Solve a system of 4 non linear equations with symbolic expressions

Hello,

Could you tell me please the way to solve this system of equations using Matlab ?

Thanks for your help,

h = ((zeta*(1-u))/(gamma*(1+eta-n*eta)-1))^(1/(alpha*theta))*((theta*(1-alpha)*(1-tau_l)*A)/(1-theta))^(1/alpha)*u^(-1);
c = (((1-beta)*n*eta*gamma)*(b+1-delta+alpha*A*(h*u)^(1-alpha)*(1-tau_k)))/(beta*(1-delta+alpha*A*(h*u)^(1-alpha)*(1-tau_k))-1+n+n*eta);
b = (A*(h*u)^(1-alpha)*(d-tau_l*(1-alpha)-alpha*tau_k)*(1-delta+alpha*A*(h*u)^(1-alpha)*(1-tau_k)))/((1+n)*gamma-1+delta-alpha*A*(h*u)^(1-alpha)*(1-tau_k));
gamma = (A*(h*u)^(1-alpha)*(1-d-((theta*(1-alpha)*(1-u)*(1-tau_l))/((1-theta)*u)))-c+1-delta)/(1+n);  

[Walter Roberson]

That is more than 4 variables. What variables are you trying to solve for?

Unless alpha is 1 or 1/2 or 1/4 you will probably not be able to find a general closed form solution.

[Nordine Bouchelia]

for h, c, b, and gamma, I'm trying to help a friend on his thesis,

thanks for your help

[Walter Roberson]

What are the permitted alpha and theta values? If they are not restricted to some simple fractions then there is no closed form solution and it would be necessary to proceed numerically.

[Nordine Bouchelia]

I'm not a specialist but there is no way to hold the other parameters constant and only solve for the values of h, c, b and gamma ?

or numerical analysis is the only way to solve ?

What do you suggest ?

Thanks

[Walter Roberson]

You have expressions raised to an unknown power.

Consider this:

syms x
syms n real
eqn = x^n + x + 1 == 0

What is the analytic solution for x in terms of n? Which integer n are there analytic solutions? Are those solutions similar or are they so different as to demand quite different analysis depending on the value of n? Are there any simple fractions 1/M integer M for which there are analytic solutions? Obviously there is a solution for n=1 but what is the solution for n=1/2?

[Nordine Bouchelia]

We have restrictions on the parameters theta and alpha, which belong to 0;1 both of them but nothing else,

Should we have to run a numerical analysis then ?

I think Octave software may help us for that,

thanks for your support

[Walter Roberson]

Let alpha = 1/5. b c gamma have terms ^(1-alpha) so that would be ^(4/5). If there are analytic solutions it would require multiplying through by something to the power of 5... leading us to quintics unless powers just happen to cancel out. There is no analytic solution for general quintics. There is a very limited set of alpha values that do not explode into quintics or worse.

Generally speaking when you have unknown powers, there are sometimes analytic solutions in terms of Wright Omega function. However I can see that these equations cannot be solved that way.

So... you might be able to solve a couple of cases analytically, but the rest are going to require numeric simulation.

[Nordine Bouchelia]

I think we'll opt for numerical simulation. Last question, I think Matlab simulink can help with numerical simulation as well as Octave right ?

Thanks for your time and explanations, in fact you were very clear.

[Walter Roberson]

At the moment I cannot think of any way that Simulink would be useful for this purpose.

I have no information about the quality of Octave implementation of relevant routines.

Answer 1

[Sam Chak]

Hi @Nordine Bouchelia

You can follow the examples in

https://www.mathworks.com/help/optim/ug/fsolve.html

to create a function m-file and write the equations in the form .

function F = root2d(x)

F(1) = exp(-exp(-(x(1) + x(2)))) - x(2)*(1 + x(1)^2);

F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5;

end

Then, in the command window, type:

fun = @root2d;
x0 = [0, 0];    % guess initial point of the solution [0, 0].
x = fsolve(fun, x0)

x =

0.3532 0.6061

If you prefer solving the problem symbolically, you can use the solve function and refer to this link:

https://www.mathworks.com/help/symbolic/solve.html