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Table 2 Characteristics of preconditioned (deflated) coefficient matrices, and of PCG and DPCG methods for solving ssSNPBLUP applied to the field dataset

From: A second-level diagonal preconditioner for single-step SNPBLUP

\(\text{Model}^{\mathrm{a}}\) Method \(k_{O}/k_{S}^{\mathrm{b}}\) \(\lambda _{min}^{\mathrm{c}}\) \(\lambda _{max}^{\mathrm{c}}\) \(\kappa ^{\mathrm{d}}\) \(\textit{N}^{\mathrm{e}}\) \(\text{Iterative time}^{\mathrm{f}}\) \(\text{Time/iter.}^{\mathrm{g}}\) \(\text{Total time}^{\mathrm{h}}\)
MS PCG 1 \(3.70\times 10^{-5}\) \(1.75\times 10^{3}\) \(4.74\times 10^{7}\) 10,000 44,808 4.5 46,081
MS PCG \(10^{-1}\) \(1.18\times 10^{-5}\) \(1.77\times 10^{2}\) \(1.51\times 10^{7}\) 10,000 51,768 5.2 53,550
MS PCG \(10^{-2}\) \(4.37\times 10^{-6}\) \(1.95\times 10^{1}\) \(4.45\times 10^{6}\) 6210 34,139 5.5 35,812
MS PCG \(10^{-3}\) \(3.99\times 10^{-6}\) 5.08 \(1.27\times 10^{6}\) 3825 19,043 5.0 20,866
MS PCG \(10^{-4}\) \(1.50\times 10^{-6}\) 5.07 \(3.37\times 10^{6}\) 7336 54,326 7.4 56,475
MS DPCG 1 \(2.86\times 10^{-5}\) 4.77 \(1.67\times 10^{5}\) 748 6527 8.7 17,229
MS DPCG \(10^{-1}\) \(1.41\times 10^{-5}\) 4.77 \(3.37\times 10^{5}\) 1211 11,864 9.8 22,947
MS DPCG \(10^{-2}\) \(9.17\times 10^{-6}\) 4.77 \(5.20\times 10^{5}\) 1778 17,030 9.6 28,615
MS DPCG \(10^{-3}\) \(7.50\times 10^{-6}\) 4.77 \(6.36\times 10^{5}\) 2569 23,676 9.2 35,497
Liu PCG 1 \(7.38\times 10^{-6}\) \(1.43\times 10^{2}\) \(1.93\times 10^{7}\) 10,000 44,122 4.4 45,083
Liu PCG \(10^{-1}\) \(3.66\times 10^{-6}\) \(1.52\times 10^{1}\) \(4.14\times 10^{6}\) 6049 31,085 5.1 32,018
Liu PCG \(10^{-2}\) \(4.29\times 10^{-6}\) 5.07 \(1.18\times 10^{6}\) 2669 13,225 5.0 13,888
Liu PCG \(10^{-3}\) \(3.51\times 10^{-6}\) 5.07 \(1.44\times 10^{6}\) 3606 20,578 5.7 21,458
Liu PCG \(10^{-4}\) \(1.69\times 10^{-6}\) 5.07 \(3.00\times 10^{6}\) 7033 33,534 4.8 34,675
Liu DPCG 1 \(5.40\times 10^{-6}\) 5.31 \(9.85\times 10^{5}\) 2877 22,791 7.9 26,521
Liu DPCG \(10^{-1}\) \(6.91\times 10^{-6}\) 4.77 \(6.90\times 10^{5}\) 1628 14,231 8.7 18,049
Liu DPCG \(10^{-2}\) \(5.23\times 10^{-6}\) 4.77 \(9.11\times 10^{5}\) 2234 23,244 10.4 28,057
Liu DPCG \(10^{-3}\) \(4.31\times 10^{-6}\) 4.77 \(1.11\times 10^{6}\) 3106 34,950 11.3 39,603
  1. \({}^{\mathrm{a}}\)MS = ssSNPBLUP model proposed by Mantysaari and Stranden [7]; Liu = ssSNPBLUP model proposed by Liu et al. [5];
  2. \({}^{\mathrm{b}}\)Parameters used for the second-level preconditioner;
  3. \({}^{\mathrm{c}}\)Smallest and largest eigenvalues of the preconditioned (deflated) coefficient matrix;
  4. \({}^{\mathrm{d}}\)Condition number of the preconditioned (deflated) coefficient matrix;
  5. \({}^{\mathrm{e}}\)Number of iterations. A number of iterations equal to 10,000 means that the method failed to converge within 10,000 iterations;
  6. \({}^{\mathrm{f}}\)Wall clock time (seconds) for the iterative process;
  7. \({}^{\mathrm{g}}\)Average wall clock time (seconds) per iteration;
  8. \({}^{\mathrm{h}}\)Wall clock time (seconds) for a complete process (including I/O operations)
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Genetics Selection Evolution

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