Finite-budget Robin-Robin transmission iteration

Inputs: Positive-definite real Schur operators $S_1,S_2$ on the nonzero finite-dimensional trace space $\mathrm{\Lambda }$, loads $f_1,f_2\in \mathrm{\Lambda }$, a parameter $\alpha >0$, initial incoming datum $r_1^0\in \mathrm{\Lambda }$, tolerance $\epsilon >0$, and integer pass budget $M\ge 1$.

All vector norms on $\mathrm{\Lambda }$ and operator norms on its endomorphisms are induced by its specified real inner product. Set $R_i:=(\alpha I-S_i)(\alpha I+S_i{)}^{-1}$, $q_i:=\mathrm{\parallel }{R}_i\mathrm{\parallel }$, and $q:=q_1{q}_2$. For each pass $m=0,\dots ,M-1$, perform in order

$(S_1+\alpha I){\lambda }_1^{m+1}=r_1^m+f_1,r_2^{m+1}=(\alpha I-S_1){\lambda }_1^{m+1}+f_1;(S_2+\alpha I){\lambda }_2^{m+1}=r_2^{m+1}+f_2,r_1^{m+1}=(\alpha I-S_2){\lambda }_2^{m+1}+f_2\mathrm{.}$

If $\mathrm{\parallel }{r}_1^{m+1}-r_1^m\mathrm{\parallel }\le (1-q)\epsilon \mathrm{/}\mathrm{\parallel }(\alpha I+S_1{)}^{-1}\mathrm{\parallel }$, return converged with trace ${\lambda }_1^{m+1}$ and pass count $m+1$. If all $M$ passes are used without this test succeeding, return budget_exhausted with the last trace ${\lambda }_1^M$ and pass count $M$. This procedure states the finite stop predicate but makes no claim about its error certificate.