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 the PCG method can converge similarly when applied to ssSNPBLUP and to ssGBLUP. The two proposed termination criteria always depicted these similar convergence behaviors, and we recommend them for comparing convergence properties of different models and for routine evaluations.@en

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. The DPCG method introduces a second-level preconditioner that annihilates the effect of the largest unfavourable eigenvalues of the ssSNPBLUP preconditioned coefficient matrix on the convergence of the iterative solver. While it solves the convergence issues of ssSNPBLUP, the DPCG method requires substantial additional computations, in comparison to the PCG method. Accordingly, the aim of this study was to develop a second-level preconditioner that decreases the largest eigenvalues of the ssSNPBLUP preconditioned coefficient matrix at a lower cost than the DPCG method, in addition to comparing its performance to the (D)PCG methods applied to two different ssSNPBLUP models.

Results
Based on the properties of the ssSNPBLUP preconditioned coefficient matrix, we proposed a second-level diagonal preconditioner that decreases the largest eigenvalues of the ssSNPBLUP preconditioned coefficient matrix under some conditions. This proposed second-level preconditioner is easy to implement in current software and does not result in additional computing costs as it can be combined with the commonly used (block-)diagonal preconditioner. Tested on two different datasets and with two different ssSNPBLUP models, the second-level diagonal preconditioner led to a decrease of the largest eigenvalues and the condition number of the preconditioned coefficient matrices. It resulted in an improvement of the convergence pattern of the iterative solver. For the largest dataset, the convergence of the PCG method with the proposed second-level diagonal preconditioner was slower than the DPCG method, but it performed better than the DPCG method in terms of total computing time.

Conclusions
The proposed second-level diagonal preconditioner can improve the convergence of the (D)PCG methods applied to two ssSNPBLUP models. Based on our results, the PCG method combined with the proposed second-level diagonal preconditioner seems to be more efficient than the DPCG method in solving ssSNPBLUP. However, the optimal combination of ssSNPBLUP and solver will most likely be situation-dependent.@en

Background: The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) method, such as single-step genomic BLUP (ssGBLUP), simultaneously analyses phenotypic, pedigree, and genomic informa- tion of genotyped and non-genotyped animals. In contrast to ssGBLUP, SNP effects are fitted explicitly as random effects in the ssSNPBLUP model. Similarly, principal components associated with the genomic information can be fitted explicitly as random effects in a single-step principal component BLUP (ssPCBLUP) model to remove noise in genomic information. Single-step genomic BLUP is solved efficiently by using the preconditioned conjugate gradi- ent (PCG) method. Unfortunately, convergence issues have been reported when solving ssSNPBLUP by using PCG. Poor convergence may be linked with poor spectral condition numbers of the preconditioned coefficient matrices of ssSNPBLUP. These condition numbers, and thus convergence, could be improved through the deflated PCG (DPCG) method, which is a two-level PCG method for ill-conditioned linear systems. Therefore, the first aim of this study was to compare the properties of the preconditioned coefficient matrices of ssGBLUP and ssSNPBLUP, and to document convergence patterns that are obtained with the PCG method. The second aim was to implement and test the effi- ciency of a DPCG method for solving ssSNPBLUP and ssPCBLUP. Results:For two dairy cattle datasets, the smallest eigenvalues obtained for ssSNPBLUP (ssPCBLUP) and ssGBLUP, both solved with the PCG method, were similar. However, the largest eigenvalues obtained for ssSNPBLUP and ssPCB- LUP were larger than those for ssGBLUP, which resulted in larger condition numbers and in slow convergence for both systems solved by the PCG method. Different implementations of the DPCG method led to smaller condition num- bers, and faster convergence for ssSNPBLUP and for ssPCBLUP, by deflating the largest unfavourable eigenvalues. Conclusions: Poor convergence of ssSNPBLUP and ssPCBLUP when solved by the PCG method are related to larger eigenvalues and larger condition numbers in comparison to ssGBLUP. These convergence issues were solved with a DPCG method that annihilates the effect of the largest unfavourable eigenvalues of the preconditioned coefficient matrix of ssSNPBLUP and of ssPCBLUP on the convergence of the PCG method. It resulted in a convergence pattern, at least, similar to that of ssGBLUP. @en