SWhat changed
Lanczos develops progressive polynomial and minimized-iteration procedures for spectral and resolvent problems, including the symmetric three-term orthogonal recurrence and a two-sided nonsymmetric construction, with numerical vibration examples and proposed integral/differential operator extensions.
Scientific scope
Lanczos develops progressive polynomial and minimized-iteration procedures for spectral and resolvent problems, including the symmetric three-term orthogonal recurrence and a two-sided nonsymmetric construction, with numerical vibration examples and proposed integral/differential operator extensions. Exact October 1950 NBS 45(4), 255-282 paper. Finite-matrix exact-arithmetic recurrence is distinct from floating-point behavior and infinite-operator convergence. The source explicitly discusses missing axes and repeated/defective eigenvalues: one starting chain need not recover the entire eigensystem. Symmetric orthogonal recurrence differs from two-sided nonsymmetric biorthogonalization, whose scalar denominators can vanish with nonzero residual vectors; nonzero initial moments alone do not make it breakdown-free. Later look-ahead theory is separate. The reported bar computations monitored and corrected orthogonality, so short recurrence alone is not a universal stability guarantee. Resonance poles require appropriate resolvent interpretation. Source claims of general Fredholm convergence are not independently certified here; operator domains, invertibility, compactness and boundary conditions require separate justification. Truncated spectral stabilization and coarse discretization are not automatic error certificates. Higher-dimensional input-output applications are proposed future work.
Evidence
Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.
Contributors
- Cornelius Lanczos · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators; exact 1950 NBS original
- NIST-deposited exact 1950 Lanczos journal identity
- Parlett, Taylor and Liu: publisher-deposited look-ahead Lanczos identity and abstract
Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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