6x6 system of multivariate quadratic equations ... non-negative real solutions
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What is the best way (what solver?) to effectively find real non-negative solutions of 6 multivariate quadratic equations (6 variables)?
f_i(x) = 0, i = 1,6, where x = [x1,x2,...,x6] and f_i(x) is the quadratic form with known real coeffs.
9 个评论
Walter Roberson
2024-11-4,17:54
To confirm, you have the form:
syms x [6 1]
syms A [6 6]
A = diag(diag(A))
x' * A * x
Bruno Luong
2024-11-4,18:20
"A always contains only one non-zero diagonal element on the ith row/column "
To me it means
Ai(j,j) == 0 is true for i ~= j and
false for i == j.
采纳的回答
Bruno Luong
2024-11-4,8:54
编辑:Bruno Luong
2024-11-4,11:12
Assuming the equations are
x' * Ai * x + bi'*x + ci = 0 for i = 1,2, ..., N = 6.
Ai are assumed to be symmetric. If not replace with Asi := 1/2*(Ai + Ai').
You could try to iteratively solve linearized problem; in pseudo code:
x = randn(N,1); % or your solution of previous step, slowly changing as stated
L = zeros(N);
r = zeros(N,1);
notconverge = true;
while notconverge
for i = 1:N
L(i,:) = (2*x'*Ai + bi');
r(i) = -(x'*Ai*x + bi'*x + ci);
end
dx = L \ r;
xold = x;
x = x + dx;
x = max(x,0); % since we want solution >= 0
notconverge = norm(x-xold,p) > tolx && ...
norm(r,q) > tolr; % select p, q, tolx and tolr approproately
end
I guess fmincon, fsolve do somesort of Newton-like or linearization internally. But still worth to investogate. Here the linearization is straight forward and fast to compute. Some intermediate vectors in computing L and r are common and can be shared.
3 个评论
Bruno Luong
2024-11-4,19:24
编辑:Bruno Luong
2024-11-5,8:17
x = 0 is ONE solution (up to 2^6 in the worst case). You have Newton actually converges to a local minima of ONE solution. Bravo.
You need to change
x = randn(N,1);
so as it would converge to a desired solution.
更多回答(2 个)
Aquatris
2024-11-1,10:13
2 个评论
John D'Errico
2024-11-1,18:56
No. There is not. We all want for things that are not possible. Probably more likely to hope for peace in the world. Yeah, right.
Torsten
2024-11-1,14:07
移动:Torsten
2024-11-1,14:08
I think there is no specialized solver for a system of quadratic equations. Thus a general nonlinear solver ("fsolve","lsqnonlin","fmincon") is the only chance you have.
3 个评论
Torsten
2024-11-1,15:58
编辑:Torsten
2024-11-1,16:04
You shouldn't think about speed at the moment. You are lucky if you get your system solved and if the solution is as expected. A system of quadratic equations is a challenge.
If this works satisfactory, you can save time if you set the solution of the last call to the nonlinear solver to the initial guess of the next call (since you say that your coefficients vary slowly).
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