Successive Overrelaxation, Block Iteration, and Method of Conjugate Gradients for Solving Equations for Multiple Trait Evaluation of Sires

Abstract

A potential difficulty with mixed model equations for multiple trait evaluation of sires is solving the equations as the number of equations increases proportionally to the number of traits. Time required to obtain inverse solutions increases by the number cubed. Thus, iterative procedures often are used. Three iterative procedures, successive overrelaxation, block iteration with relaxation, and the method of conjugate gradients, were compared for four sets of multiple trait equations for sire evaluation. Equations were solved after absorption of equations for random herd-year-season effects. Equations for two and four traits each with test and complete data sets made up the four sets of equations. The two-trait system featured high heritabilities and large negative correlations among effects whereas the four-trait system had low heritabilities and smaller negative correlations. Rate of convergence for block iteration was faster than for successive overrelaxation, especially for the four-trait system and especially for more exacting convergence criteria. The method of conjugate gradients was efficient only for test data sets (30 and 60 equations) and was not competitive with the other methods for complete data sets (1426 and 2852 equations). Test data sets accurately predicted optimum relaxation factors for successive overrelaxation for complete data sets. Optimum relaxation factor for the two-trait system was 1.5 to 1.7 and for the four-trait system was 1.3 to 1.5. Gauss-Seidel iteration took 33 to 400%Received April 30, 1984. more rounds than successive overrelaxation with the optimum relaxation factor depending on stopping criteria and data set.

Effect of more stringent convergence criterion of estimated breeding values on response to selection

“…QUAAS et al (1984) used the square root of criterion 3 and suggested that 10 −4 would be sufficient. VAN VLECK and DWYER (1985) stated that for the purpose of ranking animals, a less precise stopping point may be required than when genetic evaluations are used to predict genetic trend. Also, all previous discussions of CC has focused on genetic evaluations of real data where computing time is often not an important factor.…”

“…Meaningful comparisons of two CC for different traits can only be arrived at by experience. VAN VLECK and DWYER (1985) stated that for the purpose of ranking animals, a less precise stopping point may be required than when genetic evaluations are used to predict genetic trend. QUAAS et al (1984) used the square root of criterion 3 and suggested that 10 −4 would be sufficient.…”

Solution of multiple trait animal models with missing data on some traits

“…Several original computing techniques have been studied to make MT‐AM feasible even on very large data sets. For example, people have extended to the multiple trait context procedures such as those referred to as “iteration on data” which decreases core memory requirements (S chaeffer and K ennedy 1986), “Reduced Animal Model” which decreases the size of the linear system to solve (Q uaas and P ollak 1981), or more sophisticated iterative methods such as the method of conjugate gradients (V an V leck and D wyer 1985).…”