Invited review: Reliability computation from the animal model era to the single-step genomic model era.

The calculation of exact reliabilities involving the inversion of mixed model equations poses a heavy computational challenge when the system of equations is large. This has prompted the development of different approximation methods. We give an overview of the various methods and computational approaches in calculating reliability from the era before the animal model to the era of single-step genomic models.

Convergence behavior of single-step GBLUP and SNPBLUP for different termination criteria.

Background: The preconditioned conjugate gradient (PCG) method is the current method of choice for iterative solving of genetic evaluations. The relative difference between two successive iterates and the relative residual of the system of equations are usually chosen as a termination criterion for the PCG method in animal breeding. However, our initial analyses showed that these two commonly used termination criteria may report that a PCG method applied to a single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is not converged yet, whereas the solutions are accurate enough for practical use.

Technical note: Genetic groups in single-step single nucleotide polymorphism best linear unbiased predictor.

Genetic groups, also called unknown or phantom parents groups, are often used in dairy cattle genetic evaluations to account for selection that cannot be accounted for by known genetic relationships. With the advent of genomic evaluations, the theory of genetic groups was extended to the so-called single-step genomic BLUP (ssGBLUP). In short, genetic groups can be fitted in ssGBLUP through regression effects, or by including them in the pedigree and computing the adequate combined pedigree and genomic relationship matrix.

Computational strategies for the preconditioned conjugate gradient method applied to ssSNPBLUP, with an application to a multivariate maternal model.

Background: The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is one of the single-step evaluations that enable a simultaneous analysis of phenotypic and pedigree information of genotyped and non-genotyped animals with a large number of genotypes. The aim of this study was to develop and illustrate several computational strategies to efficiently solve different ssSNPBLUP systems with a large number of genotypes on current computers.

Results: The different developed strategies were based on simplified computations of some terms of the preconditioner, and on splitting the coefficient matrix of the different ssSNPBLUP systems into multiple parts to perform its multiplication by a vector more efficiently.

A second-level diagonal preconditioner for single-step SNPBLUP.

Background: The preconditioned conjugate gradient (PCG) method is an iterative solver of linear equations systems commonly used in animal breeding. However, the PCG method has been shown to encounter convergence issues when applied to single-step single nucleotide polymorphism BLUP (ssSNPBLUP) models. Recently, we proposed a deflated PCG (DPCG) method for solving ssSNPBLUP efficiently.