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- Title
Technical note: Avoiding 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.
- Authors
Masuda, Y.; Misztal, I.; Legarra, A.; Tsuruta, S.; Lourenco, D. A. L.; Fragomeni, B. O.; Aguilar, I.
- Abstract
This paper evaluates an efficient implementation to multiply the inverse of a numerator relationship matrix for genotyped animals (A22-1) 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-1) including genotyped animals and their ancestors. The elements of A-1 were rapidly calculated with the Henderson's rule and stored as sparse matrices in memory. Implementation of A22-1 q was by a series of sparse matrix–vector multiplications. Diagonal elements of A22-1, which were required as preconditioners in PCG, were approximated with a Monte Carlo method using 1,000 samples. The efficient implementation of A22-1 q 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, A22-1 is no longer a limiting factor in the computations.
- Subjects
GENOTYPES; CONJUGATE gradient methods; SPARSE matrices; MONTE Carlo method; ALGORITHMS
- Publication
Journal of Animal Science, 2017, Vol 95, Issue 1, p49
- ISSN
0021-8812
- Publication type
Article
- DOI
10.2527/jas.2016.0699