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How to solve real valued system of quadratic equations

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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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Re: How to solve real valued system of quadratic equations

Juan Pablo Carbajal-2
On Wed, Mar 22, 2017 at 9:11 AM, mmuetzel <[hidden email]> wrote:

> 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.
>
>
>
> --
> View this message in context: http://octave.1599824.n4.nabble.com/How-to-solve-real-valued-system-of-quadratic-equations-tp4682513.html
> Sent from the Octave - General mailing list archive at Nabble.com.
>
> _______________________________________________
> Help-octave mailing list
> [hidden email]
> https://lists.gnu.org/mailman/listinfo/help-octave

Hi,

Nothing pops up in ind mind besides diagonalization to solve the
problem. In your case you want simultaneous diagonalization (related
to the generalized eingenvalue problem), although I do not know how
feasible it is with more than 2 matrices.

Cheers

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Re: How to solve real valued system of quadratic equations

mmuetzel
Hola Juan Pablo,

and thank you for your answer.
From what you are writing, I get that there is no "readily available" function in Octave core or some of the packages that would do the simultaneous diagonalization of more than 2 matrices for me. Is that right?

Saludos
Markus
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Re: How to solve real valued system of quadratic equations

Juan Pablo Carbajal-2
On Thu, Mar 23, 2017 at 8:59 AM, mmuetzel <[hidden email]> wrote:

> Hola Juan Pablo,
>
> and thank you for your answer.
> From what you are writing, I get that there is no "readily available"
> function in Octave core or some of the packages that would do the
> simultaneous diagonalization of more than 2 matrices for me. Is that right?
>
> Saludos
> Markus
>
>
>
> --
> View this message in context: http://octave.1599824.n4.nabble.com/How-to-solve-real-valued-system-of-quadratic-equations-tp4682513p4682580.html
> Sent from the Octave - General mailing list archive at Nabble.com.
>
> _______________________________________________
> Help-octave mailing list
> [hidden email]
> https://lists.gnu.org/mailman/listinfo/help-octave

I haven't put to much thought on it, but it might be that the problem
is mathematically ill-posed.
Fro the generalized eigenvalue problem you can check, eig, eigs and gsvd.
Also these guys seem to have been faced with your problem
https://arxiv.org/pdf/1507.05703.pdf
They provide algorithms to check if the problem has a solution. Maybe
you want to implement that and share it with everybody.

If you get no mor answers form here, you can try stack overflow mathematics.

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