Riccati residual sensitivity and invariant graph conditioning retain spectral separation

By [F3], expanding the CARE polynomial at a Hermitian $X$ gives $F_c(X+E)-F_c(X)={\mathcal{L}}_c^X(E)-EGE,$ where ${\mathcal{L}}_c^X(E)=(A-GX{)}^{\ast }E+E(A-GX)$. For the DARE, set $f(X)=Q+A^{\ast }X(I+GX{)}^{-1}A$. Direct multiplication gives ${\mathrm{\Phi }}_G(X+E)-{\mathrm{\Phi }}_G(X)=(I+(X+E)G{)}^{-1}E(I+GX{)}^{-1},{\mathrm{\Phi }}_G(X)=X(I+GX{)}^{-1}\mathrm{.}$ Taking the derivative at $E=0$ and multiplying by $A^{\ast },A$ gives $Df(X)[E]=A_c(X{)}^{\ast }E{A}_c(X)$; since $F_d=f-X$, this gives $D{F}_d(X)[E]=A_c(X{)}^{\ast }E{A}_c(X)-E$.

Apply [F14] to the exact ordered unitary DARE pencil representation and the supplied perturbations of both pencil matrices. Its paired generalized Sylvester operator and contraction inequalities give the radius-$r$ right and left deflating graphs; its separate Neumann and Stein margins establish the perturbed stable split. Its normalized basis construction gives distance at most $\beta (r)\le r$, and its computable physical chart estimate retains the inverse graph-block norm and the strict margin $\mathrm{\parallel }{U}_1^{-1}{\mathrm{\parallel }}_2\beta (r)<1$. No part of this conclusion follows from QZ backward error alone.

