A two-vector LOBPCG block resolves a clustered pair in one finite Rayleigh-Ritz update

The vectors are orthonormal and have disjoint coordinate supports. Their initial Rayleigh quotients are ${\theta }_1=(99+5)\mathrm{/}100=26\mathrm{/}25$ and ${\theta }_2=((101\mathrm{/}100)99+9)\mathrm{/}100=10899\mathrm{/}10000$. Each $x_i$ has two nonzero coordinates associated with distinct diagonal entries of $A$, so $r_i=(A-{\theta }_{i}I)x_i\mathrm{\ne }0$; also $x_i^{\ast }{r}_i=x_i^{\ast }A{x}_i-{\theta }_i{x}_i^{\ast }{x}_i=0$, and $r_i$ lies in the same two-coordinate plane as $x_i$. Thus $x_i,r_i$ are independent in that plane, and the disjoint planes make $[X,R]$ rank four.

Since $T=I$, $W=R$ and the first trial space is all of ${\mathbb{C}}^4$. Its Rayleigh-Ritz values are exactly $1,101\mathrm{/}100,5,9$, and the selected block is $\mathrm{span}(e_1,e_2)$. The update is successful because the trial rank is four; the returned residuals are zero, so the algorithm returns converged. In particular, $1\le 26\mathrm{/}25$ and $101\mathrm{/}100\le 10899\mathrm{/}10000$, as required by [F2].

For $U=\mathrm{span}(e_1,e_2)$, every unit vector in $S=\mathrm{span}(x_1,x_2)$ has squared norm $1\mathrm{/}100$ in $U^{\perp }$, so ${\sin }^2\vartheta =1\mathrm{/}100$. The strict gap $101\mathrm{/}100<5$ holds, and [F3] gives ${\theta }_2(S)\le 101\mathrm{/}100+(9-101\mathrm{/}100)\mathrm{/}100=10899\mathrm{/}10000$, attained by the computed ${\theta }_2$. The block captures both clustered modes; a one-dimensional trial space cannot contain this two-dimensional invariant subspace.