Consider the linear system, $2x_1+x_2=1,x_1+2x_2=-1,$ with solution $x_1=1,x_2=-1.$

Write optimal SOR method in component form and take three steps starting from initial

guess $x^{(0)}=(0,0{)}^T.$ Present the results in a table with the following format.

column $1$: $k($iteration step$)$

column $2$: $x_1^{(k)}(1$st component of computed solution vector$)$

column $3$: $x_2^{(k)}(2$nd component of computed solution vector$)$

column $4$: $||e_k||($error norm$)$

column $5$: $||e_k|\frac{|}{|}|e_{k-1}||($ratio of successive error norms$)$

Find the iteration matrix $B_{{}^{**}}$ and compute $||B_{{}^{**}}||,(B_{{}^{**}}).$ Does the method converge?