Iterative Algorithms for Solving Mixed Model Equations

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Abstract
Iterative algorithms for obtaining solutions for sire effects and breeding value estimates from progeny with records in mixed model procedures were compared. Successive overrelaxation with adaptive acceleration and a Jacobi conjugate gradient method seem to be more generally useful for two general areas of interest: 1) sire evaluation models, where some effects are absorbed; and 2) reduced animal models, where no effects are absorbed. An inverse of the relationship matrix may be included as part of the coefficient matrix and its inclusion will not prevent convergence. Key features include: 1) scaling the equations so that all diagonal elements are 1 and the scaled coefficient matrix remains real, symmetric, and positive definite; 2) calculation of a new relaxation parameter during run time to approximate the value that will yield fastest convergence; 3) calculation of a new relaxation parameter after convergence begins to slow down; and 4) use of numerically accurate and efficient convergence criteria. The Jacobi conjugate gradient method was 55% more efficient than successive overrelaxation in solving reduced animal model equations of order 3356. Number of iterations and total execution times for all iterations were: 83, 2.89 s and 169, 6.44 s, respectively. Another reduced animal model application with equations of order 38,139 converged in 38 s (50 iterations) usign successive overrelaxation. Solutions for sire equations constrained to full rank converged more quickly than unconstrained equations.