Enhancement of the Kaczmarz algorithm with projection adjustment

Abstract

Projection adjustment is a technique that improves the rate of convergence, as measured by the number of iterations needed to achieve a given level of performance, of the Kaczmarz algorithm (KA) for iteratively solving a system of consistent linear equations, however at the cost of requiring additional time per iteration and increased storage. This hinders the applicability of the previously published Kaczmarz algorithm with projection adjustment (KAPA) to large-scale problems. An enhancement EKAPA of KAPA that uses projection adjustment only for a small subset of the equations is proposed for significantly reducing the time and storage requirements. An analysis of the behavior of EKAPA is provided. An illustration is given to show that EKAPA using a small subset of the equations for projection adjustment can achieve a speed-up over KA similar to that of KAPA in terms of the number of iterations, but requires much less computer time and storage; hence, it is more suitable for large-scale problems.