For a CARE additive residual perturbation $\mathrm{\Delta }$, write $X=X_c+E$ and let $\delta =\mathrm{\parallel }\mathrm{\Delta }{\mathrm{\parallel }}_F$. Then $F_c(X_c)=0$ and step 1.1 makes the perturbed root equation equivalent to $E=-{\mathcal{L}}_c^{-1}\mathrm{\Delta }+{\mathcal{L}}_c^{-1}(EGE)\mathrm{.}$ If $\rho$ satisfies ${\kappa }_c(\delta +\mathrm{\parallel }G{\mathrm{\parallel }}_2{\rho }^2)\le \rho$ and $2{\kappa }_c\mathrm{\parallel }G{\mathrm{\parallel }}_2\rho <1$, the map sends the closed radius-$\rho$ Hermitian ball to itself and has Lipschitz constant at most the second displayed quantity. Iteration from zero gives geometric successive differences; [F10] gives a limit in the ball, continuity gives a fixed point, and contraction gives uniqueness. Let $W_c={\int }_0^{\mathrm{\infty }}{e}^{t{A}_c^{\ast }}{e}^{t{A}_c}dt\succ 0$; then $A_c^{\ast }{W}_c+W_c{A}_c=-I$. If also $2\mathrm{\parallel }{W}_c{\mathrm{\parallel }}_2\mathrm{\parallel }G{\mathrm{\parallel }}_2\rho <1$, then $\mathrm{\parallel }EG{W}_c+W_{c}GE{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }{W}_c{\mathrm{\parallel }}_2\mathrm{\parallel }G{\mathrm{\parallel }}_2\rho <1$. Hence $(A_c-GE{)}^{\ast }{W}_c+W_c(A_c-GE)\prec 0$; an eigenvector $v$ with eigenvalue $\lambda$ gives $2\mathrm{Re}(\lambda )v^{\ast }{W}_{c}v<0$, so the perturbed loop is Hurwitz. For DARE, [F2] gives Schur stability and [F11] gives the Stein inverse ${\mathcal{L}}_d^{-1}(Y)=-{\sum }_{k\ge 0}(A_c^{\ast }{)}^{k}Y{A}_c^k$. On a sufficiently small ball, $S(X)=R+B^{\ast }XB⪰{\delta }_{S}I$, where ${\delta }_S={\lambda }_{\min }S(X_d)-b^2\rho >0$. Put $a=\mathrm{\parallel }A{\mathrm{\parallel }}_2$, $b=\mathrm{\parallel }B{\mathrm{\parallel }}_2$, and $m=\mathrm{\parallel }{X}_d{\mathrm{\parallel }}_F+\rho$. The resolvent identity gives $\mathrm{\parallel }S(X{)}^{-1}{\mathrm{\parallel }}_2\le {\delta }_S^{-1}$ and $\mathrm{\parallel }S(X{)}^{-1}-S(Y{)}^{-1}{\mathrm{\parallel }}_2\le b^2{\delta }_S^{-2}\mathrm{\parallel }X-Y{\mathrm{\parallel }}_F$. Splitting $K(X)-K(Y)$ into its $X$ and inverse-factor changes yields $\mathrm{\parallel }K(X)-K(Y){\mathrm{\parallel }}_F\le L_K\mathrm{\parallel }X-Y{\mathrm{\parallel }}_F$, where $L_K=ab\mathrm{/}{\delta }_S+a{b}^{3}m\mathrm{/}{\delta }_S^2$; hence $\mathrm{\parallel }{A}_c(X){\mathrm{\parallel }}_2\le M_c=a+a{b}^{2}m\mathrm{/}{\delta }_S$ and $\mathrm{\parallel }{A}_c(X)-A_c(Y){\mathrm{\parallel }}_2\le L_c\mathrm{\parallel }X-Y{\mathrm{\parallel }}_F$, $L_c=b{L}_K$. Since $D{F}_d(X)[E]=A_c(X{)}^{\ast }E{A}_c(X)-E$, subtraction gives $\mathrm{\parallel }D{F}_d(X)-D{F}_d(Y){\mathrm{\parallel }}_{F\to F}\le 2M_c{L}_c\mathrm{\parallel }X-Y{\mathrm{\parallel }}_F=:L_D\mathrm{\parallel }X-Y{\mathrm{\parallel }}_F$. Integrating this derivative along line segments gives $R(E)=F_d(X_d+E)-F_d(X_d)-{\mathcal{L}}_d(E)$ with $\mathrm{\parallel }R(E){\mathrm{\parallel }}_F\le L_D\mathrm{\parallel }E{\mathrm{\parallel }}_F^2\mathrm{/}2$ and $\mathrm{\parallel }R(E)-R(F){\mathrm{\parallel }}_F\le L_D\rho \mathrm{\parallel }E-F{\mathrm{\parallel }}_F$ on the ball. Thus $E=-{\mathcal{L}}_d^{-1}(\mathrm{\Delta }+R(E))$ is a self-map when ${\kappa }_d(\delta +L_D{\rho }^2\mathrm{/}2)\le \rho$ and a contraction when ${\kappa }_d{L}_D\rho <1$. The same completeness argument gives its unique local root. For a separate stability margin, take $W_d={\sum }_{k\ge 0}(A_c^{\ast }{)}^k{A}_c^k\succ 0$, so $W_d-A_c^{\ast }{W}_d{A}_c=I$ by [F11]. Put $\eta =L_c\rho$. Expanding the Stein form bounds its change by $\mathrm{\parallel }{W}_d{\mathrm{\parallel }}_2(2M_c\eta +{\eta }^2)$; choose $\rho$ so this is below $1$. Then $W_d-A_c(X{)}^{\ast }{W}_d{A}_c(X)\succ 0$, and an eigenvector gives $(1-\mathrm{\mid }\lambda {\mathrm{\mid }}^2)v^{\ast }{W}_{d}v>0$, so the perturbed loop is Schur stable. For sufficiently small $\delta >0$, take $\rho =2{\kappa }_c\delta$ in CARE when $\mathrm{\parallel }G{\mathrm{\parallel }}_2>0$; the contraction and stability inequalities hold for small enough $\delta$, giving $\mathrm{\parallel }E{\mathrm{\parallel }}_F=O(\delta )$. If $G=0$, the equation is linear and this order bound is exact. In DARE, take $\rho =2{\kappa }_d\delta$ when $L_D>0$; for sufficiently small $\delta$ this satisfies the self-map, contraction, $S(X)\succ 0$, and Stein-margin bounds, again giving $\mathrm{\parallel }E{\mathrm{\parallel }}_F=O(\delta )$. If $L_D=0$, the local equation is linear. The remainders are therefore $O({\delta }^2)$, proving the first-order terms $-{\mathcal{L}}_c^{-1}\mathrm{\Delta }$ and $-{\mathcal{L}}_d^{-1}\mathrm{\Delta }$.

