A·q_j = β_j q_{j-1} + α_j q_j + β_{j+1} q_{j+1}
The
Lanczos algorithm is an iterative Krylov subspace method for finding extreme eigenvalues of large sparse symmetric matrices. After k steps, it produces a tridiagonal k×k matrix T whose eigenvalues (Ritz values) converge to the extreme eigenvalues of the full n×n matrix A. Convergence is fastest for well-separated eigenvalues. Practical implementations must re-orthogonalize to combat floating-point loss of orthogonality. The algorithm underlies many physics solvers (quantum chemistry, lattice QCD) and graph analysis (spectral clustering, PageRank).