I have the equation $\Sigma_k(M_k*p_k)*V=E*V$, where the $M_k$ are nn real Hermitian matrices, $V$ is a nn eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$ are known exactly, and I get m energies $e$ from experiment, where (usually) $k\lt{m}\lt{n}$.
I now want to find the set of $p_k$ such that $\Sigma_m(e_m-E_m)^2$ is minimized (where $E_m$ are the "theoretical" eigenvalues following from the above equation, and $e_m$ my experimental ones - knowing beforehand which $E_m$ and $e_m$ "belong" together is another can of worms).
Since 30 years I solve this problem with INVEIX, a FORTRAN kludge which starts from an initial set $p_k$ and converges to a miminum by a steepest gradient algorithm. Kludge or not, it works splendid since 30 years :-)
Still, I'm curious: Can this minimax problem be solved more analytically, maybe even as closed form? I'm very reluctant to differentiate this matrix equation after the $p_k$...

I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are nn real Hermitian matrices, $V$ is a nn eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$ are known exactly, and I get m energies $e$ from experiment, where (usually) $k\lt{m}\lt{n}$.
I now want to find the set of $p_k$ such that $\Sigma_m(e_m-E_m)^2$ is minimized (where $E_m$ are the "theoretical" eigenvalues following from the above equation, and $e_m$ my experimental ones - knowing beforehand which $E_m$ and $e_m$ "belong" together is another can of worms).
Since 30 years I solve this problem with INVEIX, a FORTRAN kludge which starts from an initial set $p_k$ and converges to a miminum by a steepest gradient algorithm. Kludge or not, it works splendid since 30 years :-)
Still, I'm curious: Can this minimax problem be solved more analytically, maybe even as closed form? I'm very reluctant to differentiate this matrix equation after the $p_k$...