Effect of the CYBER 205 on Methods for Computing Natural Frequencies of Structures,

This report considers the generalized eigenvalue problem A - gamma MxO, 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, O10 to the 9th power operations are required to compute 50 eigenpairs. It is therefore interesting to examine the advantage that vector computers such as CYBER 205 can offer. The authors 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 X n matrix. This factorization dominates the total computation as n infinity 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 p40, n500 the factorization time is not important and the full order of magnitude difference is seen in the total CPU-time. When p40, n8000 simple Lanczos is only 4 times faster than Subspace Iteration on the CYBER 205. This confirms experience on serial computers.