부산대학교 도서관

Summary: ``We propose iterative projection methods for solving squareor rectangular consistent linear systems $\bf Ax = b$. Existingprojection methods use sketching matrices (possibly randomized) togenerate a sequence of small projected subproblems, but even thesmaller systems can be costly. We develop a process that appends onecolumn to the sketching matrix each iteration and converges in a finitenumber of iterations whether the sketch is random or deterministic. Ingeneral, our process generates orthogonal updates to the approximatesolution ${\bf x}_k$. By choosing the sketch to be the set of allprevious residuals, we obtain a simple recursive update and convergencein at most rank$(\bf A)$ iterations (in exact arithmetic). By choosinga sequence of identity columns for the sketch, we develop ageneralization of the Kaczmarz method. In experiments on large sparsesystems, our method (PLSS) with residual sketches is competitive withLSQR and LSMR and with residual and identity sketches comparesfavorably with state-of-the-art randomized methods.''MR