Table 1.

Time complexity of stochastic algorithms

Method	Overhead	Objective function evaluation
SLDF_REML$\left\{\begin{array}{l}\text{with precomputed GRM}\\ \text{with genotype matrix}\end{array}\right.$	$\mathcal{O}\left(n^2·(n_{\text{rand}}+c)·n_{\kappa }\right)$	$\mathcal{O}(n·c·n_{\kappa })$
	$\mathcal{O}(2m·n·(n_{\text{rand}}+c)·n_{\kappa })$	$\mathcal{O}(n·c·n_{\kappa })$
L_FOMC_REML	$\mathcal{O}(4m·n·n_{\text{rand}}·n_{\kappa })$	$\mathcal{O}(m·n·n_{\text{rand}})$
BOLT_LMM	$\mathcal{O}\left(n·c^2+m·c\right)$	$\mathcal{O}(4m·n·n_{\text{rand}}·n_{\kappa })$

n denotes the number of individuals, m the number of markers, and c the number of covariates. n_rand indicates the number of random probing vectors and is fixed at 15 in all numerical experiments. n_κ reflects the number of conjugate gradient iterations required to achieve convergence at a specified tolerance and can be bounded in terms of the spectral condition number of H₀. As noted in [8], implicit preconditioning of H₀ can be achieved by including the first few right singular vectors of the genotype matrix (or eigenvectors of the GRM) as covariates