Linear Algebra Algorithm for Large Scale Equations

Question

I was wondering how the following solution was proposed for the following system of equations:

$x_1$ = 1, $x_i$ = 1 - $(4x_{i-1})^{-1}$

For the following system of equations:

$x_1$ + $0.5x_2$ = 1.5

$0.5_{j-1}$ + $_j$ + $0.5_{j+1}$ = 2.0

... ... ... $0.5x_{99}$ + $x_{100}$ = 1.5

Personally, I find it really elegant that a large scale system could be so efficiently solved with a simple rule and a seed value, and I want to know how the rule was constructed with bypassing many rows.

Answer 1

While this is a very efficient rule, it does not solve the system of equations. You get $x_2=3/8$, hence $x_1 + x_2 = 11/8 \ne 1.5$.