A fast indirect method to compute functions of genomic relationships concerning genotyped and ungenotyped individuals, for diversity management

BACKGROUND: Pedigree-based management of genetic diversity in populations, e.g., using optimal contributions, involves computation of the [Formula: see text] type yielding elements (relationships) or functions (usually averages) of relationship matrices. For pedigree-based relationships [Formula: see text] , a very efficient method exists. When all the individuals of interest are genotyped, genomic management can be addressed using the genomic relationship matrix [Formula: see text] ; however, to date, the computational problem of efficiently computing [Formula: see text] has not been well studied. When some individuals of interest are not genotyped, genomic management should consider the relationship matrix [Formula: see text] that combines genotyped and ungenotyped individuals; however, direct computation of [Formula: see text] is computationally very demanding, because construction of a possibly huge matrix is required. Our work presents efficient ways of computing [Formula: see text] and [Formula: see text] , with applications on real data from dairy sheep and dairy goat breeding schemes. RESULTS: For genomic relationships, an efficient indirect computation with quadratic instead of cubic cost is [Formula: see text] , where Z is a matrix relating animals to genotypes. For the relationship matrix [Formula: see text] , we propose an indirect method based on the difference between vectors [Formula: see text] , which involves computation of [Formula: see text] and of products such as [Formula: see text] and [Formula: see text] , where [Formula: see text] is a working vector derived from [Formula: see text] . The latter computation is the most demanding but can be done using sparse Cholesky decompositions of matrix [Formula: see text] , which allows handling very large genomic and pedigree data files. Studies based on simulations reported in the literature show that the trends of average relationships in [Formula: see text] and [Formula: see text] differ as genomic selection proceeds. When selection is based on genomic relationships but management is based on pedigree data, the true genetic diversity is overestimated. However, our tests on real data from sheep and goat obtained before genomic selection started do not show this. CONCLUSIONS: We present efficient methods to compute elements and statistics of the genomic relationships [Formula: see text] and of matrix [Formula: see text] that combines ungenotyped and genotyped individuals. These methods should be useful to monitor and handle genomic diversity.