How to solve symbolic nonlinear equations with constrains

Hello!

I'd like to solve this system of equations:

function F = equationsSystem(sol)
F = [ (16*sol(1))/7 + (16*sol(2))/7 - 16/7;
- (16*sol(1)*sol(3))/7 - (16*sol(2)*sol(4))/7;
(16*sol(1)*sol(3)^2)/7 + (16*sol(2)*sol(4)^2)/7 + 2 ];
 end

My constrains are:

sol(3) > 0

sol(4) > sol(3)

I'd like to have an answer like: sol(1), sol(2), sol(4) function of sol(3). How can I do this in Matlab?

Answer 1

[Star Strider]

I suggest:

syms x y z t
sys = [2.286*x+2.286*y-2.286, -2.286*x*z-2.286*y*t];
Sol = solve(sys)
Sol_x = Sol.x
Sol_y = Sol.y

to produce this:

Sol_x =

t/(t - z)

Sol_y =

-z/(t - z)

Only x and y are common to both equations, so MATLAB solves for them in terms of the other variables, z and t.

Since z and t only appear in the second equation, use it to solve for them:

Sol_z = solve(sys(2), z)

produces:

Sol_z =

-(t*y)/x

similarly:

Sol_t = solve(sys(2), t)

produces:

Sol_t =

-(x*z)/y

[Dina Irofti]

But (Sol_x, Sol_y, Sol_z, Sol_t) is not a solution for my initial system.

I guess I have to use fsolve .

[Star Strider]

The numerical result is highly dependent on your starting points. There appear to be a number of solutions:

% Original eqn: f = [2.286*x+2.286*y-2.286, -2.286*x*z-2.286*y*t]
% p(1) = x,  p(2) = y,  p(3) = z,  p(4) = t
f = @(p)  [2.286*p(1)+2.286*p(2)-2.286, -2.286*p(1)*p(3)-2.286*p(2)*p(4)]
opts = optimoptions('fsolve', 'MaxFunEvals',10000,  'MaxIter',10000);
s = fsolve(f, 100*ones(4,1), opts)

Your system has an infinity of solutions in {z,t}. I apologise for not guessing the ones you were thinking of.

[Star Strider]

So, you want sol1, sol2, and sol4 to be functions of sol3?

Let X = sol3, and solve for sol1, sol2, and sol4.

syms X 
sol = sym('sol', [1 4])
assume(sol(4) > X)
assumeAlso(sol(4) > sol(3))
assumeAlso(sol(3) > 0)
assumeAlso(X > 0)
F = [ (16*sol(1))/7 + (16*sol(2))/7 - 16/7;
- (16*sol(1)*sol(3))/7 - (16*sol(2)*sol(4))/7;
(16*sol(1)*sol(3)^2)/7 + (16*sol(2)*sol(4)^2)/7 + 2 ];

F = subs(F, sol(3), X)

Sol = solve(F)
Sol_1 = simplify(Sol.sol1)
Sol_2 = simplify(Sol.sol2)
Sol_4 = simplify(Sol.sol4)

This produces:

Sol_1 =

-7/(8*X^2 - 7)

Sol_2 =

(8*X^2)/(8*X^2 - 7)

Sol_4 =

7/(8*X)