The algorithm for proven and young (APY) from a different perspective

The inverse of the genomic relationship matrix (G-1) is used in genomic BLUP (GBLUP) and the single-step GBLUP. The rapidly growing number of genotypes is a constraint for inverting G. The APY algorithm efficiently resolves 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 [Formula]. Dividing observations into two groups (e.g., core and non-core, genotyped and non-genotyped, etc), 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 [Formula] makes APY an approximate. This change is projected to the other blocks of G-1 as well. The application of APY is extendable to the inversion of any large symmetric matrix with a limited dimensionality at a lower computational cost. Furthermore, APY may improve the numerical condition of the matrix or the equation system.