Efficient approximations of the inverse of a part of the additive relationship matrix

Many computations as current implementations of single-step genomic BLUP require the inverse of A22, the additive relationship matrix between genotyped animals. Inversion of this matrix may be imeconsuming when the number of animals increases beyond tens of thousands. Sparsity pattern of its inverse, computed using pedigree information, may help to reduce computations. However, the inverse is sometimes extremely dense. An alternate implementation proposed in this study restricts the sparsity pattern of the inverse of A22 in order to approximate its inverse. Such approximations are then computed for 4 different orders of A22 (10,000; 25,000; 50,000 and 75,000). Using the best approximations, reliabilities of breeding values in equivalent model instead of regular animal model are slightly inferior. Moreover, computing times of these approximations are limited (e.g. 4h for an order of 75,000) and may be sped up by a factor of 3 using parallel computing.