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How do I solve the following system of equations on MATLAB when one of the elements of the variable vector is a constant? Please do give the code if possible.

More generally, if the solution is to use symbolic math, how will I go about generating large number of variables, say 12 (rather than just two) even before solving them?

For example, create a number of symbolic variables using syms, and then make the system of equations like below.

syms a1 a2



A = [matrix]

x = [1;a1;a2];

y = [1;0;0];



eqs = A*x == y



sol = solve(eqs,[a1, a2])



sol.a1

sol.a2

In case you have a system with many variables, you could define all the symbols using syms, and solve it like above.

You could also perform a parameter optimization with fminsearch. First you have to define a cost function, in a separate function file, in this example called cost_fcn.m.

function J = cost_fcn(p)



    % make sure p is a vector

    p = reshape(p, [length(p) 1]);



    % system of equations, can be linear or nonlinear

    A = magic(12); % your system, I took some arbitrary matrix

    sol = A*p;



    % the goal of the system of equations to reach, can be zero, or some other

    % vector

    goal = zeros(12,1);



    % calculate the error

    error = goal - sol;



    % Use a cost criterion, e.g. sum of squares

    J = sum(error.^2);



end

This cost function will contain your system of equations, and goal solution. This can be any kind of system. The vector p will contain the parameters that are being estimated, which will be optimized, starting from some initial guess. To do the optimization, you will have to create a script:

% initial guess, can be zeros, or some other starting point

p0 = zeros(12,1);



% do the parameter optimization

p = fminsearch(@cost_fcn, p0);

In this case p0 is the initial guess, which you provide to fminsearch. Then the values of this initial guess will be incremented, until a minimum to the cost function is found. When the parameter optimization is finished, p will contain the parameters that will result in the lowest error for your system of equations. It is however possible that this is a local minimum, if there is no exact solution to the problem.

Your system is over-constrained, meaning you have more equations than unknown, so you can't solve it. What you can do is find a least square solution, using mldivide. First re-arrange your equations so that you have all the constant terms on the right side of the equal sign, then use mldivide:

>> A = [0.0297 -1.7796; 2.2749 0.0297; 0.0297 2.2749]

A =



   0.029700  -1.779600

   2.274900   0.029700

   0.029700   2.274900



>> b = [1-2.2749; -0.0297; 1.7796]

b =



  -1.274900

  -0.029700

   1.779600



>> Ab

ans =



  -0.022191

   0.757299