A rapid method for computing the inverse of the gametic covariance matrix between relatives for a marked Quantitative Trait Locus

The inverse of the gametic covariance matrix between relatives, G(-1), for a marked quantitative trait locus (QTL) is required in best linear unbiased prediction (BLUP) of breeding values if marker data are available on a QTL. A rapid method for computing the inverse of a gametic relationship matrix for a marked QTL without building G itself is presented. The algorithm is particularly useful due to the approach taken in computing inbreeding coefficients by having to compute only few elements of G. Numerical techniques for determining, storing, and computing the required elements of G and the nonzero elements of the inverse are discussed. We show that the subset of G required for computing the inbreeding coefficients and hence the inverse is a tiny proportion of the whole matrix and can be easily stored in computer memory using sparse matrix storage techniques. We also introduce an algorithm to determine the maximum set of nonzero elements that can be found in G(-1 )and a strategy to efficiently store and access them. Finally, we demonstrate that the inverse can be efficiently built using the present techniques for very large and inbred populations.