How to solve real valued system of quadratic equations

[mmuetzel]

Hi,

I am having problems solving the following system of l quadratic equations:
a_i * M_ik,l * a_k = B_l

All M_ik,l and B_l have real (non-complex) values. M_ik,l is symmetric in
all l equations (M_ik,l = M_ki,l).
I am interested in real valued a_i that solve that system of quadratic
equations in a least-squares sense (l_max > i_max). l_max and i_max are both
of magnitude 100 (give or take).

The only idea that I theoretically came up with is to "simultaneously"
diagonalize M_ik,l for all l:
a_i * U_ij * S_jm,l * U_km * a_k = B_l

And than solve for b_j = U_ij * a_i. But I do not know how to find U_ij
(which must be real valued as well, I guess).

Is there some variant of svd that diagonalizes several matrices in the same
basis (in a least-squares sense)?
Is there another way of solving that problem?

Thanks already for any hints.

Markus

PS: Enjoy OctConf to everyone who is there.



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