WARNING: Most newer computers have more than one processor $($core$).$ By

default, Matlab will use all the processors available by assigning one $``$thread$\text{'}\text{'}$ to

each available processor. This procedure will mess up our timings, so you should

tell Matlab to use only a single thread. To do this, you should start up Matlab with

the command line

matlab $-$singleCompThread

If you do not know how to start up Matlab from the command line, use the

following command at the beginning of your session.

maxNumCompThreads$(1)$;

Matlab will warn you that this command will disappear in the future, but it should

work fine now.

Exercise $4$: The Matlab command inv$($A$)$ computes an approximation for the

inverse of a matrix A$.$ Do not worry for now just how it does it$,$ but be

assured that it uses one of the most efficient and reliable methods available.

Matlab provides the commands tic and toc to measure computational time.

The tic command starts the stopwatch, and the toc command stops it$,$ either

printing the elapsed time or returning its value as in the

expression elapsedTime$=$toc;. The times are in seconds.

$($Note: tic and toc measure elapsed time. When a computer is doing more

than one thing, the cputime function can be used to measure how much

computer time is being used for this task alone.$)$

$1.$ Copy the following code to a Matlab function m$-$file

named exer$4.$m and modify it to produce information for the table

below. Be sure to add comments and your name and the date. Include

the file with your summary.

$2.$ function elapsedTime$=$exer$4($n$)$

$3.\%$ elapsedTime$=$exer$4($n$)$

$4.\%$ comments

$5.$

$6.\%$ your name and the date

$7.$

$8.$ if mod$($n$,2)==0$

$9.$ error$(\text{'}$Please use only odd values for n$\text{'})$;

$10.$ end

$11.$

$12.$ A $=$ magic$($n$)$; $\%$ only odd n yield invertible matrices

$13.$ b $=$ ones$($n$,1)$; $\%$ the right side vector doesn't change the

time

$14.$ tic;

$15.$ Ainv $=???\%$ compute the inverse matrix

$16.$ xSolution $=???\%$ compute the solution

$17.$ elapsedTime$=$toc;

$18.$You have seen in lecture that the time required to invert

an matrix should be proportional to $.$ Fill in the following

table, where the column entitled $``$ratio$\text{'}\text{'}$ should contain the ratio of the

time for n divided by the time for the preceeding value of n$.$

$($Note: timings are not precise! Furthermore, the first time a function

is used results in Matlab reading the m$-$file and $``$compiling it$,\text{'}\text{'}$ and

that takes considerable time. The last row of this table may take

several minutes!$)$

Remark: Compute the first line of the table twice and use the second

value. The first time a function is called involves substantial overhead

that later calls do not require.

$19.$ Time to compute inverse matrices

$20.$ n time ratio

$21.161\_\_\_\_\_\_\_\_\_$

$22.321\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$23.641\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$24.1281\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$25.2561\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$26.5121\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$27.10241\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$28.$Are these solution times roughly proportional to $?$

Exercise $5$: Matlab provides a special operator, the backslash $(\backslash )$ operator,

that is designed to solve a linear system without computing the inverse. It is

used in the following way, for a matrix A and column vector b$.$

x$=$A$\backslash$b;

It may help you to remember this operator if you think of the A as being

$``$underneath$\text{'}\text{'}$ the slash. The effect of this operator is to find the solution of

the system of equations A$*$x$=$b$.$

The backslash looks strange, but there is a method to this madness. You

might wonder why you can't just write x$=$b$/$A$.$ This would put the column

vector bto the left of the matrix A and that is the wrong order of operations.

$1.$ Copy exer$4.$m to exer$5.$m and replace the inverse matrix computation

and solution with an expression using the Matlab $``\backslash \text{'}\text{'}$ command, and

fill in the following table

$2.$ Time to compute solutions

$3.$ n time ratio

$4.161\_\_\_\_\_\_\_\_\_$

$5.321\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$6.641\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$7.1281\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$8.2561\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$9.5121\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$10.10241\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$11.$Are these solution times roughly proportional to $?$

$12.$Compare the times for the inverse and solution and fill in the

following table

$13.$ Comparison of times

$14.$ n $($time for inverse$)/($time for solution$)$

$15.161\_\_\_\_\_\_\_\_\_$

$16.321\_\_\_\_\_\_\_\_\_$

$17.641\_\_\_\_\_\_\_\_\_$

$18.1281\_\_\_\_\_\_\_\_\_$

$19.2561\_\_\_\_\_\_\_\_\_$

$20.5121\_\_\_\_\_\_\_\_\_$

$21.10241\_\_\_\_\_\_\_\_\_$

$22.$Theory shows that computation of a matrix inverse should take

approximately three times as long as computation of the solution. Are

your results consistent with theory?

You should be getting the message: $``$You should never compute an inverse

when all you want to do is solve a system.$\text{'}\text{'}$

Warning: the $``\backslash \text{'}\text{'}$ symbol in Matlab will work when the matrix A is not square. In

fact, sometimes it will work when A actually is a vector. The results are not usually

what you expect and no error message is given. Be careful of this potential for error

when using $``\backslash \text{'}\text{'}.$ A similar warning is true for the $/($division$)$ symbol because

Matlab will try to "nterpret'$\text{'}$ it if you use it with matrices or vectors and you will

get $``$answers$\text{'}\text{'}$ that are not what you intend.