The question relates to iterative refinement. The idea is that the computer generates a solution to the linear system Ax=b which is inexact (due to roundoff errors), denoted by x0. You then iterate the algorithm given in (1) until it converges to (something much closer to) the exact solution, denoted x.
So k denotes iteration number. B is the matrix such that Bx0 = b, which is close to the value of A such that (B-1A ≈ I) where I is the identity matrix.
I'm stuck on trying to show that the first equation is equivalent to the second:
(1) xk+1 = xk + B-1(b - Axk)
(2) xk+1 - x = (I - B-1A)k(x0 - x)
The Attempt at a Solution
I [/B]have been working on this for a while but just can't figure out how to get to (2) or even get from (2) to (1). I can't find a way to get the exponent of k into the equation, the only manipulations I can think of is that the B-1b term can be substituted with x0. Also, that x can be written as A-1b but not sure if either of these help me.
Any tips appreciated!