TASK $1.$ Calculate the number of arithmetic operations that are required to solve a tridiagonal system

$Tx=f$ of size $n$ using the Thomas algorithm. Based on this result, determine the asymptotic

complexity of the Thomas algorithm, and compare it to the asymptotic complexity of the standard

Gaussian elimination.

TASK $2.$ Rewrite the Thomas algorithm in terms of the arrays $A,B,$ and $C$ introduced to store the matrix

$T$ efficiently.

TASK $3.$ Implement the Thomas algorithm from TASK $2$ as a Python function. The input parameters to

the function should be the coefficient matrix $T($stored as three arrays $A,B,$ and $C)$ and the right$-$hand

side vector $f.$ The output should be the solution vector $x.$ The coefficient matrix and the right$-$hand

side should be defined in the main script and passed to the function that solves the system.

TASK $4.$ Test your code by solving the linear system of size $n=10$ with the values $A=2,$ and $B=C=$

$-1.$ Set the right$-$hand side to $f=1.$ To verify the correctness of your code, compare the solution

vector obtained from the Thomas algorithm to that obtained by applying the direct solver

numpy.linalg. solve $().$ For the latter, the coefficient matrix should be assembled.

TASK $5.$ Solve five linear systems $Tx=f$ with $A=2,B=C=-1$ and $f=1$ varying the problem size

$n$ between ${10}^6$ and ${10}^8.$ Record the execution times in seconds for each case. To accomplish this task,

explore the Python function timer $()$ from the package timeit $($refer to the code for matrix

multiplication covered in lectures$).$ Plot a graph where the obtained execution times are represented

as the function of the problem size $n.$ What are your conclusions about the cost of the Thomas

algorithm?