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## Homework Statement

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 x

_{0}. 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 Bx

_{0}= b, which is close to the value of A such that (B

^{-1}A ≈ I) where I is the identity matrix.

I'm stuck on trying to show that the first equation is equivalent to the second:

## Homework Equations

(1) x

_{k+1}= x

_{k}+ B

^{-1}(b - Ax

_{k})

(2) x

_{k+1}- x = (I - B

^{-1}A)

^{k}(x

_{0}- 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

^{-1}b term can be substituted with x

_{0}. Also, that x can be written as A

^{-1}b but not sure if either of these help me.

Any tips appreciated!