I need help with Range equation and range error

I have set up these values in matlab. I need help finding the jacobian matrix for the system. Can I just use the jacobian function provided by matlab?

I also need help with solving for the expected solution uncertainty (1-sigma.

This equation should solve for x and y correct? I need to solve for this as well.

Finally how can I put this solution in a monte carlo simulation?

a = [0 10 0 10];
b = [0 0 10 10];
r = [25 45 65 85];
range_error = 0.5

(r^2)=((x-a)^2) + ((y-b)^2)); 

Answer 1

[Walter Roberson]

syms a b x y r
eqn = (x-a)^2 + (y-b)^2 - r^2

eqn = 

J = jacobian(eqn, [x y])

J = 

syms tolp toln
assume(tolp > 0 & toln > 0)
range_error = 0.5

range_error = 0.5000

eqn1 = eqn + tolp - range_error^2

eqn1 = 

eqn2 = eqn - toln - range_error^2

eqn2 = 

J1 = jacobian(eqn1, [x y])

J1 = 

J2 = jacobian(eqn2, [x y])

J2 = 

What I did was transform

%sqrt((x-a)^2 + (y-b)^2) = r + delta, -range_error <= delta <= range_error

into equalities, giving a name to the difference between the ideal match and the actual match; split it into two parts, one with a positive difference and one with a negative difference, and require that the variable be positive. Like A > B means that A = B + delta where delta > 0. MATLAB is a lot more comfortable reasoning about equalities and then eliminating the branches that would violate the assume(), than it is trying to solve inequalities.