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

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Abstract
A potential difficulty with mixed model equations for multiple trait evalua- tion of sires is solving the equations as the number of equations increases propor- tionally 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 overre- laxation, block iteration with relaxation, and the method of conjugate gradients, were compared for four sets of multiple trait equations for sire evaluation. Equa- tions 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 herit- abilities and large negative correlations among effects whereas the four-trait system had low heritabilities and smaller negative correlations. Rate of con- vergence for block iteration was faster than for successive overrelaxation, espe- cially for the four-trait system and especially for more exacting convergence criteria. The method of conjugate gra- dients 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%