An orthogonally accumulated projection method for symmetric linear system of equations (Q310195)

The authors recently published what they call an accumulated projection (AP) method [\textit{W. Peng} and \textit{Q. Lin}, ``A non-Krylov subspace method for solving large and sparse linear system of equations'', Numer. Math. Theory Methods Appl. 9, No. 2, 289--314 (2016; \url{doi:10.4208/nmtma.2016.y14014})]. The idea is to approximate the solution of a system \(Ax=b\) by its projection on a sequence of growing subspaces. In this paper that sequence is constructed by an orthogonally AP method, which means that a previous subspace is extended with a vector orthogonal to it. This is a Krylov subspace algorithm like Lanczos and conjugate gradient and it uses a three-term recurrence relation for the orthogonal basis for the column space of \(A\). Loss of orthogonality is dealt with by a restarting procedure. Eight successive versions of the algorithm are formulated from the basic version to versions that are taking different issues into account like breakdown, non-symmetric \(A\), and a variant where the matrix is extended with additional rows.