This note records the current native Krylov default policy. Explicit user parameters always win. Eigensolver dimension defaults apply when krylov_dimension or retained_ritz_count is zero.

For the full algorithm inventory, scalar coverage, parameter list, and internal tuning values, see Krylov Algorithms.

Dimensions

The public parameters are:

parameter	meaning
`eigenvalue_count` / `nev`	Number of requested converged eigenpairs.
`krylov_dimension` / `ncv`	Maximum Lanczos or Arnoldi subspace dimension.
`retained_ritz_count` / `nkeep`	Native restart dimension retained between cycles.

The required restarted relation is:

nev <= nkeep < ncv <= problem_dimension

For non-restarted solves, nkeep is ignored.

Symmetric and Hermitian Problems

Symmetric/Hermitian Lanczos uses:

ncv = min(problem_dimension, max(20, 2*nev + 1))

nkeep = nev

This keeps the ARPACK-style implicit restart behavior by default: the solver retains the requested invariant subspace and uses the rest of the Krylov space for shifts. It is usually a good default for extremal Hermitian eigenvalues.

For clustered spectra, increase either or both of:

ncv

nkeep

Increasing nkeep gives thick-restart guard vectors, but weakens each restart filter because the number of discarded directions is ncv - nkeep.

Nonsymmetric Problems

Real and complex nonsymmetric Arnoldi use roomier defaults:

ncv = min(problem_dimension, max(20, 6*nev + 8))

nkeep = min(ncv - 1, max(2*nev, nev + 4))

Nonsymmetric restarts are more fragile than Hermitian Lanczos restarts because the projected problem can be nonnormal and, in real arithmetic, complex conjugate pairs occupy real Schur two-planes. The default therefore keeps guard Schur dimensions and gives the restart a larger search space.

This is intentionally more conservative than the minimal ARPACK guideline ncv > 2*nev. It costs more vector storage and local projected dense work, but has a better chance of converging without the user already knowing the matrix.

Iteration Limits

max_iterations is still an iteration budget, not a cure for a cramped subspace. Raising it can eventually converge many hard cases, but a too-small ncv may waste many restart cycles or stagnate. For difficult nonsymmetric problems, first try a larger ncv, then increase max_iterations.

External Comparisons

ARPACK comparison adapters can use the same default ncv formulas when krylov_dimension == 0. ARPACK does not expose a separate nkeep, so retained_ritz_count is necessarily a native-only control. Those adapters are benchmark/oracle infrastructure and should live outside core Uni20.

GMRES and refinement

Parameter	Default
GMRES restart length	30, capped by problem dimension and iteration budget
GMRES maximum Arnoldi steps	1000
GMRES relative residual tolerance	`sqrt(epsilon)` at inner arithmetic precision
GMRES absolute residual tolerance	Zero
Outer refinement maximum corrections	10
Outer relative residual tolerance	Working-precision epsilon
Outer absolute residual tolerance	Zero

These are configurable Uni20 defaults, not universal tolerances prescribed by the refinement literature. Acceptance uses recomputed residuals of the original operator. Optional operator norm and conditioning information is absent by default and adds diagnostics without changing these stopping tolerances. See GMRES and refinement for full semantics.