Solving non linear equation "Circle" equations
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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]
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채택된 답변
Matt J
2021년 9월 15일
편집: Matt J
2021년 9월 15일
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)
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
추가 답변 (2개)
Walter Roberson
2021년 9월 15일
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.xa
sol.ya
r3_needed = sqrt(lhs(vpa(subs(eqn(3), sol))))
r3_needed - r3
RR = double([R(1:2); r3_needed(2)]);
viscircles([xc, yc], RR, 'Color', 'k')
hold on
scatter(sol.xa, sol.ya, 60, 'r+')
hold off
댓글 수: 0
Bjorn Gustavsson
2021년 9월 15일
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).
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Bjorn Gustavsson
2021년 9월 15일
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.
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