Hermitian Lanczos method descriptor.

Usage

lanczos(
  max_subspace = NULL,
  max_restarts = NULL,
  block = 1L,
  check_stride = 0L,
  reorthogonalize = TRUE,
  completeness = NULL
)

Arguments

max_subspace: Optional maximum active Krylov subspace size m (the ARPACK ncv). Must be at least k + 1 (k + block for block Lanczos). The native thick-restart path keeps the active basis bounded by this value across restart cycles; unrestarted reference paths build at most this many Lanczos vectors. This is the only subspace-size control: the solve's maxit argument is an iteration (restart) limit.
max_restarts: Optional non-negative integer giving the maximum number of thick-restart cycles allowed before stopping with whatever has converged. Default 100L. Equivalent to the solve's maxit argument on thick-restart routes; supplying both with different values is an error.
block: Native block size. 1L selects the scalar path; for a matrix-free operator this remains the reference Hermitian Lanczos boundary. Values greater than one select the native block Krylov path where supported, including real Hermitian matrix-free callbacks.
check_stride: Native block thick-restart mid-sweep convergence stride. 0L (default) evaluates convergence once per full sweep (legacy). A positive N evaluates convergence every N block iterations within a sweep, letting a warm start that converges after a few blocks stop early instead of paying a full cold-sized sweep. Mid-sweep checks never consume extra operator applications and never change results at check_stride = 0L. This control applies only to native block paths.
reorthogonalize: Whether to apply full reorthogonalization. The native path always reorthogonalizes (DGKS x2) and ignores this flag; it is preserved for the R reference solver's public API.
completeness: Target-completeness check run after a certified Hermitian solve (every target except interval(), which verifies its own set; smallest_magnitude() is checked as nearest(0) and both_ends() one end at a time). Unless the check verifies the set, certificate(fit)$passed is FALSE (with residual_passed = TRUE). "inertia" proves completeness deterministically by counting eigenvalues with an $LDL^T$ factorisation of A - t B (see eigen_count()); it needs an explicit dense or sparse matrix (a matrix-free operator with n at most getOption("eigencore.completeness_materialize_limit", 2000) is materialised by n applies) and reports "inertia_verified", "inertia_failed" (repaired when possible, otherwise passed = FALSE) or "inertia_inconclusive" (an eigenvalue cluster straddles the target edge within the residual bound and the counts cannot show that a correct choice was returned; a repeated eigenvalue at the edge is verified as a tie, recorded in certificate(fit)$completeness$tie). "probe" runs a short block Lanczos process on the operator deflated against the returned eigenvectors (from a fixed-seed start; the global random stream is not touched) and, if it finds a more-preferred eigenvalue outside the returned set (for example a missed copy of a repeated eigenvalue), repairs the result with a deflated complement solve; it is probabilistic (it can prove a set incomplete but not complete) and is the check for large matrix-free operators; for nearest() targets it runs on the route's own $(A - \sigma I)^{-1}$ after shift-invert, otherwise on $(A - \sigma I)^2$. Generalized problems are probed in the standard space $R^{-T} A R^{-1}$, $B = R^T R$. "auto" uses the inertia check when the matrix is explicit (or materialisable) and its predicted factorisation time is at most max(getOption("eigencore.completeness_inertia_seconds", 0.5), getOption("eigencore.completeness_inertia_ratio", 1) * solve time), and the probe otherwise. "none" skips the check. NULL (default) uses getOption("eigencore.target_completeness", "auto"). The outcome is recorded in certificate(fit)$target_completeness; see the "Certificates" vignette.

Value

An eigencore_method descriptor selecting Lanczos iteration.