Load carrying capacity of AISI9310 steel spur gears. II: Kotorii, H. and Takada, J. J. Mech. Eng. Lab. (Jpn) Jan. 1992 46 (1) 58–71 (in Japanese)

Abstract

We consider the generalized eigenvalue problem. (A − λM)x = 0, where A and M are large, sparse, symmetric matrices. For large problems finding only a few eigenpairs involves a major computational task. In a typical example from structural dynamic analysis with matrices of order 8000, O(109) operations are required to compute 50 eigenpairs. It is therefore interesting to examine the advantage that vector computers such as CYBER 205 can offer.We adopted our best versions of the Subspace Iteration Method and the simple Lanczos Method in order to take advantage of the special vector processor of the CYBER 205. Both techniques lend themselves to vectorization. Our extensive comparisons support the following general statements. Both methods require the triangular factorization of the same large n by n matrix. This factorization dominates the total computation as n → ∞ provided that the number of wanted eigenpairs, p, remains fixed (independent of n). However, simple Lanczos is at least an order of magnitude more efficient (in CPU-time) for the remainder of the computation. For p = 40, n = 500 the factorization time is not important and the full order of magnitude difference is seen in the total CPU-time. When p = 40, n = 8000 simple Lanczos is only 4 times faster than Subspace Iteration on the CYBER 205. This confirms experience on serial computers.For problems that cannot fit into primary storage, input/output becomes increasingly important. We found that the cost of input/output dominated over the CPU-cost for a problem that required twice the available primary storage on our CYBER 205. However, this will depend on the billing algorithm of the computer center. We conclude that problems which have a substantial overhead in reading and writing the matrices, should not be solved by the simple Lanczos Method, but by a Block Lanczos Method.