We are interested in solving $Ax=b.$ In this assignment, you will compare the

computing time of finding matrix inverses and finding matrix factorizations for large

sparse matrices.

The following line of code produces a commonly$-$used tridiagonal matrix of size $n($the

one$-$dimensional discrete Laplacian$)$:

e$=$ones $(n,1)$; A$\_$n$=$spdiags $([e-2**ee],-1$:$1,n,n$;

Part A: Complete the following, using the above line of code, to generate sparse $nn$

matrices.

For each $n$ that you are considering, generate a large sparse matrix $A_n$ and

measure the average computing time that Matlab's inv $()$ algorithm requires to find

the inverse of $A_{-}.$ Be sure to specify how many executions of inv $()$ you are

averaging over to obtain the average time. In the computing time estimate, do not

include the cost of constructing $A_n.$

Plot the average computing time $\frac{?}{\text{b}\text{a}\text{r}}(v)s$ for a variety of $n$ to visualize the relationship

between $n$ and computing time. Choose appropriate scales for the axes in your plot.

Analyze the plot to determine how the time for computing inv $()$ varies with $n$ for

large $n.$ Use big$-$Oh notation to describe the relationship and explain your findings.