Coarse‐arithmetic coordinate descent for the resolution of semidefinite linear systems

Abstract

We propose an inexact coordinate descent method to solve a consistent equation of the type Ax = b where A is a real symmetric and positive semidefinite matrix. The admissible range of inexactitude allows to reduce every multiplication present in the computation to a scaling by a signed power of 2 for the sake of hardware simplification. Although the resulting descent is nonideal, the numerical experiments show that its rate of convergence remains close to that of exact coordinate descent, whether the coordinate selection is cyclic or random.Meanwhile, our algorithm outperforms both methods with an additional low complexity "weak greedy" selection of the coordinates, also based on scaling by signed powers of 2. Our method is based on an advanced use of varying relaxation coefficients in the Gauss-Seidel iteration, with special theoretical considerations when A is singular.