On estimating the largest eigenvalue with the Lanczos algorithm

Parlett, B. N.; Simon, H.; Stringer, L. M.

doi:10.1090/S0025-5718-1982-0637293-9

On estimating the largest eigenvalue with the Lanczos algorithm
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by B. N. Parlett, H. Simon and L. M. Stringer PDF

Math. Comp. 38 (1982), 153-165 Request permission

Abstract:

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.

References

Dianne P. O’Leary, G. W. Stewart, and James S. Vandergraft, Estimating the largest eigenvalue of a positive definite matrix, Math. Comp. 33 (1979), no. 148, 1289–1292. MR 537973, DOI 10.1090/S0025-5718-1979-0537973-X
Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 570116
B. N. Parlett and J. K. Reid, Tracking the progress of the Lanczos algorithm for large symmetric eigenproblems, IMA J. Numer. Anal. 1 (1981), no. 2, 135–155. MR 616327, DOI 10.1093/imanum/1.2.135

Efficient and Optimal Methods for Finding the Largest Eigenvalue of a Real Symmetric Matrix