bibkey: lanczos1950iteration authors: Cornelius Lanczos year: 1950 title: An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators doi: null url: https://nvlpubs.nist.gov/nistpubs/jres/045/jresv45n4p255_A1b.pdf claim: Krylov orthogonal recurrence specializes to symmetric tridiagonal reduction; the response application uses an explicitly normalized exact-arithmetic convention. strata_touched: [] license: citation-only triage: anchor

Symmetric tridiagonal recurrence

The primary publication is Journal of Research of the National Bureau of Standards 45(4), 255–282, Research Paper RP2133. Sections VII–VIII, pp. 267–270, equations (77)–(78) and the symmetric specialization (97), supply the orthogonal recurrence and tridiagonal reduction. The paper’s coefficient conventions are not identical to the normalized convention below.

The Static endpoint application, Theorem 29.8 uses $q_{-1}=0$, ${\beta }_0=0$, $q_0=b/\Vert b\Vert$ only when $b\ne 0$. The residual norm defines ${\beta }_{j+1}$ and is normalized only when nonzero. First breakdown ${\beta }_q=0$ means that $q$ directions span the cyclic subspace. A zero port has dimension zero with no starting direction. For the specified diagonal-three endpoint chain, $q_j=(-1{)}^j{e}_{j+1}$, ${\alpha }_j=3$, ${\beta }_j=1$ for $1\le j<n$, and ${\beta }_0={\beta }_n=0$. The continued fraction uses squared off-diagonals.

These are exact-arithmetic and supplied-port statements. Finite-precision breakdown, loss of orthogonality, numerical stopping and noisy observation require additional analysis. The Jacobi chain represents the same linear response; it does not supply native FIB arithmetic operations.