Chapter 2.6, Problem 6CP

Let A be the $n\times n$ matrix with $n=1000$

and entries $A\left(i,i+1\right)=A\left(i+1,i\right)=1/2,A\left(i,i+2\right)=A\left(i+2,i\right)=1/2$

for all $i$ that fit within the matrix. (a) Print the nonzero structure spy (A). (b) Let $x_e$

be the vector of n ones. Set

$b=A{x}_e$, and apply the Conjugate Gradient Method, without preconditioner, with the Jacobi preconditioner, and with the Gauss-Seidel preconditioner. Compare errors of the three runs in a plot versus step number.