Principal matrix logarithm and square root by branch-aware Schur–Parlett

Input a finite complex matrix $A\in {\mathbb{C}}^{n\times n}$ with $n\ge 1$, a branch request selecting principal logarithm or principal square root, finite Schur and Schur–Parlett controls, an initial partition of $\{1,\dots ,n\}$ into nonempty contiguous intervals for the Schur–Parlett call, a finite table of exact Jordan certificates keyed by contiguous intervals, and a finite table of supplied exact derivatives $f^{(k)}(\lambda )$ of the requested principal branch at the eigenvalues, for the orders required by those certificates, including merged blocks. These are exact input data of that same branch, not values computed from field arithmetic. A required absent derivative returns missing-jordan-data. The first Schur reduction is used only to certify the branch domain; no Jordan derivative data are needed if the branch check fails.

1. Run Finite-budget computed Francis Schur reduction with explicit statuses in its complex branch. Propagate any non-Schur status. Treat its returned $Q,T$ as candidates and verify exactly that $Q^{\ast }Q=I$, $A=QT{Q}^{\ast }$, and $T$ is upper triangular. If a check fails, return schur-certificate-failure.
2. The diagonal entries of this exact upper-triangular $T$ are the eigenvalues of $A$. Test each $\lambda$. If $\lambda \in (-\mathrm{\infty },0]$, return branch-domain-failure; if an exact sign comparison is unavailable, return branch-domain-undecided. This test occurs before any holomorphic matrix evaluation.
3. For the logarithm request, select the scalar branch $f(z)=\mathrm{Log}z$. For the square-root request, use $f(z)=z_{\mathrm{Log}}^{}$. Then call Finite Schur–Parlett recurrence with block merging and inconclusive statuses with $A$, $f$, its controls, the supplied initial partition, the Jordan-certificate table, and the supplied exact branch derivatives, forwarding their finite table to every diagonal evaluator. Propagate any Schur, Jordan-data, Sylvester, range, or budget status. On parlett-success, return its factored primary value with status branch-aware-candidate.

The procedure returns no floating-point error bound. The exact branch check ensures the scalar function is holomorphic at every eigenvalue.