Beschreibung:
<jats:title>Summary</jats:title><jats:p>Single‐step genomic <jats:styled-content style="fixed-case">BLUP</jats:styled-content> (ssGBLUP) requires a dense matrix of the size equal to the number of genotyped animals in the coefficient matrix of mixed model equations (<jats:styled-content style="fixed-case">MME</jats:styled-content>). When the number of genotyped animals is high, solving time of <jats:styled-content style="fixed-case">MME</jats:styled-content> will be dominated by this matrix. The matrix is the difference of two inverse relationship matrices: genomic (<jats:bold>G</jats:bold>) and pedigree (<jats:bold>A</jats:bold><jats:sub>22</jats:sub>). Different approaches were used to ease computations, reduce computing time and improve numerical stability. Inverse of <jats:bold>A</jats:bold><jats:sub>22</jats:sub> can be computed as <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jbg12257-math-0001.png" xlink:title="urn:x-wiley:09312668:media:jbg12257:jbg12257-math-0001" /> where <jats:bold>A</jats:bold><jats:sup><jats:italic>ij</jats:italic></jats:sup>, <jats:italic>i</jats:italic>,<jats:italic> j</jats:italic> = 1,2, are sparse sub‐matrices of <jats:bold>A</jats:bold><jats:sup>−1</jats:sup>, and numbers 1 and 2 refer to non‐genotyped and genotyped animals, respectively. Inversion of <jats:bold>A</jats:bold><jats:sup>11</jats:sup> was avoided by three alternative approaches: iteration on pedigree (<jats:styled-content style="fixed-case">IOP</jats:styled-content>), matrix iteration in memory (<jats:styled-content style="fixed-case">IM</jats:styled-content>), and Cholesky decomposition by <jats:styled-content style="fixed-case">CHOLMOD</jats:styled-content> library (<jats:styled-content style="fixed-case">CM</jats:styled-content>). For the inverse of <jats:bold>G</jats:bold>, the <jats:styled-content style="fixed-case">APY</jats:styled-content> (algorithm for proven and young) approach using Cholesky decomposition was formulated. Different approaches to choose the <jats:styled-content style="fixed-case">APY</jats:styled-content> core were compared. These approaches were tested on a joint genetic evaluation of the Nordic Holstein cattle for fertility traits and had 81,031 genotyped animals. Computing time per iteration was 1.19 min by regular ss<jats:styled-content style="fixed-case">GBLUP</jats:styled-content>, 1.49 min by <jats:styled-content style="fixed-case">IOP</jats:styled-content>, 1.32 min by <jats:styled-content style="fixed-case">IM</jats:styled-content>, and 1.21 min by <jats:styled-content style="fixed-case">CM</jats:styled-content>. In comparison with the regular ss<jats:styled-content style="fixed-case">GBLUP</jats:styled-content>, the total computing time decreased due to omitting the inversion of the relationship matrix <jats:bold>A</jats:bold><jats:sub>22</jats:sub>. When <jats:styled-content style="fixed-case">APY</jats:styled-content> used 10,000 (20,000) animals in the core, the computing time per iteration was at most 0.44 (0.63) min by all the <jats:styled-content style="fixed-case">APY</jats:styled-content> alternatives. A core of 10,000 animals in <jats:styled-content style="fixed-case">APY</jats:styled-content> gave <jats:styled-content style="fixed-case">GEBV</jats:styled-content>s sufficiently close to those by regular ss<jats:styled-content style="fixed-case">GBLUP</jats:styled-content> but needed only 25% of the total computing time. The developed approaches to invert the two relationship matrices are expected to allow much higher number of genotyped animals than was used in this study.</jats:p>