Rapid Inversion of Dominance Relationship Matrices for Noninbred Populations by Including Sire by Dam Subclass Effects

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Abstract
For estimation of dominance effects and dominance variance, the inverse of a dominance relationship matrix is required. Dominance effects can be partitioned into sire x dam or sire x maternal grandsire subclass effects that are inherited and residuals within subclass that are not inherited. The subclass effects have immediate use in predicting performance of offspring from prospective matings. A rapid method for directly computing the inverse relationship matrix of subclass effects is presented. The procedure is similar to Henderson's simple method of computing an inverse additive genetic relationship matrix. The inverse relationship matrix among subclass effects consists of a contribution from each subclass of coefficients of a matrix of maximum size 9 x 9. The algorithm can be modified to compute the inverse of the relationship matrix among sire x dam or sire x maternal grandsire subclasses and among individual dominance effects. Computing cost increases approximately linearly with dimensions of inverses. Dimensions could be several times the number of subclasses in the data because subclasses without records but providing relationship ties must be added. Computation of the inverse relationship matrix among 136,827 sire x maternal grandsire subclasses in a population of 765,868 animals required 163 central processing unit seconds on an IBM 3090 and less than 4 Mbytes of memory.