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. |
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 _______________________________________________ Help-octave mailing list [hidden email] https://lists.gnu.org/mailman/listinfo/help-octave |
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 |
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. _______________________________________________ Help-octave mailing list [hidden email] https://lists.gnu.org/mailman/listinfo/help-octave |
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