Certified Hermitian eigenvalue intervals from residual and Gram bounds

Input $n\ge 1$, a nonempty family of Hermitian $n\times n$ matrices enclosed by complex rectangles, a point matrix $V\in {\mathbb{C}}^{n\times n}$, a real diagonal matrix $\mathrm{\Theta }\in {\mathbb{R}}^{n\times n}$, and proposed nonnegative bounds $\delta ,\rho ,M_A,M_{\mathrm{\Theta }},\tau$. Treat point data as exact dyadics and the family as any subset of its enclosure.

1. With outward interval evaluation, enclose $V^{\ast }V-I$ and every $AV-V\mathrm{\Theta }$ for the full family. Form Frobenius upper bounds $d,r$ for these two matrices. Also form Frobenius upper bounds $m_A$ for every $A$ in the family and $m_{\mathrm{\Theta }}$ for $\mathrm{\Theta }$. If any enclosure or square-root operation is non-finite, return inconclusive.

2. Verify by outward scalar comparisons that $d\le \delta <1$, $r\le \rho$, $m_A\le M_A$, $m_{\mathrm{\Theta }}\le M_{\mathrm{\Theta }}$, and that $1-\sqrt{1-\delta }\le \tau$. If any comparison is undecided or false, return inconclusive with the failed check. Otherwise outward-compute a finite upper bound $\bar{\eta }\ge \rho +(M_A+M_{\mathrm{\Theta }})\tau$. For each increasingly sorted diagonal entry ${\theta }_{(k)}$, outward-compute finite endpoints ${\mathrm{\ell }}_k\le {\theta }_{(k)}-\bar{\eta }$ and $u_k\ge {\theta }_{(k)}+\bar{\eta }$, and set $I_k=[{\mathrm{\ell }}_k,u_k]$. If the radius bound or any final endpoint cannot be enclosed finitely, return inconclusive; otherwise return certified with radius bound $\bar{\eta }$ and intervals $I_1,\dots ,I_n$.

The checks use the Frobenius norm as an upper bound for the operator 2-norm. An inconclusive check makes no statement about the true spectrum.