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Successful certified Bartels-Stewart substitution returns the exact Sylvester solution

Statement

Suppose Finite-budget Bartels-Stewart solve for a Sylvester equation returns success for $A, B, C$ . Then its returned $X$ satisfies $A X + X B = C$ . If $σ (A) \cap (- σ (B)) = \emptyset$ , this $X$ is the unique solution.

Facts & Assumptions

Given: A successful output $X = U Y V^{*}$ from the algorithm, with its verified exact Schur identities and exact local block solves; the delegated rounded Schur calls were accepted only after those checks.

[F1]

Finite-budget Bartels-Stewart solve for a Sylvester equation: Success requires exact Schur certificates, and the block traversal solves $T_{i i} Y_{i j} + Y_{i j} S_{j j} = D_{i j} - \sum_{k > i} T_{i k} Y_{k j} - \sum_{ℓ < j} Y_{i ℓ} S_{ℓ j}$ for every block pair.

[F2]

Sylvester operator and Frobenius separation number: $S_{A, B} (X) = A X + X B$ .

[F3]

A Sylvester equation is uniquely solvable exactly when the spectra are disjoint: The spectral-disjointness condition is equivalent to unique solvability for every right-hand side.

Proof

1.1F1algebra

By [F1], $A = U T U^{*}$ , $B = V S V^{*}$ , and $D = U^{*} C V$ hold exactly. Multiplying $A X + X B$ on the left by $U^{*}$ and on the right by $V$ gives $T Y + Y S$ .

2.1F1step 1.1algebra

In the $(i, j)$ block, $T Y + Y S$ equals $T_{i i} Y_{i j} + Y_{i j} S_{j j} + \sum_{k > i} T_{i k} Y_{k j} + \sum_{ℓ < j} Y_{i ℓ} S_{ℓ j}$ . The successful local solve from [F1] makes this block equal $D_{i j}$ , so every block of $T Y + Y S$ equals the corresponding block of $D$ .

3.1F2step 1.1step 2.1algebra

Therefore $T Y + Y S = D = U^{*} C V$ . Multiplying by $U$ on the left and $V^{*}$ on the right yields $A (U Y V^{*}) + (U Y V^{*}) B = C$ . By [F2], this is $S_{A, B} (X) = C$ for the returned $X = U Y V^{*}$ .

4.1F3step 3.1∎

Under the stated spectral-disjointness condition, [F3] gives uniqueness for this right-hand side; hence the returned exact solution is the unique one.

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Sources

Valeria Simoncini, Computational Methods for Linear Matrix Equations, SIAM Review 58(3), 2016 (standard reference, not scraped)
Elias Jarlebring, Numerical Methods for Lyapunov Equations, lecture notes in numerical linear algebra, 2017 (standard reference, not scraped)