- #1

- 2

- 0

## Homework Statement

Show that [itex] {_{n+2}}x = _nx + M^{-1}(b-A{_nx}) [/itex]

## Homework Equations

[itex] {_{n+1}}x = _nx + M_1^{-1}(b-A{_nx}) [/itex]

[itex] {_{n+2}}x = {_{n+1}}x + M_2^{-1}(b-A{_{n+1}x}) [/itex]

[itex] M = M_1(M_1+M_2-A)^{-1}M_2 [/itex]

## The Attempt at a Solution

I tried was adding the two equations, which gives [itex] _{n+2}x = M_2^{-1}(b-A{_{n+1}x}) + _nx + M_1^{-1}(b-A{_nx}) [/itex]. But [itex] M_2^{-1}(b-A{_{n+1}}x) [/itex] devolves into a huge mess once I expand it. Subtraction is less helpful, as you are left without any [itex] _nx [/itex]. Directly substituting for [itex]_{n+1}x [/itex] in the second equation is even less helpful, as you are left with a huge mess of terms that can't really be factored. What am I missing?