Abstract
The inverse of the genomic relationship matrix (G-1) is used in the single-step genomic BLUP, which incorporates genomic, pedigree, and phenotype information for simultaneous genetic evaluation of genotyped and non-genotyped individuals. The rapidly growing number of genotypes is a constraint for inverting a huge G. The APY algorithm is an efficient method of solving this issue. Matrix G has a limited dimensionality. Dividing individuals into core and non-core, G-1 is approximated via the inverse partition of G for core individuals. The quality of the approximation depends on the core size and composition. The APY algorithm conditions genomic breeding values of the non-core individuals to those of the core individuals, leading to a diagonal block of G-1 for non-core individuals 
. Dividing observations into two groups (e.g., core and non-core, or genotyped and non-genotyped), any symmetric matrix can be expressed in APY and APY inverse expressions, equal to the matrix itself and its inverse, respectively. The change of Gnn to 
makes APY an approximate. The application of APY is extendable to the inversion of any large symmetric matrix with a limited dimensionality at a lower computational cost. Possible applications are: computing the pedigree relationship matrix (A) from the APY inverse of A-1, a diagonal block of A (same as the previous one, but avoiding unnecessary calculations), and the block of the block-diagonal preconditioner matrix corresponding to marker effects for iterative solving of marker effect model equations. Furthermore, APY may improve the matrix’s numerical condition.
Competing Interest Statement
MAN is employed at Livestock Improvement Corporation, Hamilton, New Zealand. He declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
