Problem $1$: $(60$p$)$

Solve a system of linear equations $Ax=b$ using Gaussian elimination with and without partial pivoting.

Use a matrix size of $10001000$ initialized with random numbers in the interval $-0\mathrm{.}5,0\mathrm{.}5.$ Initialize the

right hand side by computing $b=Az$ with all $z$ values $=1.($Thus$,$ the solution is in fact known.$).$

Write your own MATLAB function that performs the initialization of $A,b$ and $z.$ The matrix size is chosen

with the intent to make the running time long enough for a reasonably good time measurement, but not

excessively long. Use tic and toc to measure execution time.

i$)$ Write your own MATLAB function in the spirit of the pseudocode in the lecture slides $($from the

book$)$ that performs Gaussian Elimination without partial pivoting and apply it to solve for $x$ in

$Ax=b$ with A and $b$ generated by your code as described above. Then compute the mean squared

error $($MSE$)\frac{(x-1{)}^2,1}{N},$ and the square root thereof $($SMSE$).$ Measure the time to compute $x.$

Do not include the time to initialize $A,b$ and $z,$ and the time to compute the average mean

squared error $,$ and the square root thereof. Thus, only measure the time to compute

$x$ once A and $b$ are known. $(20$p$)$

a$.$ Carry this out in single precision. Report SMSE and time. Note that by default all variables are

represented in double precision. Single precision must be specified explicitly $(20$p$)$

b$.$ Carry this out in double precision. Report SMSE and time. $(20$p$)$

ii$)$ Write your own MATLAB function that performs Gaussian Elimination with partial pivoting. The

coding should be at a comparable level as in a$)($i$.$e$.,$ not by using high level MATLAB function

calls$).$ Then carry out the same set of computations and measurements as in a$)$ for the same A

and $b($Do not reinitialize A and $b).$

iii$)$ Solve $Ax=b$ using the MATLAB built in function Linsolve for the same A and $b$ as in a$)$ and b$)$ using

double precision, compute MSE and SMSE and measure the time. Report SMSE and time.

Problem $2(40$p$)$:

Write a MATLAB function that initializes the matrix such that $a(i,j)=\frac{1}{i+j-1}$ for $i,j=1,2,$dots,$1000.$

Initialize the vector $z$ with all values $1$ as in Problem $1$ and compute $b=Az$ as in Problem $1.$

a$)$ Use your MATLAB function for Gaussian Elimination with partial pivoting to compute $x$ in $Ax=b,$

the MSE and SMSE and measure the time. $(20$p$)$

i$.$ Carry this out in single$-$precision and report SMSE and time.

ii$.$ Carry this out in double precision and report SMSE and time.

b$)$ Use the MATLAB Linsolve function to compute $x,$ then compute MSE and SMSE and measure the

time. Report SMSE and time. $(20$p$)$

The matrix A is known as the Hilbert matrix, which is ill$-$conditioned.