Solving non linear equation "Circle" equations

Hello I have a 20 non linear equations that I want to solve, I started with only three to check if it would be working but I always get an empty vector. I do not get any answers.

syms xa ya
eq1 = xa^2 + ya^2 == 84.4^2;
eq2 = (xa-40)^2 + ya^2 == 68^2;
eq3 = (xa-100.66)^2 + (ya-49.07)^2 == 52.59^2;
sol = solve([eq1, eq2, eq3],[xa,ya])

for example the above 3 circles intersect at one point which should be (51.24, 67.06) but I always get an empty object. I know I can solve each 2 of them seperately and compare results, but my main object is to solve 20 equations with 15 unknowns at once.

but even for 2 unknowns I do not get answers.

sol = 

  struct with fields:

    xa: [0×1 sym]
    ya: [0×1 sym]

[Matt J]

for example the above 3 circles intersect at one point which should be (51.24, 67.06)

This point doesn't even solve the first equation:

xa=51.24;
ya=67.06;

xa^2 + ya^2 == 84.4^2

ans = logical
   0

[Ahmed Ashour]

it does

51.24 ^ 2 + 67.06 ^ 2 = 84.4^2

[Matt J]

Nope. Let's compute the difference,

(51.24 ^ 2 + 67.06 ^ 2) - 84.4^2 

ans = -0.7788

[Ahmed Ashour]

yes because I approximated the point, it can be zero if I write the complete number. this point I already got by solving pairs of the equations together.

[Matt J]

So, write the complete number and we will check that too.

Answer 1

[Matt J]

Because you cannot be certain of an exact 3-way (or n-way) intersection, you need to use a numerical least squares solver like fsolve.

opts=optimoptions(@fsolve,'StepTol',1e-8,'FunctionTol',1e-8,'OptimalityTol',1e-8);

[xy,Fxy]=fsolve(@modelEq,[51.24, 67.06],opts)

Warning: Trust-region-dogleg algorithm of FSOLVE cannot handle non-square systems; using Levenberg-Marquardt algorithm instead.

Equation solved, solver stalled.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance and the vector of function values
is near zero as measured by the value of the function tolerance.

xy = 1×2

51.2421 67.0642

Fxy = 1×3

0.0036 -0.0043 0.0027

function F=modelEq(p)

    xa=p(1); ya=p(2);
    
    eq1 = xa^2 + ya^2 - 84.4^2;
    eq2 = (xa-40)^2 + ya^2 - 68^2;
    eq3 = (xa-100.66)^2 + (ya-49.07)^2 - 52.592^2;
    
    F=[eq1, eq2, eq3];

end

[Ahmed Ashour]

Thanks for your elaboration. it is all clear now.

it was my mistake from the beginning as I wanted to avoid numerical solutions because I do not want to bias the system with my initial guesses.

Answer 2

[Walter Roberson]

syms xa ya R3
xc = [0; 40; 100.66];
yc = [0; 0; 49.07];
r3 = 52.59;
R = [84.4; 68; R3];
eqn = (xa-xc).^2 + (ya-yc).^2 == R.^2;
sol = solve(eqn(1:2), [xa, ya])

sol = struct with fields:
    xa: [2×1 sym]
    ya: [2×1 sym]

sol.xa

ans = 

sol.ya

ans = 

r3_needed = sqrt(lhs(vpa(subs(eqn(3), sol))))

r3_needed = 

r3_needed - r3

ans = 

RR = double([R(1:2); r3_needed(2)]);

viscircles([xc, yc], RR, 'Color', 'k')

ans = 
  Group with properties:

    Children: [2×1 Line]
     Visible: on
     HitTest: on

  Show all properties

hold on
scatter(sol.xa, sol.ya, 60, 'r+')
hold off

Answer 3

[Bjorn Gustavsson]

If the 20 equations are of the same form you will have a vastly overdetermined problem (that in general will not have any exact solution which will lead you to some kind of least-squares like solutions). The most natural first-stab might be to simply solve pairwise problems and then check if you have any common solution to all (or some subsets).

[Ahmed Ashour]

I get your point but why would I have 20 equations with only 2 unknowns.

there are more unknowns but I already substitued with some here to narrow down the problem and figure out where the issue is.

so instead of

(xa-40)^2 + ya^2 == 68^2;

it can be

(xa-x2)^2 + ya^2 == r2^2;

but I am trying to at least find the intersection point correctly for now.

[Bjorn Gustavsson]

Why indeed? Because the way you described the problem...

So you want to find out the issue with your real problem or the problem you posted? The first problem is due to the fact that Matt J showed - the representation of the numbers you wrote doesn't match the first equation, and most likely the symbolic interpretation of the exact numbers you were thinking of doesn't either. Then you have your real problem of 20 nonlinear equations in (what you now revealed to be) some number of unknowns that is larger than 2. You don't give us much to work with, do you? We could provide a couple of guesses:

1, you have at least some equation something like this:

2, your system of equations ends up with some polynomial of degree higher than 5.

[Ahmed Ashour]

I get it now, sorry for the confusion I made.