How can I solve this non-linear system?

EquationOne = @(x1,x2) 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EquationTwo = @(x1,x2) 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[darova]

Do you have any attempts? What about numerical approach? What about contour?

[Yves Adou]

I tried the folling code for solving:

z = optimvar('z',2);

Eq1 = fcn2optimexpr(@(z)EquationsOne(z(1),z(2)),z) == 0;

Eq2 = fcn2optimexpr(@(z)EquationsTwo(z(1),z(2)),z) == 0;

prob = eqnproblem;

prob.Equations.Eq1 = Eq1;

prob.Equations.Eq2 = Eq2;

show(prob)

z0.z = [0 0];

[sol, fval, exitflag] = solve(prob,z0)

and I obained the following:

Answer 1

[Vidhi Agarwal]

Hi Yves Adou,

I recognize that you're encountering an issue in solving 2 complex nonlinear equations. “fsolve” function from the Optimization Toolbox is commonly used for this purpose. Here’s the steps you can follow to solve the problem:

1. Define the equation as anonymous functions in MATLAB.
2. “fsolve" requires the equations to be in a single function if you have multiple equations. You can combine them into a single function that returns a vector.
3. Before you call “fsolve”, you need to provide an initial guess for the values of “x1” and “x2”. If you don't have a good initial guess, you can start with zeros or any arbitrary values.

Here is the sample code demonstrating the same:

function solve
    % Nested functions for the equations
    function f1 = EquationOne(x)
        f1 = x(1)^2 + x(2)^2 - 4;
    end
    function f2 = EquationTwo(x)
        f2 = x(1) * x(2) - 1;
    end
    % Combined function for fsolve
    function F = mySystem(x)
        F(1) = EquationOne(x);
        F(2) = EquationTwo(x);
    end
    % Initial guess: Adjusted based on the problem's feedback
    x0 = [1.2, 0.8]; % Adjusted initial guess for potentially better convergence
    % Options for fsolve: Adjust function tolerance and step tolerance
    options = optimoptions('fsolve', ...
                           'FunctionTolerance', 1e-10, ...
                           'StepTolerance', 1e-10, ...
                           'Algorithm', 'trust-region', ...
                           'Display', 'iter'); % Showing iteration process for better insights.
    % Solve the system with adjusted settings
    [x, fval, exitflag, output] = fsolve(@mySystem, x0, options);
    % Displaying the results
    disp('Solution:');
    disp(x);
    disp('Function values at the solution:');
    disp(fval);
end

Output of the following code will look like this:

For better understanding of “fsolve”, and optimization toolbox please refer to the following documentation:

• https://www.mathworks.com/help/optim/ug/fsolve.html
• https://www.mathworks.com/products/optimization.html[HS1]

Hope this helps!