Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method

In many applications it is important to have reliable approximations for the extreme eigenvalues of a symmetric or Hermitian matrix. A method which is often used to compute these eigenvalues is the Lanczos method. Unfortunately 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 derive probabilistic bounds for the extreme eigenvalues. 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, an 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.