Abstract
The aim of this study was to compare iterative and direct solvers for estimation of marker effects in genomic selection. One iterative and two direct methods were used: Gauss-Seidel with Residual Update, Cholesky Decomposition and Gentleman-Givens rotations. For resembling different scenarios with respect to number of markers and of genotyped animals, a simulated data set divided into 25 subsets was used. Number of markers ranged from 1,200 to 5,925 and number of animals ranged from 1,200 to 5,865. Methods were also applied to real data comprising 3081 individuals genotyped for 45181 SNPs. Results from simulated data showed that the iterative solver was substantially faster than direct methods for larger numbers of markers. Use of a direct solver may allow for computing (co)variances of SNP effects. When applied to real data, performance of the iterative method varied substantially, depending on the level of ill-conditioning of the coefficient matrix. From results with real data, Gentleman-Givens rotations would be the method of choice in this particular application as it provided an exact solution within a fairly reasonable time frame (less than two hours). It would indeed be the preferred method whenever computer resources allow its use.
Highlights
Most applications in animal breeding involve the solution of systems of equations with very large numbers of unknowns
As the number of animals increased, processing time with Gauss-Seidel with Residual Update (GSRU) increased at a higher rate than with GG and Cholesky Decomposition (CHD)
As the number of markers increases, the advantage of GSRU over the direct methods persists over an increasing number of animals, up to the point when it is the fastest method for any number of animals
Summary
Most applications in animal breeding involve the solution of systems of equations with very large numbers of unknowns. With multiple or single trait animal models or random regression test-day models the number of parameters increases as a function of the number of animals being evaluated, which are not so rarely in the hundreds of thousands or above. The coefficient matrices that arise from these kinds of problems are too large to be stored in high speed memory. For this reason, iterative solvers, both on data or on the mixed model equations, have gained high popularity and are widely employed in genetic evaluation of livestock. When an iterative algorithm is applied for the solution of a linear system, one cannot obtain the elements of the inverse of the coefficient matrix. Prediction error variances and standard errors of estimates have to be calculated in an indirect way and are approximated values
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