Solving large test-day models by iteration on data and preconditioned conjugate gradient

A preconditioned conjugate gradient method was implemented into an iteratio n on a program for data estimation of breeding values, and its convergence characteristics were studied. An algorithm was used as a reference in which one fixed effect was solved by Gauss-Seidel method, and other effects were solved by a second-order Jacobi method. Implementation of the precondition ed conjugate gradient required storing four vectors (size equal to number o f unknowns in the mixed model equations) in random access memory and readin g the data at each round of iteration. The preconditioner comprised diagona l blocks of the coefficient matrix. Comparison of algorithms was based on s olutions of mixed model equations obtained by a single-trait animal model a nd a single-trait, random regression test-day model. Data sets for both mod els used milk yield records of primiparous Finnish dairy cows. Animal model data comprised 665,629 lactation milk yields and random regression test-da y model data of 6,732,765 test-day milk yields. Both models included pedigr ee information of 1,099,622 animals. The animal model {random regression te st-day model} required 122 {305} rounds of iteration to converge with the r eference algorithm, but only 88 {149} were required with the preconditioned conjugate gradient. To solve the random regression test-day model with the preconditioned conjugate gradient required 237 megabytes of random access memory and took 14% of the computation time needed by the reference algorit hm.