On estimating the largest eigenvalue with the Lanczos algorithm

The Lanczos algorithm applied to a positive definite matrix produces good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations. In this note we utilize this behavior and develop a simple algorithm which computes the largest eigenvalue. The algorithm is especially economical if the order of the matrix is large and the accuracy requirements are low. The phenomenon of misconvergence is discussed. Some simple extensions of the algorithm are also indicated. Finally, some numerical examples and a comparison with the power method are given.