A method to approximate the inverse of a part of the additive relationship matrix

Single-step genomic predictions need the inverse of the part of the additive relationship matrix between genotyped animals (A22). Gains in computing time are feasible with an algorithm that sets up the sparsity pattern of A22inv (SP algorithm) using pedigree searches, when A22inv is close to sparse. The objective of this study is to present a modification of the SP algorithm (RSP algorithm) and to assess its use in approximating A22inv when the actual A22inv is dense. The RSP algorithm sets up a restricted sparsity pattern of A22inv by limiting the pedigree search to a maximum number of searched branches. We have tested its use on four different simulated genotyped populations, from 10 000 to 75 000 genotyped animals. Accuracy of approximation is tested by replacing the actual A22inv by its approximation in an equivalent mixed model including only genotyped animals. Results show that limiting the pedigree search to four branches is enough to provide accurate approximations of A22inv, which contain approximately 80% of zeros. Computing approximations is not expensive in time but may require a great amount of memory (at maximum, approximately 81 min and approximately 55 Gb of RAM for 75 000 genotyped animals using parallel processing on four threads).