Extrapolation and Convergence Criteria with Jacobi and Gauss-Seidel Iteration in Animal Models

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Abstract
A population of individuals was simulated to study convergence rate of an iterative method, a mix of Gauss-Seidel and second-order Jacobi, for solving mixed model equations for an animal model. The solutions drifted for many iterations and their accuracy for converged solutions was far from that suggested by criteria such as the difference between right-hand and left-hand sides or a modified difference between consecutive solutions. The drift in later rounds of iteration closely followed a geometric progression, and formulas were derived for estiamting: 1) the true solutions via exponential extrapolation, 2) relationships between various convergence criteria, and 3) number of rounds needed to increase the accuracy of solutions by an arbitrarily specified factor. A range of relaxation factors was studied. The accuracy of solutions was very sensitive to the value of this factor in the absence of extrapolation. The optimal relaxation factor was lower when solutions were extrapolated, but its value was not as critical in this case.