how I solve the two equation and two unknown variables using 'levenberg-marquardt' method?

I wnat to solve follow two equations.

F = @(X) [d(1,3).*(abs(X(1)./(X(2).^2.*ar))+d(2,3).*(X(1)./(X(2).^2.*ar))).^d(3,3).*(bg_loc_M)+d(4,3).*(X(2)).^d(5,3)-l./X(2);

abs(c(1,3)).*(bg_loc_M).^c(2,3).*X(2).^c(3,3).*(ar).^c(4,3).*exp(-abs(c(5,3)).*abs(X(1)+c(6,3).*abs(X(1))).^abs(c(7,3)))-foc];

here, I don't know X(1) & X(2), and kown others

how I solve the X(1) and X(2)?

I want to use levenberg-marquardt method.

I did follow.

F = @(X) [d(1,3).*(abs(X(1)./(X(2).^2.*ar))+d(2,3).*(X(1)./(X(2).^2.*ar))).^d(3,3).*(bg_loc_M)+d(4,3).*(X(2)).^d(5,3)-l./X(2);

abs(c(1,3)).*(bg_loc_M).^c(2,3).*X(2).^c(3,3).*(ar).^c(4,3).*exp(-abs(c(5,3)).*abs(X(1)+c(6,3).*abs(X(1))).^abs(c(7,3)))-foc];

x0=[0 l];

opts.Algorithm = 'levenberg-marquardt';

opts.TolX = 1e-10;

recal=fsolve(F,x0,opts);

But, results of X(1) & X(2) are different real value.....

I need your advice.

Answer 1

[Nipun]

Hi SungMin,

I understand that you want to solve the following system of equations using the Levenberg-Marquardt method in MATLAB, but the results for 𝑋(1) and 𝑋(2) are not as expected.

Here's a revised approach to ensure you correctly use the fsolve function with the Levenberg-Marquardt algorithm:

% Define the function
F = @(X) [d(1,3)*(abs(X(1)/(X(2)^2*ar))+d(2,3)*(X(1)/(X(2)^2*ar)))^d(3,3)*(bg_loc_M)+d(4,3)*(X(2))^d(5,3)-l/X(2);
          abs(c(1,3))*(bg_loc_M)^c(2,3)*X(2)^c(3,3)*(ar)^c(4,3)*exp(-abs(c(5,3))*abs(X(1)+c(6,3)*abs(X(1)))^abs(c(7,3)))-foc];

% Initial guess
x0 = [0, l];

% Set options for fsolve
opts = optimoptions('fsolve', 'Algorithm', 'levenberg-marquardt', 'TolX', 1e-10, 'Display', 'iter');

% Solve the system of equations
[X, fval, exitflag, output] = fsolve(F, x0, opts);

% Display the results
disp(['X(1) = ', num2str(X(1))]);
disp(['X(2) = ', num2str(X(2))]);

% Check the output status
if exitflag <= 0
    disp('The solver did not converge to a solution.');
else
    disp('The solver converged to a solution.');
end

Ensure that the initial guess x0 is close to the expected solution and that all parameters (d, ar, bg_loc_M, l, c, foc) are correctly defined.

You can refer to the MathWorks documentation for more details on fsolve and its options: https://www.mathworks.com/help/optim/ug/fsolve.html

Hope this helps.

Regards,

Nipun