A Krylov-subspace algorithm for Hermitian (symmetric) matrices that generates a tridiagonal matrix representation via short three-term recurrences, enabling efficient extraction of extreme eigenvalues and approximate spectral information for large sparse operators.

Principle

Apply successive Lanczos three-term recurrences to build an orthonormal basis of the Krylov subspace while generating a symmetric tridiagonal projection whose eigenvalues approximate those of the original Hermitian operator; exploit short recurrences for low per-step cost but monitor numerical orthogonality loss.

Demonstration

Computing the largest few eigenvalues of a large sparse symmetric stiffness matrix by running Lanczos iterations with selective full reorthogonalization or partial reorthogonalization to avoid spurious (ghost) eigenvalues caused by finite-precision breakdown.

Misapplication

Running Lanczos indefinitely without reorthogonalization in finite precision, which leads to loss of orthogonality and spurious repeated eigenvalues (ghosts) or incorrect multiplicities in the computed spectrum.

Consequence

Provides a compact tridiagonal representation from which a few extreme eigenpairs can be obtained cheaply, enabling spectral approximation, preconditioner construction, and model reduction for large Hermitian problems.

Reversal

Use of full orthogonalization methods like Arnoldi with complete Gram-Schmidt for general nonsymmetric problems, or direct dense eigensolvers that do not exploit sparsity and require much more memory and computation.

Boundary

Applies primarily to Hermitian (real symmetric) or complex Hermitian operators; behavior in finite precision requires attention (reorthogonalization strategies); not directly applicable to general nonsymmetric matrices without modification (then Arnoldi is preferred).

Semantic Tension

Tension with Arnoldi/other Krylov methods: Lanczos uses short three-term recurrences optimal for Hermitian problems but is vulnerable to numerical loss of orthogonality; Arnoldi handles non-Hermitian cases with longer recurrences and greater stability at higher cost.

Synthesis

An efficient method for Hermitian spectral approximation that trades short, low-cost recurrences for vulnerability to finite-precision orthogonality loss, producing a tridiagonal projection whose eigenvalues approximate the operator's dominant spectrum.