Put $T_X=[I0;X_{c}I]$. Multiplying the Hamiltonian in [F4] on the right by $T_X$ and then on the left by $T_X^{-1}=[I0;-X_{c}I]$, and using the CARE identity in [F3], gives $T_X^{-1}H{T}_X=\left[\begin{array}{cc}{A}_c& -G\\ 0& -A_c^{\ast }\end{array}\right]\mathrm{.}$ The Sylvester map $\mathcal{S}(Y)=-A_c^{\ast }Y-Y{A}_c$ has smallest singular value $\sigma =\mathrm{sep}(-A_c^{\ast },A_c)>0$: [F9] applies because $\sigma (-A_c^{\ast })$ lies in the right half-plane while $-\sigma (-A_c)=\sigma (A_c)$ lies in the left half-plane; [F6] gives inverse norm ${\kappa }_s=1\mathrm{/}\sigma$. In the perturbed coordinates, write $Z=[Z_{ij}]$ and seek an invariant graph $[I;Y]$. Its lower-block graph equation is $-A_c^{\ast }Y-Y{A}_c+Z_{21}+Z_{22}Y-Y{Z}_{11}+YGY-Y{Z}_{12}Y=0.$ Hence $Y$ is a fixed point of $\mathcal{T}(Y)=-{\mathcal{S}}^{-1}(Z_{21}+Z_{22}Y-Y{Z}_{11}+YGY-Y{Z}_{12}Y)\mathrm{.}$ On the Frobenius ball $\mathrm{\parallel }Y{\mathrm{\parallel }}_F\le 2\theta$, each block norm of $Z$ is at most $\zeta$. The map norm is at most ${\kappa }_s[\zeta +2\zeta \mathrm{\parallel }Y{\mathrm{\parallel }}_F+(\mathrm{\parallel }G{\mathrm{\parallel }}_2+\zeta )\mathrm{\parallel }Y{\mathrm{\parallel }}_F^2]$, hence at most $\theta +4{\theta }^2+4\mu {\theta }^2+4{\theta }^3\le 2\theta$ under the displayed smallness condition. Its Lipschitz constant is at most $2\theta +4\mu \theta +4{\theta }^2<1$. Completeness [F10] and iteration therefore give a unique fixed point in the ball. Returning through $T_X$ turns $[I;Y]$ into $[I;X_c+Y]$, proving the graph bound. Under [A2], the restriction has $n$ stable eigenvalues and the perturbed matrix has exactly $n$ stable eigenvalues in total. In a basis beginning with this invariant graph, the perturbed matrix is block upper triangular with this stable restriction as its first diagonal block; the quotient block therefore has no stable eigenvalues. A stable generalized eigenvector must project to a stable generalized eigenvector of the quotient, which is impossible, so every stable generalized eigenspace lies in the graph. Both spaces have dimension $n$, hence the graph is the stable spectral subspace.

Since ${\tilde{U}}_1=U_1+\mathrm{\Delta }{U}_1=U_1(I+U_1^{-1}\mathrm{\Delta }{U}_1)$ and $\mathrm{\parallel }{U}_1^{-1}\mathrm{\Delta }{U}_1{\mathrm{\parallel }}_2<1$, the Neumann geometric series makes the second factor invertible and gives $\mathrm{\parallel }{\tilde{U}}_1^{-1}{\mathrm{\parallel }}_2\le (\mathrm{\parallel }{U}_1^{-1}{\mathrm{\parallel }}_2^{-1}-\mathrm{\parallel }\mathrm{\Delta }{U}_1{\mathrm{\parallel }}_2{)}^{-1}$. Using $U_2=X{U}_1$, ${\tilde{U}}_2{\tilde{U}}_1^{-1}-X=(\mathrm{\Delta }{U}_2-X\mathrm{\Delta }{U}_1){\tilde{U}}_1^{-1}\mathrm{.}$ Submultiplicativity [F12] proves the displayed basis-chart bound. This derivation uses only the graph equation, so it applies to both the CARE invariant graph and the DARE deflating graph; it does not convert a pencil backward error into a deflating-subspace error.