Let $T_{}in{R}^{nn}$ be the tridiagonal matrix defined as

$T_{}=\left[\begin{array}{ccccc}& -1& ?& ?& ?\\ -1& & -1& ?& ?\\ ?& -1& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& ?\\ ?& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& -1& ?\\ ?& -1& & ?& ?\end{array}\right]$

$($a$)$ Using only information about the entries of $T_{},$ for what values of

$$ can we guarantee that $T_{}$ is non$-$singular with only positive eigenvalues?

$($b$)$

For what values of $$ will the Jacobi FPI converge to the solution

of $T_{}$vec$(x)=$vec$(b)$ for any initial approximation vec$(x{)}_0$ and any right$-$hand side vec$(b)?($Hint:

study the iteration matrix.$)$

$($c$)$

For one such value of $$ from $($b$),n=5,$ and vec$(x{)}_0=$vec$(0)$ write a script

which applies $1000$ iterations of both Jacobi and Forward Gauss$-$Seidel for this

matrix and vec$(b)=$vec$(1)$ the vector of ones. For vec$(r{)}_i=$vec$(b)-T_{}$vec$(x{)}_i,$ plot $||vec(r{)}_i|\frac{|}{|}|vec(r{)}_0||$ for all $i$

for both Jacobi and Forward Gauss$-$Seidel.