Restricted Maximum Likelihood Estimation of Variance Components in Animal Model Using Sparse Matrix Inversion and a Supercomputer

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The estimation of variance components by EM-type REML algorithms requires repeated inversion of the coefficient matrix of the mixed model equations. In the animal model, the rank of the coefficient matrix is usually larger than the number of records. If inversion is done by dense matrix algorithms, equations are limited to fewer than 2000. This study investigated the inversion of large coefficient matrices by direct sparse matrix solvers in analyzing conformation final scores of selected Holstein herds. The largest model contained 36,771 animal effects, 30,739 permanent environmental effects, 6102 herd × sire interactions, 532 herd-time classes, and 55 unknown parent groups, resulting in a coefficient matrix of order 74,199. Inversion was by sparse matrix package SMPAK, partly vectorized for the Cray-2 supercomputer. The convergence rate was accelerated by modified Aitken's extrapolation. For most rounds, the traces were based on a sample of inverse elements, resulting in up to 25 times lower cost per round. The largest model required 8 MWords (64 Mbytes) memory and used 5h central processing unit time for one inverse or 10h CPU time for complete run. Estimates were .67, 6.08, 4.62, and 3.47 for variance of herd by sire, permanent environment, animal, and residual, respectively.