Rapid inversion of additive by additive relationship matrices by including sire-dam combination effects

Inverses of relationship matrices are useful for prediction of individual additive or nonadditive genetic merits and for estimation of variance components. An algorithm to form inverses of additive by additive relationship matrices rapidly from lists of individuals and their parents was developed. The algorithm uses simple recurrences among additive by additive and sire-dam combination effects to construct inverses for noninbred or inbred populations. Dimensions of matrices produced may be several times the number of individuals in the population because combination effects for sire-dam subclasses must be included in matrices. Rules of inheritance of sire-dam combination effects are the same as for dominance combination effects. Cost of forming inverses increases linearly with number of individuals. Each individual contributes 36 or fewer nonzero coefficients, and each sire-dam subclass contributes an additional 81 or fewer nonzero coefficients to the matrix. Computation of inverse of the relationship matrix due to 1003 sires and maternal grandsires of 765,868 cows required forming a matrix of order 137,830 and 4 Mbytes of memory.