Randomized admissible block coordinate descent methods for computing extreme eigenpairs of symmetric matrices

For iteratively computing the smallest eigenpair of a huge-scale symmetric matrix, we construct a randomized admissible block coordinate descent (BCD) method by first partitioning the matrix into a number of blocks with respect to its columns, then computing its next iterate through updating the current iterate along with a randomly selected block sub-vector of the affine coordinate direction, and finally obtaining the step-length through minimizing the Rayleigh quotient of the next iterate. This iteration method is indeed a blockwise variant of the admissibly randomized coordinate descent (CD) method proposed and analyzed recently by Bai & Chen (2025, Admissibly randomized coordinate descent methods for computing extreme eigenpairs of symmetric matrices. Numer. Linear Algebra Appl., 32, e70016:1–15), and it can also be considered as a randomized variant of the block CD method. For this class of iteration methods, we rigorously analyze its local and semilocal convergence properties, and solidly demonstrate its computational advantages over the admissibly randomized CD method, as well as the BCD method by numerical experiments.