Solving efficiently large single-step genomic best linear unbiased prediction models.

Single-step genomic BLUP (ssGBLUP) requires a dense matrix of the size equal to the number of genotyped animals in the coefficient matrix of mixed model equations (MME). When the number of genotyped animals is high, solving time of MME will be dominated by this matrix. The matrix is the difference of two inverse relationship matrices: genomic (G) and pedigree (A22 ). Different approaches were used to ease computations, reduce computing time and improve numerical stability. Inverse of A22 can be computed as A22-1=A22-A21A11-1A12 where Aij , i, j = 1,2, are sparse sub-matrices of A-1 , and numbers 1 and 2 refer to non-genotyped and genotyped animals, respectively. Inversion of A11 was avoided by three alternative approaches: iteration on pedigree (IOP), matrix iteration in memory (IM), and Cholesky decomposition by CHOLMOD library (CM). For the inverse of G, the APY (algorithm for proven and young) approach using Cholesky decomposition was formulated. Different approaches to choose the APY core were compared. These approaches were tested on a joint genetic evaluation of the Nordic Holstein cattle for fertility traits and had 81,031 genotyped animals. Computing time per iteration was 1.19 min by regular ssGBLUP, 1.49 min by IOP, 1.32 min by IM, and 1.21 min by CM. In comparison with the regular ssGBLUP, the total computing time decreased due to omitting the inversion of the relationship matrix A22 . When APY used 10,000 (20,000) animals in the core, the computing time per iteration was at most 0.44 (0.63) min by all the APY alternatives. A core of 10,000 animals in APY gave GEBVs sufficiently close to those by regular ssGBLUP but needed only 25% of the total computing time. The developed approaches to invert the two relationship matrices are expected to allow much higher number of genotyped animals than was used in this study.