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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. Therefore, the aim of this study was to propose two termination criteria that have been (partly) developed in other fields, but are new in animal breeding, and to compare their behavior to that of the two termination criteria widely used in animal breeding for the PCG method applied to ssSNPBLUP. The convergence patterns of ssSNPBLUP were also compared to the convergence patterns of single-step genomic BLUP (ssGBLUP). Results Building upon previous work, we propose two termination criteria that take the properties of the system of equations into account. These two termination criteria are directly related to the relative error of the iterates with respect to the true solutions. Based on pig and dairy cattle datasets, we show that the preconditioned coefficient matrices of ssSNPBLUP and ssGBLUP have similar properties when using a second-level preconditioner for ssSNPBLUP. Therefore, the PCG method applied to ssSNPBLUP and ssGBLUP converged similarly based on the relative error of the iterates with respect to the true solutions. This similar convergence behavior between ssSNPBLUP and ssGBLUP was observed for both proposed termination criteria. This was, however, not the case for the termination criterion defined as the relative residual when applied to the dairy cattle evaluations. Conclusion Our results showed that, Numerical Analysis

Improving longevity helps to increase the profit of the farmer, and it is seen as an important measure of improved animal welfare and sustainability of the sector. Breeding values for longevity have been published since in 1999 in the Netherlands. For AI-companies and farmers it is necessary that breeding values are accurately estimated and will remain stable for the rest of life. However, current breeding values for longevity of bulls seem to fluctuate more than expected. The main aim of this thesis was to revisit the genetics of longevity and develop a genetic evaluation model for longevity, where breeding values reflect the true breeding value quicker during early life and therefore breeding values become more stable. Genetic parameters were estimated for survival up to 72 months after first calving with a random regression model (RRM). Survival rates were higher in early life than later in life (99 vs. 95%). Survival was genetically not the same trait across the entire lifespan, because genetic correlations differ from unity between different time intervals, especially when intervals were further apart. Survival in the first year after first calving was investigated more in depth. Survival of heifers has improved considerably in the past 25 years, initially due to the focus on a high milk production. More recently, the importance of a high milk production for survival has been reduced. Therefore functional survival was defined as survival adjusted for within-herd production level. For survival the optimum age at first calving was around 24 months, whereas for functional survival calving before 24 months resulted in a higher survival. Over years, genetic correlations between survival in different 5-yr intervals were below unity, whereas for functional survival genetic correlations did not indicate that survival changed over years. This suggested that a genetic evaluation using historical data should analyze functional survival rather than survival. A new genetic