02919naa a2200289 a 450000100080000000500110000800800410001902400310006010000140009124502380010526000090034350003050035252016710065765300160232865300220234465300140236665300340238065300380241465300180245270000160247070000160248670000160250270000210251870000200253970000160255977300540257510577432019-11-25       2017    bl uuuu     u00u1 u    #d7 a10.2527/jas.2016.06992DOI1 aMASUDA, Y  aTechnical notebAvoiding the direct inversion of the numerator relationship matrix for genotyped animals in single-step genomic best linear unbiased prediction solved with the preconditioned conjugate gradient.h[electronic resource]  c2017  aArticle history: Received: July 05, 2016; Accepted: Aug 16, 2016; Published: February 2, 2017. This research was partially funded by the United States Department of Agriculture?s National Institute of Food and Agriculture (Agriculture and Food Research Initiative competitive grant 2015-67015-22936).  aABSTRACT. This paper evaluates an efficient implementation to multiply the inverse of a numerator relationship matrix for genotyped animals () by a vector (q). The computation is required for solving mixed model equations in single-step genomic BLUP (ssGBLUP) with the preconditioned conjugate gradient (PCG). The inverse can be decomposed into sparse matrices that are blocks of the sparse inverse of a numerator relationship matrix (A&#8722;1) including genotyped animals and their ancestors. The elements of A&#8722;1 were rapidly calculated with the Henderson?s rule and stored as sparse matrices in memory. Implementation of  was by a series of sparse matrix?vector multiplications. Diagonal elements of , which were required as preconditioners in PCG, were approximated with a Monte Carlo method using 1,000 samples. The efficient implementation of was compared with explicit inversion of A22 with 3 data sets including about 15,000, 81,000, and 570,000 genotyped animals selected from populations with 213,000, 8.2 million, and 10.7 million pedigree animals, respectively. The explicit inversion required 1.8 GB, 49 GB, and 2,415 GB (estimated) of memory, respectively, and 42 s, 56 min, and 13.5 d (estimated), respectively, for the computations. The efficient implementation required <1 MB, 2.9 GB, and 2.3 GB of memory, respectively, and <1 sec, 3 min, and 5 min, respectively, for setting up. Only <1 sec was required for the multiplication in each PCG iteration for any data sets. When the equations in ssGBLUP are solved with the PCG algorithm,  is no longer a limiting factor in the computations.  Copyright © 2016. American Society of Animal Science.  aCOMPUTATION  aGENOMIC SELECTION  aINVERSION  aNUMERATOR RELATIONSHIP MATRIX  aPRECONDITIONED CONJUGATE GRADIENT  aSPARSE MATRIX1 aMISZTAL, I.1 aLEGARRA, A.1 aTSURUTA, S.1 aLOURENCO, D.A.L.1 aFRAGOMENI, B.O.1 aAGUILAR, I.  tJournal of Animal Science, 2017gv. 95(1): 49-52.