This is intended to be a more generic question not about a specific system. Given a hermitian matrix $H(x_1,\dots,x_n)$ depending non-linearly on some real parameters $x_1,\dots,x_n$. We want these to be determined numerically such that $$H=0$$ and assume we already know that a set of parameters always exists such that this holds.

Would it be more efficient to directly solve the above equation or to optimise its Frobenius norm $||H||_F^2$ (which would just be a scalar)? Also, which method would you apply?

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    $\begingroup$ Could you clarify what you're solving for? It looks as though you're seeking the zero matrix. $\endgroup$ – Richard Jul 25 '19 at 14:49
  • $\begingroup$ @Richard Indeed, solve for $x_1,\dots,x_n$ such that $H$ vanishes. $\endgroup$ – user32355 Jul 25 '19 at 15:40
  • $\begingroup$ How is that different from setting all of $x_1,\ldots,x_n$ to zero? $\endgroup$ – Richard Jul 25 '19 at 15:48
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    $\begingroup$ It really doesn't matter that you have a matrix -- you could just think of a matrix as a vector with funnily-arranged entries. So this is no different to solving the vector equation $\vec F(\vec x)=0$, which is always done using a Newton method (or variants thereof). $\endgroup$ – Wolfgang Bangerth Jul 25 '19 at 15:56
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    $\begingroup$ Optimize the Frobenius norm might be a bad idea since the objective function can have local extrema that are different from zero. $\endgroup$ – nicoguaro Jul 25 '19 at 16:47

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