]
2/3 1/3 1/3
2 A Bmin Fmax Fmin
(C10)
Wolfgang