Part $2$: Solving tridiagonal systems. Consider the solution of linear systems

$Ax=f$

with $fin{R}^n$ given, unknown solution $xin{R}^n$ and tridiagonal matrix

$A=([a_1,b_1,],[c_1,a_2,b_2,],[,$ddots,ddots,ddots,$],[,c_{n-2},a_{n-1},b_{n-1}],[,c_{n-1},a_n])in{R}^{nn}$

with $a_i>0,b_i<mn\; id="EditorElement\_1481993">0</mn>$ and You will investigate a particular variant of the LU$-$decomposition tailored

to the tridiagonal structure of $A.$

Exercise $5.$ Assume the ansatz $A=LU$ with

$($i$)$ Give an algorithm for the computation of $l_i,u_i,v_i,1in-1$ and $u_n.$ You may assume that the

algorithm can be executed, i$.$e$.,$ no division by zero occurs.

Hint: To get an idea, perform the multiplication $LU$ by hand for $n=4.$

Remark: If $A$ is strictly diagonally dominant, i$.$e$.,a_i>|c_{i-1}|+|b_i|,$ with $c_0=0=b_n,$ then the

algorithm can be executed $($sufficient condition$).$

$($ii$)$ Give the number of multiplications as well as divisions required by the algorithm in $($i$).$

$($iii$)$ Show that $A$ is invertible.