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). 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. 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.

中文翻译：

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不同步骤终止条件下单步GBLUP和SNPBLUP的收敛行为

预处理的共轭梯度（PCG）方法是当前迭代选择遗传评价的方法。通常选择两次连续迭代之间的相对差和方程组的相对残差作为PCG方法在动物育种中的终止标准。但是，我们的初步分析表明，这两个常用的终止标准可能报告说，应用于单步单核苷酸多态性最佳线性无偏预测（ssSNPBLUP）的PCG方法尚未收敛，而解决方案对于实际使用而言足够准确。因此，这项研究的目的是提出两个（部分）在其他领域已经制定但在动物育种中是新的终止标准，并将其行为与动物育种中针对ssSNPBLUP的PCG方法广泛使用的两个终止标准的行为进行比较。还比较了ssSNPBLUP的收敛模式与单步基因组BLUP（ssGBLUP）的收敛模式。在先前工作的基础上，我们提出了两个终止标准，其中考虑了方程组系统的特性。这两个终止标准与迭代相对于真实解的相对误差直接相关。基于猪和奶牛的数据集，我们表明，当对ssSNPBLUP使用二级预处理器时，ssSNPBLUP和ssGBLUP的预处理系数矩阵具有相似的属性。所以，基于迭代次数相对于真实解的相对误差，应用于ssSNPBLUP和ssGBLUP的PCG方法类似地收敛。对于两个建议的终止标准，在ssSNPBLUP和ssGBLUP之间都观察到了类似的收敛行为。但是，终止标准定义为应用于奶牛评估时的相对残差并非如此。我们的结果表明，将PCG方法应用于ssSNPBLUP和ssGBLUP可以类似地收敛。提出的两个终止标准始终描绘了这些相似的收敛行为，我们建议将它们用于比较不同模型的收敛特性和进行常规评估。但是，终止标准并非定义为应用于奶牛评估时的相对残差。我们的结果表明，将PCG方法应用于ssSNPBLUP和ssGBLUP可以类似地收敛。提出的两个终止标准始终描绘了这些相似的收敛行为，我们建议将它们用于比较不同模型的收敛特性和进行常规评估。但是，终止标准并非定义为应用于奶牛评估时的相对残差。我们的结果表明，将PCG方法应用于ssSNPBLUP和ssGBLUP可以类似地收敛。提出的两个终止标准始终描绘了这些相似的收敛行为，我们建议将它们用于比较不同模型的收敛特性和进行常规评估。