Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available during the execution of the Lanczos process. Four different types of bounds are obtained using Lanczos, Ritz, and Chebyshev polynomials. These bounds are compared theoretically and numerically. Furthermore we show how one can determine, after each Lanczos step, a probabilistic upper bound for the number of steps still needed (without performing these steps) to obtain an approximation to the largest or smallest eigenvalue within a prescribed tolerance.