Use of the preconditioned conjugate gradient algorithm as a generic solverfor mixed-model equations in animal breeding applications

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applica tions was evaluated with 16 test problems. The problems included single- an d multiple-trait analyses, with data on-beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differenc es between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering-within random effects. The preconditioned conjugate gradient program implemented with dou ble precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converg e for seven large models. The preconditioned conjugate gradient and success ive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to impl ement with the iteration on data for general models. However, successive ov errelaxation requires specific programming for each set of models. On avera ge, the preconditioned conjugate gradient algorithm converged in three time s fewer rounds of iteration than successive overrelaxation. With straightfo rward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive over relaxation. However, programs using the preconditioned conjugate gradient a lgorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can i nfluence rankings. The preconditioned conjugate gradient implemented with i teration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.