Application of vector extrapolation methods to consistent singular linear systems (Q751179)

The paper is principally concerned with the solution of consistent systems of linear equations \((*)\quad Bx=f,\) in which the square matrix B is allowed to be singular, by iterative methods of the form \(x(i+1)=x(i)+\omega \{f-Bx(i)\}\) (i\(\geq 0)\). The main result derived is as follows. Let k be the smallest integer for which, for some n, \(\sum c(i)\{x(n+i+1)-x(n+i)\}\) (0\(\leq i\leq k)\) with \(1=\sum c(i)\) (0\(\leq i\leq k)\), is zero. Then \((**)\quad x=\sum c(i)x(n+i)\) (0\(\leq i\leq k)\) is a solution of (*). Linear combinations of iterated vectors of the form (**) arise in a number of vector convergence acceleration schemes.