Computing the condensed conditional gametic QTL relationship matrix and its inverse

Abstract. The inverse of the conditional gametic relationship matrix (G^-1) for a marked quantitative trait locus (MQTL) is required for estimation of gametic effects in best linear unbiased prediction (BLUP) of breeding values if marker data are available. Calculation of the "condensed" gametic relationship matrix G* – a version of G where linear dependencies have been removed – and its inverse G^*-1 is described using a series of simplified equations following a known algorithm. The software program COBRA (covariance between relatives for a marked QTL) is introduced, and techniques for storing and computing the condensed gametic relationship matrix G* and the non-zero elements of its inverse are discussed. The program operates with both simple pedigrees and those augmented by transmission probabilities derived from marker data. Using sparse matrix storage techniques, G* and its inverse can be efficiently stored in computer memory. COBRA is written in FORTRAN 90/95 and runs on a variety of computers. Pedigree data and information for a single MQTL in the German Holstein population are used to test the efficiency of the program.